hdmap14lem4a - Mathbox for Norm Megill

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Theorem hdmap14lem4a 41348

Description: Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 41347 to provide a slightly simpler definition later. (Contributed by NM, 31-May-2015.)

**Hypotheses**
Ref	Expression
hdmap14lem1.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap14lem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap14lem1.v	⊢ 𝑉 = (Base‘𝑈)
hdmap14lem1.t	⊢ · = ( ·_𝑠 ‘𝑈)
hdmap14lem3.o	⊢ 0 = (0_g‘𝑈)
hdmap14lem1.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmap14lem1.b	⊢ 𝐵 = (Base‘𝑅)
hdmap14lem1.z	⊢ 𝑍 = (0_g‘𝑅)
hdmap14lem1.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap14lem2.e	⊢ ∙ = ( ·_𝑠 ‘𝐶)
hdmap14lem1.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap14lem2.p	⊢ 𝑃 = (Scalar‘𝐶)
hdmap14lem2.a	⊢ 𝐴 = (Base‘𝑃)
hdmap14lem2.q	⊢ 𝑄 = (0_g‘𝑃)
hdmap14lem1.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmap14lem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap14lem3.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap14lem1.f	⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍}))

**Assertion**
Ref	Expression
hdmap14lem4a	⊢ (𝜑 → (∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))

Distinct variable groups: 𝐴,𝑔 ∙ ,𝑔 𝑔,𝐹 𝑄,𝑔 𝑆,𝑔 · ,𝑔 𝑔,𝑋 𝜑,𝑔 Allowed substitution hints: 𝐵(𝑔) 𝐶(𝑔) 𝑃(𝑔) 𝑅(𝑔) 𝑈(𝑔) 𝐻(𝑔) 𝐾(𝑔) 𝐿(𝑔) 𝑉(𝑔) 𝑊(𝑔) 0 (𝑔) 𝑍(𝑔)

**Proof of Theorem hdmap14lem4a**
Step	Hyp	Ref	Expression
1		hdmap14lem1.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap14lem1.u	. . . . . . . . 9 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap14lem1.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap14lem3.o	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑈)
5		hdmap14lem1.c	. . . . . . . . 9 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
6		eqid 2727	. . . . . . . . 9 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
7		eqid 2727	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		hdmap14lem1.s	. . . . . . . . 9 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
9		hdmap14lem1.k	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10	1, 2, 9	dvhlmod 40587	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
11		hdmap14lem1.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍}))
12	11	eldifad 3959	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
13		hdmap14lem3.x	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3959	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
15		hdmap14lem1.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (Scalar‘𝑈)
16		hdmap14lem1.t	. . . . . . . . . . . 12 ⊢ · = ( ·_𝑠 ‘𝑈)
17		hdmap14lem1.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑅)
18	3, 15, 16, 17	lmodvscl 20766	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 · 𝑋) ∈ 𝑉)
19	10, 12, 14, 18	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ 𝑉)
20		eldifsni 4796	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐵 ∖ {𝑍}) → 𝐹 ≠ 𝑍)
21	11, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ≠ 𝑍)
22		eldifsni 4796	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 )
23	13, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ 0 )
24		hdmap14lem1.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (0_g‘𝑅)
25	1, 2, 9	dvhlvec 40586	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LVec)
26	3, 16, 15, 17, 24, 4, 25, 12, 14	lvecvsn0 21002	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 · 𝑋) ≠ 0 ↔ (𝐹 ≠ 𝑍 ∧ 𝑋 ≠ 0 )))
27	21, 23, 26	mpbir2and 711	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 · 𝑋) ≠ 0 )
28		eldifsn 4793	. . . . . . . . . 10 ⊢ ((𝐹 · 𝑋) ∈ (𝑉 ∖ { 0 }) ↔ ((𝐹 · 𝑋) ∈ 𝑉 ∧ (𝐹 · 𝑋) ≠ 0 ))
29	19, 27, 28	sylanbrc 581	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ (𝑉 ∖ { 0 }))
30	1, 2, 3, 4, 5, 6, 7, 8, 9, 29	hdmapnzcl 41322	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) ∈ ((Base‘𝐶) ∖ {(0_g‘𝐶)}))
31		eldifsni 4796	. . . . . . . 8 ⊢ ((𝑆‘(𝐹 · 𝑋)) ∈ ((Base‘𝐶) ∖ {(0_g‘𝐶)}) → (𝑆‘(𝐹 · 𝑋)) ≠ (0_g‘𝐶))
32	30, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) ≠ (0_g‘𝐶))
33	32	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ {𝑄}) → (𝑆‘(𝐹 · 𝑋)) ≠ (0_g‘𝐶))
34		elsni 4647	. . . . . . . 8 ⊢ (𝑔 ∈ {𝑄} → 𝑔 = 𝑄)
35	34	oveq1d 7439	. . . . . . 7 ⊢ (𝑔 ∈ {𝑄} → (𝑔 ∙ (𝑆‘𝑋)) = (𝑄 ∙ (𝑆‘𝑋)))
36	1, 5, 9	lcdlmod 41069	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ LMod)
37	1, 2, 3, 5, 7, 8, 9, 14	hdmapcl 41307	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶))
38		hdmap14lem2.p	. . . . . . . . 9 ⊢ 𝑃 = (Scalar‘𝐶)
39		hdmap14lem2.e	. . . . . . . . 9 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
40		hdmap14lem2.q	. . . . . . . . 9 ⊢ 𝑄 = (0_g‘𝑃)
41	7, 38, 39, 40, 6	lmod0vs 20783	. . . . . . . 8 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑋) ∈ (Base‘𝐶)) → (𝑄 ∙ (𝑆‘𝑋)) = (0_g‘𝐶))
42	36, 37, 41	syl2anc 582	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∙ (𝑆‘𝑋)) = (0_g‘𝐶))
43	35, 42	sylan9eqr 2789	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ {𝑄}) → (𝑔 ∙ (𝑆‘𝑋)) = (0_g‘𝐶))
44	33, 43	neeqtrrd 3011	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ {𝑄}) → (𝑆‘(𝐹 · 𝑋)) ≠ (𝑔 ∙ (𝑆‘𝑋)))
45	44	neneqd 2941	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ {𝑄}) → ¬ (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
46	45	nrexdv 3145	. . 3 ⊢ (𝜑 → ¬ ∃𝑔 ∈ {𝑄} (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
47		reuun2 4316	. . 3 ⊢ (¬ ∃𝑔 ∈ {𝑄} (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) → (∃!𝑔 ∈ ((𝐴 ∖ {𝑄}) ∪ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
48	46, 47	syl 17	. 2 ⊢ (𝜑 → (∃!𝑔 ∈ ((𝐴 ∖ {𝑄}) ∪ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
49		hdmap14lem2.a	. . . 4 ⊢ 𝐴 = (Base‘𝑃)
50	38, 49, 40	lmod0cl 20776	. . 3 ⊢ (𝐶 ∈ LMod → 𝑄 ∈ 𝐴)
51		difsnid 4816	. . 3 ⊢ (𝑄 ∈ 𝐴 → ((𝐴 ∖ {𝑄}) ∪ {𝑄}) = 𝐴)
52		reueq1 3411	. . 3 ⊢ (((𝐴 ∖ {𝑄}) ∪ {𝑄}) = 𝐴 → (∃!𝑔 ∈ ((𝐴 ∖ {𝑄}) ∪ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
53	36, 50, 51, 52	4syl 19	. 2 ⊢ (𝜑 → (∃!𝑔 ∈ ((𝐴 ∖ {𝑄}) ∪ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
54	48, 53	bitr3d 280	1 ⊢ (𝜑 → (∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2936 ∃wrex 3066 ∃!wreu 3370 ∖ cdif 3944 ∪ cun 3945 {csn 4630 ‘cfv 6551 (class class class)co 7424 Basecbs 17185 Scalarcsca 17241 ·_𝑠 cvsca 17242 0_gc0g 17426 LModclmod 20748 LSpanclspn 20860 HLchlt 38826 LHypclh 39461 DVecHcdvh 40555 LCDualclcd 41063 HDMapchdma 41269

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-riotaBAD 38429

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-tpos 8236 df-undef 8283 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-n0 12509 df-z 12595 df-uz 12859 df-fz 13523 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-sca 17254 df-vsca 17255 df-0g 17428 df-mre 17571 df-mrc 17572 df-acs 17574 df-proset 18292 df-poset 18310 df-plt 18327 df-lub 18343 df-glb 18344 df-join 18345 df-meet 18346 df-p0 18422 df-p1 18423 df-lat 18429 df-clat 18496 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-grp 18898 df-minusg 18899 df-sbg 18900 df-subg 19083 df-cntz 19273 df-oppg 19302 df-lsm 19596 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-oppr 20278 df-dvdsr 20301 df-unit 20302 df-invr 20332 df-dvr 20345 df-drng 20631 df-lmod 20750 df-lss 20821 df-lsp 20861 df-lvec 20993 df-lsatoms 38452 df-lshyp 38453 df-lcv 38495 df-lfl 38534 df-lkr 38562 df-ldual 38600 df-oposet 38652 df-ol 38654 df-oml 38655 df-covers 38742 df-ats 38743 df-atl 38774 df-cvlat 38798 df-hlat 38827 df-llines 38975 df-lplanes 38976 df-lvols 38977 df-lines 38978 df-psubsp 38980 df-pmap 38981 df-padd 39273 df-lhyp 39465 df-laut 39466 df-ldil 39581 df-ltrn 39582 df-trl 39636 df-tgrp 40220 df-tendo 40232 df-edring 40234 df-dveca 40480 df-disoa 40506 df-dvech 40556 df-dib 40616 df-dic 40650 df-dih 40706 df-doch 40825 df-djh 40872 df-lcdual 41064 df-mapd 41102 df-hvmap 41234 df-hdmap1 41270 df-hdmap 41271

This theorem is referenced by: hdmap14lem4 41349

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