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Theorem iscrngd 20228

Description: Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)

**Hypotheses**
Ref	Expression
isringd.b	⊢ (𝜑 → 𝐵 = (Base‘𝑅))
isringd.p	⊢ (𝜑 → + = (+_g‘𝑅))
isringd.t	⊢ (𝜑 → · = (._r‘𝑅))
isringd.g	⊢ (𝜑 → 𝑅 ∈ Grp)
isringd.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
isringd.a	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
isringd.d	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
isringd.e	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
isringd.u	⊢ (𝜑 → 1 ∈ 𝐵)
isringd.i	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = 𝑥)
isringd.h	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = 𝑥)
iscrngd.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) = (𝑦 · 𝑥))

**Assertion**
Ref	Expression
iscrngd	⊢ (𝜑 → 𝑅 ∈ CRing)

Distinct variable groups: 𝑥, 1 𝑥,𝑦,𝑧,𝐵 𝜑,𝑥,𝑦,𝑧 𝑥,𝑅,𝑦,𝑧 Allowed substitution hints: + (𝑥,𝑦,𝑧) · (𝑥,𝑦,𝑧) 1 (𝑦,𝑧)

**Proof of Theorem iscrngd**
Step	Hyp	Ref	Expression
1		isringd.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
2		isringd.p	. . 3 ⊢ (𝜑 → + = (+_g‘𝑅))
3		isringd.t	. . 3 ⊢ (𝜑 → · = (._r‘𝑅))
4		isringd.g	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
5		isringd.c	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
6		isringd.a	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
7		isringd.d	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
8		isringd.e	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
9		isringd.u	. . 3 ⊢ (𝜑 → 1 ∈ 𝐵)
10		isringd.i	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = 𝑥)
11		isringd.h	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = 𝑥)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	isringd 20227	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
13		eqid 2728	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
14		eqid 2728	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 14	mgpbas 20080	. . . 4 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
16	1, 15	eqtrdi 2784	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝑅)))
17		eqid 2728	. . . . 5 ⊢ (._r‘𝑅) = (._r‘𝑅)
18	13, 17	mgpplusg 20078	. . . 4 ⊢ (._r‘𝑅) = (+_g‘(mulGrp‘𝑅))
19	3, 18	eqtrdi 2784	. . 3 ⊢ (𝜑 → · = (+_g‘(mulGrp‘𝑅)))
20	16, 19, 5, 6, 9, 10, 11	ismndd 18716	. . 3 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd)
21		iscrngd.c	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
22	16, 19, 20, 21	iscmnd 19749	. 2 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
23	13	iscrng 20180	. 2 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ CMnd))
24	12, 22, 23	sylanbrc 582	1 ⊢ (𝜑 → 𝑅 ∈ CRing)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 +_gcplusg 17233 ._rcmulr 17234 Grpcgrp 18890 CMndccmn 19735 mulGrpcmgp 20074 Ringcrg 20173 CRingccrg 20174

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-cmn 19737 df-mgp 20075 df-ring 20175 df-cring 20176

This theorem is referenced by: cncrng 21316 cncrngOLD 21317 rloccring 32997

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