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Mirrors > Home > MPE Home > Th. List > ringidval | Structured version Visualization version GIF version |
Description: The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
ringidval.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
ringidval.u | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
ringidval | ⊢ 1 = (0g‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ur 20115 | . . . . 5 ⊢ 1r = (0g ∘ mulGrp) | |
2 | 1 | fveq1i 6892 | . . . 4 ⊢ (1r‘𝑅) = ((0g ∘ mulGrp)‘𝑅) |
3 | fnmgp 20069 | . . . . 5 ⊢ mulGrp Fn V | |
4 | fvco2 6989 | . . . . 5 ⊢ ((mulGrp Fn V ∧ 𝑅 ∈ V) → ((0g ∘ mulGrp)‘𝑅) = (0g‘(mulGrp‘𝑅))) | |
5 | 3, 4 | mpan 689 | . . . 4 ⊢ (𝑅 ∈ V → ((0g ∘ mulGrp)‘𝑅) = (0g‘(mulGrp‘𝑅))) |
6 | 2, 5 | eqtrid 2779 | . . 3 ⊢ (𝑅 ∈ V → (1r‘𝑅) = (0g‘(mulGrp‘𝑅))) |
7 | 0g0 18617 | . . . 4 ⊢ ∅ = (0g‘∅) | |
8 | fvprc 6883 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (1r‘𝑅) = ∅) | |
9 | fvprc 6883 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
10 | 9 | fveq2d 6895 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (0g‘(mulGrp‘𝑅)) = (0g‘∅)) |
11 | 7, 8, 10 | 3eqtr4a 2793 | . . 3 ⊢ (¬ 𝑅 ∈ V → (1r‘𝑅) = (0g‘(mulGrp‘𝑅))) |
12 | 6, 11 | pm2.61i 182 | . 2 ⊢ (1r‘𝑅) = (0g‘(mulGrp‘𝑅)) |
13 | ringidval.u | . 2 ⊢ 1 = (1r‘𝑅) | |
14 | ringidval.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
15 | 14 | fveq2i 6894 | . 2 ⊢ (0g‘𝐺) = (0g‘(mulGrp‘𝑅)) |
16 | 12, 13, 15 | 3eqtr4i 2765 | 1 ⊢ 1 = (0g‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∅c0 4318 ∘ ccom 5676 Fn wfn 6537 ‘cfv 6542 0gc0g 17414 mulGrpcmgp 20067 1rcur 20114 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-1cn 11190 ax-addcl 11192 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-nn 12237 df-slot 17144 df-ndx 17156 df-base 17174 df-0g 17416 df-mgp 20068 df-ur 20115 |
This theorem is referenced by: dfur2 20117 srgidcl 20132 srgidmlem 20134 issrgid 20137 srgpcomp 20151 srg1expzeq1 20158 srgbinom 20164 ringidcl 20195 ringidmlem 20197 isringid 20200 prds1 20252 pwspjmhmmgpd 20257 xpsring1d 20262 oppr1 20282 unitsubm 20318 rngidpropd 20347 dfrhm2 20406 isrhm2d 20419 rhm1 20421 c0rhm 20464 c0rnghm 20465 subrgsubm 20517 issubrg3 20532 isdomn3 21241 cnfldexp 21325 expmhm 21362 nn0srg 21363 rge0srg 21364 fermltlchr 21452 freshmansdream 21501 assamulgscmlem1 21825 mplcoe3 21969 mplcoe5 21971 mplbas2 21973 evlslem1 22021 evlsgsummul 22031 mhppwdeg 22067 ply1scltm 22193 lply1binomsc 22223 evls1gsummul 22237 evl1gsummul 22272 madetsumid 22356 mat1mhm 22379 scmatmhm 22429 mdet0pr 22487 mdetunilem7 22513 smadiadetlem4 22564 mat2pmatmhm 22628 pm2mpmhm 22715 chfacfscmulgsum 22755 chfacfpmmulgsum 22759 cpmadugsumlemF 22771 efsubm 26478 amgmlem 26915 amgm 26916 wilthlem2 26994 wilthlem3 26995 dchrelbas3 27164 dchrzrh1 27170 dchrmulcl 27175 dchrn0 27176 dchrinvcl 27179 dchrfi 27181 dchrabs 27186 sumdchr2 27196 rpvmasum2 27438 psgnid 32812 cnmsgn0g 32861 altgnsg 32864 urpropd 32933 frobrhm 32935 rrgsubm 32947 erlbr2d 32972 erler 32973 rloccring 32978 rloc0g 32979 rloc1r 32980 rlocf1 32981 zringfrac 32990 znfermltl 33072 evls1fldgencl 33344 iistmd 33493 aks6d1c1p6 41570 evl1gprodd 41573 idomnnzpownz 41587 idomnnzgmulnz 41588 aks6d1c5lem2 41593 deg1gprod 41596 deg1pow 41597 pwsgprod 41746 evlsvvvallem 41766 evlsvvval 41768 evlselv 41792 mhphf 41802 mon1psubm 42599 deg1mhm 42600 amgmwlem 48207 amgmlemALT 48208 |
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