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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 20179 | . . . . 5 ⊢ 1r = (0g ∘ mulGrp) | |
| 2 | 1 | fveq1i 6907 | . . . 4 ⊢ (1r‘𝑅) = ((0g ∘ mulGrp)‘𝑅) |
| 3 | fnmgp 20139 | . . . . 5 ⊢ mulGrp Fn V | |
| 4 | fvco2 7006 | . . . . 5 ⊢ ((mulGrp Fn V ∧ 𝑅 ∈ V) → ((0g ∘ mulGrp)‘𝑅) = (0g‘(mulGrp‘𝑅))) | |
| 5 | 3, 4 | mpan 690 | . . . 4 ⊢ (𝑅 ∈ V → ((0g ∘ mulGrp)‘𝑅) = (0g‘(mulGrp‘𝑅))) |
| 6 | 2, 5 | eqtrid 2789 | . . 3 ⊢ (𝑅 ∈ V → (1r‘𝑅) = (0g‘(mulGrp‘𝑅))) |
| 7 | 0g0 18677 | . . . 4 ⊢ ∅ = (0g‘∅) | |
| 8 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (1r‘𝑅) = ∅) | |
| 9 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
| 10 | 9 | fveq2d 6910 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (0g‘(mulGrp‘𝑅)) = (0g‘∅)) |
| 11 | 7, 8, 10 | 3eqtr4a 2803 | . . 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 6909 | . 2 ⊢ (0g‘𝐺) = (0g‘(mulGrp‘𝑅)) |
| 16 | 12, 13, 15 | 3eqtr4i 2775 | 1 ⊢ 1 = (0g‘𝐺) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ∘ ccom 5689 Fn wfn 6556 ‘cfv 6561 0gc0g 17484 mulGrpcmgp 20137 1rcur 20178 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-slot 17219 df-ndx 17231 df-base 17248 df-0g 17486 df-mgp 20138 df-ur 20179 |
| This theorem is referenced by: dfur2 20181 srgidcl 20196 srgidmlem 20198 issrgid 20201 srgpcomp 20215 srg1expzeq1 20222 srgbinom 20228 ringidcl 20262 ringidmlem 20265 isringid 20268 prds1 20320 pwspjmhmmgpd 20325 xpsring1d 20330 oppr1 20350 unitsubm 20386 rngidpropd 20415 dfrhm2 20474 isrhm2d 20487 rhm1 20489 c0rhm 20534 c0rnghm 20535 subrgsubm 20585 issubrg3 20600 isdomn3 20715 cnfldexp 21417 expmhm 21454 nn0srg 21455 rge0srg 21456 fermltlchr 21544 freshmansdream 21593 frobrhm 21594 assamulgscmlem1 21919 mplcoe3 22056 mplcoe5 22058 mplbas2 22060 evlslem1 22106 evlsgsummul 22116 mhppwdeg 22154 psdpw 22174 ply1scltm 22284 ply1idvr1 22298 lply1binomsc 22315 evls1gsummul 22329 evl1gsummul 22364 madetsumid 22467 mat1mhm 22490 scmatmhm 22540 mdet0pr 22598 mdetunilem7 22624 smadiadetlem4 22675 mat2pmatmhm 22739 pm2mpmhm 22826 chfacfscmulgsum 22866 chfacfpmmulgsum 22870 cpmadugsumlemF 22882 efsubm 26593 amgmlem 27033 amgm 27034 wilthlem2 27112 wilthlem3 27113 dchrelbas3 27282 dchrzrh1 27288 dchrmulcl 27293 dchrn0 27294 dchrinvcl 27297 dchrfi 27299 dchrabs 27304 sumdchr2 27314 rpvmasum2 27556 psgnid 33117 cnmsgn0g 33166 altgnsg 33169 urpropd 33236 isunit3 33245 elrgspnlem2 33247 erlbr2d 33268 erler 33269 rloccring 33274 rloc0g 33275 rloc1r 33276 rlocf1 33277 domnprodn0 33279 rrgsubm 33287 znfermltl 33394 unitprodclb 33417 ssdifidlprm 33486 rprmdvdspow 33561 rprmdvdsprod 33562 1arithidomlem1 33563 1arithidom 33565 1arithufdlem3 33574 1arithufdlem4 33575 dfufd2lem 33577 zringfrac 33582 evl1deg1 33601 evl1deg2 33602 evl1deg3 33603 assarrginv 33687 evls1fldgencl 33720 iistmd 33901 aks6d1c1p6 42115 evl1gprodd 42118 idomnnzpownz 42133 idomnnzgmulnz 42134 aks6d1c5lem2 42139 deg1gprod 42141 deg1pow 42142 aks5lem2 42188 unitscyglem5 42200 domnexpgn0cl 42533 abvexp 42542 pwsgprod 42554 evlsvvvallem 42571 evlsvvval 42573 evlselv 42597 mhphf 42607 mon1psubm 43211 deg1mhm 43212 amgmwlem 49321 amgmlemALT 49322 |
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