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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 20151 | . . . . 5 ⊢ 1r = (0g ∘ mulGrp) | |
2 | 1 | fveq1i 6897 | . . . 4 ⊢ (1r‘𝑅) = ((0g ∘ mulGrp)‘𝑅) |
3 | fnmgp 20105 | . . . . 5 ⊢ mulGrp Fn V | |
4 | fvco2 6994 | . . . . 5 ⊢ ((mulGrp Fn V ∧ 𝑅 ∈ V) → ((0g ∘ mulGrp)‘𝑅) = (0g‘(mulGrp‘𝑅))) | |
5 | 3, 4 | mpan 688 | . . . 4 ⊢ (𝑅 ∈ V → ((0g ∘ mulGrp)‘𝑅) = (0g‘(mulGrp‘𝑅))) |
6 | 2, 5 | eqtrid 2777 | . . 3 ⊢ (𝑅 ∈ V → (1r‘𝑅) = (0g‘(mulGrp‘𝑅))) |
7 | 0g0 18643 | . . . 4 ⊢ ∅ = (0g‘∅) | |
8 | fvprc 6888 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (1r‘𝑅) = ∅) | |
9 | fvprc 6888 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
10 | 9 | fveq2d 6900 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (0g‘(mulGrp‘𝑅)) = (0g‘∅)) |
11 | 7, 8, 10 | 3eqtr4a 2791 | . . 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 6899 | . 2 ⊢ (0g‘𝐺) = (0g‘(mulGrp‘𝑅)) |
16 | 12, 13, 15 | 3eqtr4i 2763 | 1 ⊢ 1 = (0g‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∅c0 4322 ∘ ccom 5682 Fn wfn 6544 ‘cfv 6549 0gc0g 17440 mulGrpcmgp 20103 1rcur 20150 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-1cn 11203 ax-addcl 11205 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-nn 12251 df-slot 17170 df-ndx 17182 df-base 17200 df-0g 17442 df-mgp 20104 df-ur 20151 |
This theorem is referenced by: dfur2 20153 srgidcl 20168 srgidmlem 20170 issrgid 20173 srgpcomp 20187 srg1expzeq1 20194 srgbinom 20200 ringidcl 20231 ringidmlem 20233 isringid 20236 prds1 20288 pwspjmhmmgpd 20293 xpsring1d 20298 oppr1 20318 unitsubm 20354 rngidpropd 20383 dfrhm2 20442 isrhm2d 20455 rhm1 20457 c0rhm 20500 c0rnghm 20501 subrgsubm 20553 issubrg3 20568 isdomn3 21282 cnfldexp 21366 expmhm 21403 nn0srg 21404 rge0srg 21405 fermltlchr 21493 freshmansdream 21542 assamulgscmlem1 21866 mplcoe3 22015 mplcoe5 22017 mplbas2 22019 evlslem1 22067 evlsgsummul 22077 mhppwdeg 22114 ply1scltm 22242 lply1binomsc 22272 evls1gsummul 22286 evl1gsummul 22321 madetsumid 22424 mat1mhm 22447 scmatmhm 22497 mdet0pr 22555 mdetunilem7 22581 smadiadetlem4 22632 mat2pmatmhm 22696 pm2mpmhm 22783 chfacfscmulgsum 22823 chfacfpmmulgsum 22827 cpmadugsumlemF 22839 efsubm 26547 amgmlem 26987 amgm 26988 wilthlem2 27066 wilthlem3 27067 dchrelbas3 27236 dchrzrh1 27242 dchrmulcl 27247 dchrn0 27248 dchrinvcl 27251 dchrfi 27253 dchrabs 27258 sumdchr2 27268 rpvmasum2 27510 psgnid 32931 cnmsgn0g 32980 altgnsg 32983 urpropd 33053 frobrhm 33054 erlbr2d 33075 erler 33076 rloccring 33081 rloc0g 33082 rloc1r 33083 rlocf1 33084 domnprodn0 33086 rrgsubm 33093 znfermltl 33198 unitprodclb 33222 ssdifidlprm 33291 rprmdvdspow 33366 rprmdvdsprod 33367 1arithidomlem1 33368 1arithidom 33370 1arithufdlem3 33379 1arithufdlem4 33380 dfufd2lem 33382 zringfrac 33387 evl1deg1 33405 evl1deg3 33406 evls1fldgencl 33508 iistmd 33654 aks6d1c1p6 41736 evl1gprodd 41739 idomnnzpownz 41754 idomnnzgmulnz 41755 aks6d1c5lem2 41760 deg1gprod 41762 deg1pow 41763 aks5lem2 41809 pwsgprod 41931 evlsvvvallem 41948 evlsvvval 41950 evlselv 41974 mhphf 41984 mon1psubm 42774 deg1mhm 42775 amgmwlem 48426 amgmlemALT 48427 |
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