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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 20126 | . . . . 5 ⊢ 1r = (0g ∘ mulGrp) | |
2 | 1 | fveq1i 6893 | . . . 4 ⊢ (1r‘𝑅) = ((0g ∘ mulGrp)‘𝑅) |
3 | fnmgp 20080 | . . . . 5 ⊢ mulGrp Fn V | |
4 | fvco2 6990 | . . . . 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 18623 | . . . 4 ⊢ ∅ = (0g‘∅) | |
8 | fvprc 6884 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (1r‘𝑅) = ∅) | |
9 | fvprc 6884 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
10 | 9 | fveq2d 6896 | . . . 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 6895 | . 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 3463 ∅c0 4318 ∘ ccom 5676 Fn wfn 6538 ‘cfv 6543 0gc0g 17420 mulGrpcmgp 20078 1rcur 20125 |
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 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-1cn 11196 ax-addcl 11198 |
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 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-nn 12243 df-slot 17150 df-ndx 17162 df-base 17180 df-0g 17422 df-mgp 20079 df-ur 20126 |
This theorem is referenced by: dfur2 20128 srgidcl 20143 srgidmlem 20145 issrgid 20148 srgpcomp 20162 srg1expzeq1 20169 srgbinom 20175 ringidcl 20206 ringidmlem 20208 isringid 20211 prds1 20263 pwspjmhmmgpd 20268 xpsring1d 20273 oppr1 20293 unitsubm 20329 rngidpropd 20358 dfrhm2 20417 isrhm2d 20430 rhm1 20432 c0rhm 20475 c0rnghm 20476 subrgsubm 20528 issubrg3 20543 isdomn3 21252 cnfldexp 21336 expmhm 21373 nn0srg 21374 rge0srg 21375 fermltlchr 21463 freshmansdream 21512 assamulgscmlem1 21836 mplcoe3 21983 mplcoe5 21985 mplbas2 21987 evlslem1 22035 evlsgsummul 22045 mhppwdeg 22082 ply1scltm 22209 lply1binomsc 22239 evls1gsummul 22253 evl1gsummul 22288 madetsumid 22381 mat1mhm 22404 scmatmhm 22454 mdet0pr 22512 mdetunilem7 22538 smadiadetlem4 22589 mat2pmatmhm 22653 pm2mpmhm 22740 chfacfscmulgsum 22780 chfacfpmmulgsum 22784 cpmadugsumlemF 22796 efsubm 26503 amgmlem 26940 amgm 26941 wilthlem2 27019 wilthlem3 27020 dchrelbas3 27189 dchrzrh1 27195 dchrmulcl 27200 dchrn0 27201 dchrinvcl 27204 dchrfi 27206 dchrabs 27211 sumdchr2 27221 rpvmasum2 27463 psgnid 32863 cnmsgn0g 32912 altgnsg 32915 urpropd 32984 frobrhm 32985 rrgsubm 33000 erlbr2d 33027 erler 33028 rloccring 33033 rloc0g 33034 rloc1r 33035 rlocf1 33036 znfermltl 33126 rprmdvdspow 33296 rprmdvdsprod 33297 zringfrac 33301 evls1fldgencl 33415 iistmd 33560 aks6d1c1p6 41641 evl1gprodd 41644 idomnnzpownz 41659 idomnnzgmulnz 41660 aks6d1c5lem2 41665 deg1gprod 41668 deg1pow 41669 pwsgprod 41830 evlsvvvallem 41859 evlsvvval 41861 evlselv 41885 mhphf 41895 mon1psubm 42692 deg1mhm 42693 amgmwlem 48347 amgmlemALT 48348 |
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