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Mirrors > Home > ILE Home > Th. List > ringidvalg | 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 |
---|---|
ringidvalg | ⊢ (𝑅 ∈ 𝑉 → 1 = (0g‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2748 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | df-ur 12969 | . . . . 5 ⊢ 1r = (0g ∘ mulGrp) | |
3 | 2 | fveq1i 5512 | . . . 4 ⊢ (1r‘𝑅) = ((0g ∘ mulGrp)‘𝑅) |
4 | fnmgp 12959 | . . . . 5 ⊢ mulGrp Fn V | |
5 | fvco2 5581 | . . . . 5 ⊢ ((mulGrp Fn V ∧ 𝑅 ∈ V) → ((0g ∘ mulGrp)‘𝑅) = (0g‘(mulGrp‘𝑅))) | |
6 | 4, 5 | mpan 424 | . . . 4 ⊢ (𝑅 ∈ V → ((0g ∘ mulGrp)‘𝑅) = (0g‘(mulGrp‘𝑅))) |
7 | 3, 6 | eqtrid 2222 | . . 3 ⊢ (𝑅 ∈ V → (1r‘𝑅) = (0g‘(mulGrp‘𝑅))) |
8 | 1, 7 | syl 14 | . 2 ⊢ (𝑅 ∈ 𝑉 → (1r‘𝑅) = (0g‘(mulGrp‘𝑅))) |
9 | ringidval.u | . 2 ⊢ 1 = (1r‘𝑅) | |
10 | ringidval.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
11 | 10 | fveq2i 5514 | . 2 ⊢ (0g‘𝐺) = (0g‘(mulGrp‘𝑅)) |
12 | 8, 9, 11 | 3eqtr4g 2235 | 1 ⊢ (𝑅 ∈ 𝑉 → 1 = (0g‘𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∘ ccom 4627 Fn wfn 5207 ‘cfv 5212 0gc0g 12653 mulGrpcmgp 12957 1rcur 12968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1re 7896 ax-addrcl 7899 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-inn 8909 df-2 8967 df-3 8968 df-ndx 12448 df-slot 12449 df-sets 12452 df-plusg 12531 df-mulr 12532 df-mgp 12958 df-ur 12969 |
This theorem is referenced by: dfur2g 12971 srgidcl 12985 srgidmlem 12987 issrgid 12990 srgpcomp 12999 srg1expzeq1 13004 ringidcl 13029 ringidmlem 13031 isringid 13034 oppr1g 13077 unitsubm 13113 rngidpropdg 13138 |
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