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Mirrors > Home > MPE Home > Th. List > grpinvval | Structured version Visualization version GIF version |
Description: The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) |
Ref | Expression |
---|---|
grpinvval.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinvval.p | ⊢ + = (+g‘𝐺) |
grpinvval.o | ⊢ 0 = (0g‘𝐺) |
grpinvval.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvval | ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7413 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋)) | |
2 | 1 | eqeq1d 2728 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 )) |
3 | 2 | riotabidv 7363 | . 2 ⊢ (𝑥 = 𝑋 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
4 | grpinvval.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
5 | grpinvval.p | . . 3 ⊢ + = (+g‘𝐺) | |
6 | grpinvval.o | . . 3 ⊢ 0 = (0g‘𝐺) | |
7 | grpinvval.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
8 | 4, 5, 6, 7 | grpinvfval 18908 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) |
9 | riotaex 7365 | . 2 ⊢ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) ∈ V | |
10 | 3, 8, 9 | fvmpt 6992 | 1 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 ℩crio 7360 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 0gc0g 17394 invgcminusg 18864 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-riota 7361 df-ov 7408 df-minusg 18867 |
This theorem is referenced by: grplinv 18919 isgrpinv 18923 xrsinvgval 32683 ringinvval 32886 ressply1invg 33153 linvh 41476 primrootsunit1 41477 |
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