Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > grpoinvval | Structured version Visualization version GIF version |
Description: The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpinvfval.1 | ⊢ 𝑋 = ran 𝐺 |
grpinvfval.2 | ⊢ 𝑈 = (GId‘𝐺) |
grpinvfval.3 | ⊢ 𝑁 = (inv‘𝐺) |
Ref | Expression |
---|---|
grpoinvval | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvfval.1 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
2 | grpinvfval.2 | . . . 4 ⊢ 𝑈 = (GId‘𝐺) | |
3 | grpinvfval.3 | . . . 4 ⊢ 𝑁 = (inv‘𝐺) | |
4 | 1, 2, 3 | grpoinvfval 28291 | . . 3 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
5 | 4 | fveq1d 6665 | . 2 ⊢ (𝐺 ∈ GrpOp → (𝑁‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴)) |
6 | oveq2 7156 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴)) | |
7 | 6 | eqeq1d 2821 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑦𝐺𝐴) = 𝑈)) |
8 | 7 | riotabidv 7108 | . . 3 ⊢ (𝑥 = 𝐴 → (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)) |
9 | eqid 2819 | . . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) | |
10 | riotaex 7110 | . . 3 ⊢ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ V | |
11 | 8, 9, 10 | fvmpt 6761 | . 2 ⊢ (𝐴 ∈ 𝑋 → ((𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)) |
12 | 5, 11 | sylan9eq 2874 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ↦ cmpt 5137 ran crn 5549 ‘cfv 6348 ℩crio 7105 (class class class)co 7148 GrpOpcgr 28258 GIdcgi 28259 invcgn 28260 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-ginv 28264 |
This theorem is referenced by: grpoinvcl 28293 grpoinv 28294 |
Copyright terms: Public domain | W3C validator |