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Mirrors > Home > MPE Home > Th. List > grpoinvfval | Structured version Visualization version GIF version |
Description: The inverse function of a group. (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 |
---|---|
grpoinvfval | ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvfval.3 | . 2 ⊢ 𝑁 = (inv‘𝐺) | |
2 | grpinvfval.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
3 | rnexg 7907 | . . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V) | |
4 | 2, 3 | eqeltrid 2830 | . . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V) |
5 | mptexg 7230 | . . . 4 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐺 ∈ GrpOp → (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V) |
7 | rneq 5934 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺) | |
8 | 7, 2 | eqtr4di 2784 | . . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋) |
9 | oveq 7422 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥)) | |
10 | fveq2 6893 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺)) | |
11 | grpinvfval.2 | . . . . . . . 8 ⊢ 𝑈 = (GId‘𝐺) | |
12 | 10, 11 | eqtr4di 2784 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = 𝑈) |
13 | 9, 12 | eqeq12d 2742 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = (GId‘𝑔) ↔ (𝑦𝐺𝑥) = 𝑈)) |
14 | 8, 13 | riotaeqbidv 7375 | . . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) |
15 | 8, 14 | mpteq12dv 5236 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
16 | df-ginv 30425 | . . . 4 ⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)))) | |
17 | 15, 16 | fvmptg 6999 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V) → (inv‘𝐺) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
18 | 6, 17 | mpdan 685 | . 2 ⊢ (𝐺 ∈ GrpOp → (inv‘𝐺) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
19 | 1, 18 | eqtrid 2778 | 1 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ↦ cmpt 5228 ran crn 5675 ‘cfv 6546 ℩crio 7371 (class class class)co 7416 GrpOpcgr 30419 GIdcgi 30420 invcgn 30421 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-ginv 30425 |
This theorem is referenced by: grpoinvval 30453 grpoinvf 30462 |
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