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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 7603 | . . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V) | |
4 | 2, 3 | eqeltrid 2914 | . . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V) |
5 | mptexg 6975 | . . . 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 5799 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺) | |
8 | 7, 2 | syl6eqr 2871 | . . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋) |
9 | oveq 7151 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥)) | |
10 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺)) | |
11 | grpinvfval.2 | . . . . . . . 8 ⊢ 𝑈 = (GId‘𝐺) | |
12 | 10, 11 | syl6eqr 2871 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = 𝑈) |
13 | 9, 12 | eqeq12d 2834 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = (GId‘𝑔) ↔ (𝑦𝐺𝑥) = 𝑈)) |
14 | 8, 13 | riotaeqbidv 7106 | . . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) |
15 | 8, 14 | mpteq12dv 5142 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
16 | df-ginv 28199 | . . . 4 ⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)))) | |
17 | 15, 16 | fvmptg 6759 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V) → (inv‘𝐺) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
18 | 6, 17 | mpdan 683 | . 2 ⊢ (𝐺 ∈ GrpOp → (inv‘𝐺) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
19 | 1, 18 | syl5eq 2865 | 1 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ↦ cmpt 5137 ran crn 5549 ‘cfv 6348 ℩crio 7102 (class class class)co 7145 GrpOpcgr 28193 GIdcgi 28194 invcgn 28195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 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 7103 df-ov 7148 df-ginv 28199 |
This theorem is referenced by: grpoinvval 28227 grpoinvf 28236 |
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