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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 7924 | . . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V) | |
| 4 | 2, 3 | eqeltrid 2845 | . . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V) |
| 5 | mptexg 7241 | . . . 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 5947 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺) | |
| 8 | 7, 2 | eqtr4di 2795 | . . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋) |
| 9 | oveq 7437 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥)) | |
| 10 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺)) | |
| 11 | grpinvfval.2 | . . . . . . . 8 ⊢ 𝑈 = (GId‘𝐺) | |
| 12 | 10, 11 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = 𝑈) |
| 13 | 9, 12 | eqeq12d 2753 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = (GId‘𝑔) ↔ (𝑦𝐺𝑥) = 𝑈)) |
| 14 | 8, 13 | riotaeqbidv 7391 | . . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) |
| 15 | 8, 14 | mpteq12dv 5233 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
| 16 | df-ginv 30514 | . . . 4 ⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)))) | |
| 17 | 15, 16 | fvmptg 7014 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V) → (inv‘𝐺) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
| 18 | 6, 17 | mpdan 687 | . 2 ⊢ (𝐺 ∈ GrpOp → (inv‘𝐺) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
| 19 | 1, 18 | eqtrid 2789 | 1 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 GrpOpcgr 30508 GIdcgi 30509 invcgn 30510 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-ginv 30514 |
| This theorem is referenced by: grpoinvval 30542 grpoinvf 30551 |
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