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Mirrors > Home > MPE Home > Th. List > gimcnv | Structured version Visualization version GIF version |
Description: The converse of a group isomorphism is a group isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
gimcnv | ⊢ (𝐹 ∈ (𝑆 GrpIso 𝑇) → ◡𝐹 ∈ (𝑇 GrpIso 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2726 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | 1, 2 | ghmf 19143 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
4 | frel 6715 | . . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹) | |
5 | dfrel2 6181 | . . . . . 6 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
6 | 4, 5 | sylib 217 | . . . . 5 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → ◡◡𝐹 = 𝐹) |
7 | 3, 6 | syl 17 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ◡◡𝐹 = 𝐹) |
8 | id 22 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
9 | 7, 8 | eqeltrd 2827 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ◡◡𝐹 ∈ (𝑆 GrpHom 𝑇)) |
10 | 9 | anim1ci 615 | . 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → (◡𝐹 ∈ (𝑇 GrpHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 GrpHom 𝑇))) |
11 | isgim2 19188 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpIso 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆))) | |
12 | isgim2 19188 | . 2 ⊢ (◡𝐹 ∈ (𝑇 GrpIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 GrpHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 GrpHom 𝑇))) | |
13 | 10, 11, 12 | 3imtr4i 292 | 1 ⊢ (𝐹 ∈ (𝑆 GrpIso 𝑇) → ◡𝐹 ∈ (𝑇 GrpIso 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ◡ccnv 5668 Rel wrel 5674 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 Basecbs 17151 GrpHom cghm 19136 GrpIso cgim 19180 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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-csb 3889 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-iun 4992 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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-ghm 19137 df-gim 19182 |
This theorem is referenced by: gicsym 19198 reloggim 26484 abliso 32698 |
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