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Mirrors > Home > MPE Home > Th. List > gimcnv | Structured version Visualization version GIF version |
Description: The converse of a bijective group homomorphism is a bijective group homomorphism. (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 2732 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2732 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | 1, 2 | ghmf 19090 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
4 | frel 6719 | . . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹) | |
5 | dfrel2 6185 | . . . . . 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 2833 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ◡◡𝐹 ∈ (𝑆 GrpHom 𝑇)) |
10 | 9 | anim1ci 616 | . 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → (◡𝐹 ∈ (𝑇 GrpHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 GrpHom 𝑇))) |
11 | isgim2 19133 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpIso 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆))) | |
12 | isgim2 19133 | . 2 ⊢ (◡𝐹 ∈ (𝑇 GrpIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 GrpHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 GrpHom 𝑇))) | |
13 | 10, 11, 12 | 3imtr4i 291 | 1 ⊢ (𝐹 ∈ (𝑆 GrpIso 𝑇) → ◡𝐹 ∈ (𝑇 GrpIso 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ◡ccnv 5674 Rel wrel 5680 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 GrpHom cghm 19083 GrpIso cgim 19125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-ghm 19084 df-gim 19127 |
This theorem is referenced by: gicsym 19142 reloggim 26098 abliso 32184 |
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