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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 2823 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2823 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | 1, 2 | ghmf 18364 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
4 | frel 6521 | . . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹) | |
5 | dfrel2 6048 | . . . . . 6 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
6 | 4, 5 | sylib 220 | . . . . 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 2915 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ◡◡𝐹 ∈ (𝑆 GrpHom 𝑇)) |
10 | 9 | anim1ci 617 | . 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → (◡𝐹 ∈ (𝑇 GrpHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 GrpHom 𝑇))) |
11 | isgim2 18407 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpIso 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆))) | |
12 | isgim2 18407 | . 2 ⊢ (◡𝐹 ∈ (𝑇 GrpIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 GrpHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 GrpHom 𝑇))) | |
13 | 10, 11, 12 | 3imtr4i 294 | 1 ⊢ (𝐹 ∈ (𝑆 GrpIso 𝑇) → ◡𝐹 ∈ (𝑇 GrpIso 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ◡ccnv 5556 Rel wrel 5562 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 GrpHom cghm 18357 GrpIso cgim 18399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-ghm 18358 df-gim 18401 |
This theorem is referenced by: gicsym 18416 reloggim 25184 abliso 30685 |
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