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Mirrors > Home > MPE Home > Th. List > gimco | Structured version Visualization version GIF version |
Description: The composition of group isomorphisms is a group isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
gimco | ⊢ ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpIso 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgim2 19296 | . . 3 ⊢ (𝐹 ∈ (𝑇 GrpIso 𝑈) ↔ (𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ ◡𝐹 ∈ (𝑈 GrpHom 𝑇))) | |
2 | isgim2 19296 | . . 3 ⊢ (𝐺 ∈ (𝑆 GrpIso 𝑇) ↔ (𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐺 ∈ (𝑇 GrpHom 𝑆))) | |
3 | ghmco 19267 | . . . . 5 ⊢ ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈)) | |
4 | cnvco 5899 | . . . . . 6 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹) | |
5 | ghmco 19267 | . . . . . . 7 ⊢ ((◡𝐺 ∈ (𝑇 GrpHom 𝑆) ∧ ◡𝐹 ∈ (𝑈 GrpHom 𝑇)) → (◡𝐺 ∘ ◡𝐹) ∈ (𝑈 GrpHom 𝑆)) | |
6 | 5 | ancoms 458 | . . . . . 6 ⊢ ((◡𝐹 ∈ (𝑈 GrpHom 𝑇) ∧ ◡𝐺 ∈ (𝑇 GrpHom 𝑆)) → (◡𝐺 ∘ ◡𝐹) ∈ (𝑈 GrpHom 𝑆)) |
7 | 4, 6 | eqeltrid 2843 | . . . . 5 ⊢ ((◡𝐹 ∈ (𝑈 GrpHom 𝑇) ∧ ◡𝐺 ∈ (𝑇 GrpHom 𝑆)) → ◡(𝐹 ∘ 𝐺) ∈ (𝑈 GrpHom 𝑆)) |
8 | 3, 7 | anim12i 613 | . . . 4 ⊢ (((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (◡𝐹 ∈ (𝑈 GrpHom 𝑇) ∧ ◡𝐺 ∈ (𝑇 GrpHom 𝑆))) → ((𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ ◡(𝐹 ∘ 𝐺) ∈ (𝑈 GrpHom 𝑆))) |
9 | 8 | an4s 660 | . . 3 ⊢ (((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ ◡𝐹 ∈ (𝑈 GrpHom 𝑇)) ∧ (𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐺 ∈ (𝑇 GrpHom 𝑆))) → ((𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ ◡(𝐹 ∘ 𝐺) ∈ (𝑈 GrpHom 𝑆))) |
10 | 1, 2, 9 | syl2anb 598 | . 2 ⊢ ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → ((𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ ◡(𝐹 ∘ 𝐺) ∈ (𝑈 GrpHom 𝑆))) |
11 | isgim2 19296 | . 2 ⊢ ((𝐹 ∘ 𝐺) ∈ (𝑆 GrpIso 𝑈) ↔ ((𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ ◡(𝐹 ∘ 𝐺) ∈ (𝑈 GrpHom 𝑆))) | |
12 | 10, 11 | sylibr 234 | 1 ⊢ ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpIso 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ◡ccnv 5688 ∘ ccom 5693 (class class class)co 7431 GrpHom cghm 19243 GrpIso cgim 19288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-ghm 19244 df-gim 19290 |
This theorem is referenced by: gictr 19307 |
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