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Mirrors > Home > MPE Home > Th. List > ghmabl | Structured version Visualization version GIF version |
Description: The image of an abelian group 𝐺 under a group homomorphism 𝐹 is an abelian group. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.) |
Ref | Expression |
---|---|
ghmabl.x | ⊢ 𝑋 = (Base‘𝐺) |
ghmabl.y | ⊢ 𝑌 = (Base‘𝐻) |
ghmabl.p | ⊢ + = (+g‘𝐺) |
ghmabl.q | ⊢ ⨣ = (+g‘𝐻) |
ghmabl.f | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
ghmabl.1 | ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) |
ghmabl.3 | ⊢ (𝜑 → 𝐺 ∈ Abel) |
Ref | Expression |
---|---|
ghmabl | ⊢ (𝜑 → 𝐻 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ghmabl.f | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) | |
2 | ghmabl.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
3 | ghmabl.y | . . 3 ⊢ 𝑌 = (Base‘𝐻) | |
4 | ghmabl.p | . . 3 ⊢ + = (+g‘𝐺) | |
5 | ghmabl.q | . . 3 ⊢ ⨣ = (+g‘𝐻) | |
6 | ghmabl.1 | . . 3 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) | |
7 | ghmabl.3 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
8 | ablgrp 18584 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
10 | 1, 2, 3, 4, 5, 6, 9 | ghmgrp 17926 | . 2 ⊢ (𝜑 → 𝐻 ∈ Grp) |
11 | ablcmn 18585 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
12 | 7, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
13 | 2, 3, 4, 5, 1, 6, 12 | ghmcmn 18623 | . 2 ⊢ (𝜑 → 𝐻 ∈ CMnd) |
14 | isabl 18583 | . 2 ⊢ (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd)) | |
15 | 10, 13, 14 | sylanbrc 578 | 1 ⊢ (𝜑 → 𝐻 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 –onto→wfo 6133 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 Grpcgrp 17809 CMndccmn 18579 Abelcabl 18580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-cmn 18581 df-abl 18582 |
This theorem is referenced by: efabl 24734 |
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