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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 18903 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
10 | 1, 2, 3, 4, 5, 6, 9 | ghmgrp 18215 | . 2 ⊢ (𝜑 → 𝐻 ∈ Grp) |
11 | ablcmn 18905 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
12 | 7, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
13 | 2, 3, 4, 5, 1, 6, 12 | ghmcmn 18945 | . 2 ⊢ (𝜑 → 𝐻 ∈ CMnd) |
14 | isabl 18902 | . 2 ⊢ (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd)) | |
15 | 10, 13, 14 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝐻 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 –onto→wfo 6322 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Grpcgrp 18095 CMndccmn 18898 Abelcabl 18899 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-cmn 18900 df-abl 18901 |
This theorem is referenced by: efabl 25142 |
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