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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 18840 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
10 | 1, 2, 3, 4, 5, 6, 9 | ghmgrp 18161 | . 2 ⊢ (𝜑 → 𝐻 ∈ Grp) |
11 | ablcmn 18842 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
12 | 7, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
13 | 2, 3, 4, 5, 1, 6, 12 | ghmcmn 18881 | . 2 ⊢ (𝜑 → 𝐻 ∈ CMnd) |
14 | isabl 18839 | . 2 ⊢ (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd)) | |
15 | 10, 13, 14 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐻 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 –onto→wfo 6346 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Grpcgrp 18041 CMndccmn 18835 Abelcabl 18836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-riota 7103 df-ov 7148 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-cmn 18837 df-abl 18838 |
This theorem is referenced by: efabl 25061 |
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