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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 19703 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
10 | 1, 2, 3, 4, 5, 6, 9 | ghmgrp 18992 | . 2 ⊢ (𝜑 → 𝐻 ∈ Grp) |
11 | ablcmn 19705 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
12 | 7, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
13 | 2, 3, 4, 5, 1, 6, 12 | ghmcmn 19749 | . 2 ⊢ (𝜑 → 𝐻 ∈ CMnd) |
14 | isabl 19702 | . 2 ⊢ (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd)) | |
15 | 10, 13, 14 | sylanbrc 582 | 1 ⊢ (𝜑 → 𝐻 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 –onto→wfo 6534 ‘cfv 6536 (class class class)co 7404 Basecbs 17151 +gcplusg 17204 Grpcgrp 18861 CMndccmn 19698 Abelcabl 19699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-riota 7360 df-ov 7407 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-cmn 19700 df-abl 19701 |
This theorem is referenced by: efabl 26435 |
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