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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 19747 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
10 | 1, 2, 3, 4, 5, 6, 9 | ghmgrp 19029 | . 2 ⊢ (𝜑 → 𝐻 ∈ Grp) |
11 | ablcmn 19749 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
12 | 7, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
13 | 2, 3, 4, 5, 1, 6, 12 | ghmcmn 19793 | . 2 ⊢ (𝜑 → 𝐻 ∈ CMnd) |
14 | isabl 19746 | . 2 ⊢ (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd)) | |
15 | 10, 13, 14 | sylanbrc 581 | 1 ⊢ (𝜑 → 𝐻 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 –onto→wfo 6551 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 Grpcgrp 18897 CMndccmn 19742 Abelcabl 19743 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fo 6559 df-fv 6561 df-riota 7382 df-ov 7429 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-cmn 19744 df-abl 19745 |
This theorem is referenced by: efabl 26504 |
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