Theorem abliso 33114

Description: The image of an Abelian group by a group isomorphism is also Abelian. (Contributed by Thierry Arnoux, 8-Mar-2018.)

Assertion

Ref	Expression
abliso	⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)

Proof of Theorem abliso

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gimghm 19233	. . . 4 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2		ghmgrp2 19188	. . . 4 ⊢ (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
3	1, 2	syl 17	. . 3 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝑁 ∈ Grp)
4	3	adantl 481	. 2 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Grp)
5	4	grpmndd 18916	. . 3 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Mnd)
6		simpll 767	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑀 ∈ Abel)
7		eqid 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
8		eqid 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
9	7, 8	gimf1o 19232	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
10		f1ocnv 6787	. . . . . . . . . . 11 ⊢ (𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) → ◡𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀))
11		f1of 6775	. . . . . . . . . . 11 ⊢ (◡𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀) → ◡𝐹:(Base‘𝑁)⟶(Base‘𝑀))
12	9, 10, 11	3syl 18	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → ◡𝐹:(Base‘𝑁)⟶(Base‘𝑀))
13	12	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ◡𝐹:(Base‘𝑁)⟶(Base‘𝑀))
14		simprl 771	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑥 ∈ (Base‘𝑁))
15	13, 14	ffvelcdmd 7032	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘𝑥) ∈ (Base‘𝑀))
16		simprr 773	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑦 ∈ (Base‘𝑁))
17	13, 16	ffvelcdmd 7032	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘𝑦) ∈ (Base‘𝑀))
18		eqid 2737	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
19	7, 18	ablcom 19768	. . . . . . . 8 ⊢ ((𝑀 ∈ Abel ∧ (◡𝐹‘𝑥) ∈ (Base‘𝑀) ∧ (◡𝐹‘𝑦) ∈ (Base‘𝑀)) → ((◡𝐹‘𝑥)(+g‘𝑀)(◡𝐹‘𝑦)) = ((◡𝐹‘𝑦)(+g‘𝑀)(◡𝐹‘𝑥)))
20	6, 15, 17, 19	syl3anc 1374	. . . . . . 7 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ((◡𝐹‘𝑥)(+g‘𝑀)(◡𝐹‘𝑦)) = ((◡𝐹‘𝑦)(+g‘𝑀)(◡𝐹‘𝑥)))
21		gimcnv 19236	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → ◡𝐹 ∈ (𝑁 GrpIso 𝑀))
22	21	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ◡𝐹 ∈ (𝑁 GrpIso 𝑀))
23		gimghm 19233	. . . . . . . . 9 ⊢ (◡𝐹 ∈ (𝑁 GrpIso 𝑀) → ◡𝐹 ∈ (𝑁 GrpHom 𝑀))
24	22, 23	syl 17	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ◡𝐹 ∈ (𝑁 GrpHom 𝑀))
25		eqid 2737	. . . . . . . . 9 ⊢ (+g‘𝑁) = (+g‘𝑁)
26	8, 25, 18	ghmlin 19190	. . . . . . . 8 ⊢ ((◡𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (◡𝐹‘(𝑥(+g‘𝑁)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑀)(◡𝐹‘𝑦)))
27	24, 14, 16, 26	syl3anc 1374	. . . . . . 7 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘(𝑥(+g‘𝑁)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑀)(◡𝐹‘𝑦)))
28	8, 25, 18	ghmlin 19190	. . . . . . . 8 ⊢ ((◡𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (◡𝐹‘(𝑦(+g‘𝑁)𝑥)) = ((◡𝐹‘𝑦)(+g‘𝑀)(◡𝐹‘𝑥)))
29	24, 16, 14, 28	syl3anc 1374	. . . . . . 7 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘(𝑦(+g‘𝑁)𝑥)) = ((◡𝐹‘𝑦)(+g‘𝑀)(◡𝐹‘𝑥)))
30	20, 27, 29	3eqtr4d 2782	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘(𝑥(+g‘𝑁)𝑦)) = (◡𝐹‘(𝑦(+g‘𝑁)𝑥)))
31	30	fveq2d 6839	. . . . 5 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(◡𝐹‘(𝑥(+g‘𝑁)𝑦))) = (𝐹‘(◡𝐹‘(𝑦(+g‘𝑁)𝑥))))
32	9	ad2antlr 728	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
33	3	ad2antlr 728	. . . . . . 7 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑁 ∈ Grp)
34	8, 25	grpcl 18911	. . . . . . 7 ⊢ ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥(+g‘𝑁)𝑦) ∈ (Base‘𝑁))
35	33, 14, 16, 34	syl3anc 1374	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g‘𝑁)𝑦) ∈ (Base‘𝑁))
36		f1ocnvfv2 7226	. . . . . 6 ⊢ ((𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) ∧ (𝑥(+g‘𝑁)𝑦) ∈ (Base‘𝑁)) → (𝐹‘(◡𝐹‘(𝑥(+g‘𝑁)𝑦))) = (𝑥(+g‘𝑁)𝑦))
37	32, 35, 36	syl2anc 585	. . . . 5 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(◡𝐹‘(𝑥(+g‘𝑁)𝑦))) = (𝑥(+g‘𝑁)𝑦))
38	8, 25	grpcl 18911	. . . . . . 7 ⊢ ((𝑁 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝑦(+g‘𝑁)𝑥) ∈ (Base‘𝑁))
39	33, 16, 14, 38	syl3anc 1374	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑦(+g‘𝑁)𝑥) ∈ (Base‘𝑁))
40		f1ocnvfv2 7226	. . . . . 6 ⊢ ((𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) ∧ (𝑦(+g‘𝑁)𝑥) ∈ (Base‘𝑁)) → (𝐹‘(◡𝐹‘(𝑦(+g‘𝑁)𝑥))) = (𝑦(+g‘𝑁)𝑥))
41	32, 39, 40	syl2anc 585	. . . . 5 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(◡𝐹‘(𝑦(+g‘𝑁)𝑥))) = (𝑦(+g‘𝑁)𝑥))
42	31, 37, 41	3eqtr3d 2780	. . . 4 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g‘𝑁)𝑦) = (𝑦(+g‘𝑁)𝑥))
43	42	ralrimivva 3181	. . 3 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g‘𝑁)𝑦) = (𝑦(+g‘𝑁)𝑥))
44	8, 25	iscmn 19758	. . 3 ⊢ (𝑁 ∈ CMnd ↔ (𝑁 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g‘𝑁)𝑦) = (𝑦(+g‘𝑁)𝑥)))
45	5, 43, 44	sylanbrc 584	. 2 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ CMnd)
46		isabl 19753	. 2 ⊢ (𝑁 ∈ Abel ↔ (𝑁 ∈ Grp ∧ 𝑁 ∈ CMnd))
47	4, 45, 46	sylanbrc 584	1 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ^◡ccnv 5624  ⟶wf 6489  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  +_gcplusg 17214  Mndcmnd 18696  Grpcgrp 18903   GrpHom cghm 19181   GrpIso cgim 19226  CMndccmn 19749  Abelcabl 19750

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-map 8769  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-grp 18906  df-ghm 19182  df-gim 19228  df-cmn 19751  df-abl 19752

This theorem is referenced by: (None)