Theorem abliso 32983

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 19202	. . . 4 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2		ghmgrp2 19157	. . . 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 18884	. . 3 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Mnd)
6		simpll 766	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑀 ∈ Abel)
7		eqid 2730	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
8		eqid 2730	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
9	7, 8	gimf1o 19201	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
10		f1ocnv 6814	. . . . . . . . . . 11 ⊢ (𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) → ◡𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀))
11		f1of 6802	. . . . . . . . . . 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 727	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ◡𝐹:(Base‘𝑁)⟶(Base‘𝑀))
14		simprl 770	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑥 ∈ (Base‘𝑁))
15	13, 14	ffvelcdmd 7059	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘𝑥) ∈ (Base‘𝑀))
16		simprr 772	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑦 ∈ (Base‘𝑁))
17	13, 16	ffvelcdmd 7059	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘𝑦) ∈ (Base‘𝑀))
18		eqid 2730	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
19	7, 18	ablcom 19735	. . . . . . . 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 1373	. . . . . . 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 19205	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → ◡𝐹 ∈ (𝑁 GrpIso 𝑀))
22	21	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ◡𝐹 ∈ (𝑁 GrpIso 𝑀))
23		gimghm 19202	. . . . . . . . 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 2730	. . . . . . . . 9 ⊢ (+g‘𝑁) = (+g‘𝑁)
26	8, 25, 18	ghmlin 19159	. . . . . . . 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 1373	. . . . . . 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 19159	. . . . . . . 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 1373	. . . . . . 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 2775	. . . . . 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 6864	. . . . 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 727	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
33	3	ad2antlr 727	. . . . . . 7 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑁 ∈ Grp)
34	8, 25	grpcl 18879	. . . . . . 7 ⊢ ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥(+g‘𝑁)𝑦) ∈ (Base‘𝑁))
35	33, 14, 16, 34	syl3anc 1373	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g‘𝑁)𝑦) ∈ (Base‘𝑁))
36		f1ocnvfv2 7254	. . . . . 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 584	. . . . 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 18879	. . . . . . 7 ⊢ ((𝑁 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝑦(+g‘𝑁)𝑥) ∈ (Base‘𝑁))
39	33, 16, 14, 38	syl3anc 1373	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑦(+g‘𝑁)𝑥) ∈ (Base‘𝑁))
40		f1ocnvfv2 7254	. . . . . 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 584	. . . . 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 2773	. . . 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 19725	. . 3 ⊢ (𝑁 ∈ CMnd ↔ (𝑁 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g‘𝑁)𝑦) = (𝑦(+g‘𝑁)𝑥)))
45	5, 43, 44	sylanbrc 583	. 2 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ CMnd)
46		isabl 19720	. 2 ⊢ (𝑁 ∈ Abel ↔ (𝑁 ∈ Grp ∧ 𝑁 ∈ CMnd))
47	4, 45, 46	sylanbrc 583	1 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ^◡ccnv 5639  ⟶wf 6509  –1-1-onto→wf1o 6512  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  +_gcplusg 17226  Mndcmnd 18667  Grpcgrp 18871   GrpHom cghm 19150   GrpIso cgim 19195  CMndccmn 19716  Abelcabl 19717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-map 8803  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18874  df-ghm 19151  df-gim 19197  df-cmn 19718  df-abl 19719

This theorem is referenced by: (None)