Theorem ghmf1o 13852

Description: A bijective group homomorphism is an isomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
ghmf1o.x	⊢ 𝑋 = (Base‘𝑆)
ghmf1o.y	⊢ 𝑌 = (Base‘𝑇)

Assertion

Ref	Expression
ghmf1o	⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)))

Proof of Theorem ghmf1o

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

Step	Hyp	Ref	Expression
1		ghmgrp2 13823	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
2		ghmgrp1 13822	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
3	1, 2	jca 306	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
4	3	adantr 276	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
5		f1ocnv 5593	. . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
6	5	adantl 277	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹:𝑌–1-1-onto→𝑋)
7		f1of 5580	. . . . 5 ⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋)
8	6, 7	syl 14	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹:𝑌⟶𝑋)
9		simpll 527	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
10	8	adantr 276	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ◡𝐹:𝑌⟶𝑋)
11		simprl 529	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑌)
12	10, 11	ffvelcdmd 5779	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (◡𝐹‘𝑥) ∈ 𝑋)
13		simprr 531	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌)
14	10, 13	ffvelcdmd 5779	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (◡𝐹‘𝑦) ∈ 𝑋)
15		ghmf1o.x	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑆)
16		eqid 2229	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
17		eqid 2229	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
18	15, 16, 17	ghmlin 13825	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (◡𝐹‘𝑥) ∈ 𝑋 ∧ (◡𝐹‘𝑦) ∈ 𝑋) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))) = ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑇)(𝐹‘(◡𝐹‘𝑦))))
19	9, 12, 14, 18	syl3anc 1271	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))) = ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑇)(𝐹‘(◡𝐹‘𝑦))))
20		simplr 528	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝐹:𝑋–1-1-onto→𝑌)
21		f1ocnvfv2 5914	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
22	20, 11, 21	syl2anc 411	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
23		f1ocnvfv2 5914	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑦 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦)
24	20, 13, 23	syl2anc 411	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦)
25	22, 24	oveq12d 6031	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑇)(𝐹‘(◡𝐹‘𝑦))) = (𝑥(+g‘𝑇)𝑦))
26	19, 25	eqtrd 2262	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑇)𝑦))
27	9, 2	syl 14	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑆 ∈ Grp)
28	15, 16	grpcl 13581	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ (◡𝐹‘𝑥) ∈ 𝑋 ∧ (◡𝐹‘𝑦) ∈ 𝑋) → ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)) ∈ 𝑋)
29	27, 12, 14, 28	syl3anc 1271	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)) ∈ 𝑋)
30		f1ocnvfv 5915	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)) ∈ 𝑋) → ((𝐹‘((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑇)𝑦) → (◡𝐹‘(𝑥(+g‘𝑇)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))))
31	20, 29, 30	syl2anc 411	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((𝐹‘((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑇)𝑦) → (◡𝐹‘(𝑥(+g‘𝑇)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))))
32	26, 31	mpd 13	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (◡𝐹‘(𝑥(+g‘𝑇)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)))
33	32	ralrimivva 2612	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (◡𝐹‘(𝑥(+g‘𝑇)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)))
34	8, 33	jca 306	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (◡𝐹:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (◡𝐹‘(𝑥(+g‘𝑇)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))))
35		ghmf1o.y	. . . 4 ⊢ 𝑌 = (Base‘𝑇)
36	35, 15, 17, 16	isghm 13820	. . 3 ⊢ (◡𝐹 ∈ (𝑇 GrpHom 𝑆) ↔ ((𝑇 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (◡𝐹:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (◡𝐹‘(𝑥(+g‘𝑇)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)))))
37	4, 34, 36	sylanbrc 417	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 GrpHom 𝑆))
38	15, 35	ghmf 13824	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌)
39	38	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋⟶𝑌)
40	39	ffnd 5480	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑋)
41	35, 15	ghmf 13824	. . . . 5 ⊢ (◡𝐹 ∈ (𝑇 GrpHom 𝑆) → ◡𝐹:𝑌⟶𝑋)
42	41	adantl 277	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → ◡𝐹:𝑌⟶𝑋)
43	42	ffnd 5480	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → ◡𝐹 Fn 𝑌)
44		dff1o4 5588	. . 3 ⊢ (𝐹:𝑋–1-1-onto→𝑌 ↔ (𝐹 Fn 𝑋 ∧ ◡𝐹 Fn 𝑌))
45	40, 43, 44	sylanbrc 417	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)
46	37, 45	impbida 598	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ^◡ccnv 4722   Fn wfn 5319  ⟶wf 5320  –1-1-onto→wf1o 5323  ‘cfv 5324  (class class class)co 6013  Basecbs 13072  +_gcplusg 13150  Grpcgrp 13573   GrpHom cghm 13817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-inn 9134  df-2 9192  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576  df-ghm 13818

This theorem is referenced by:  rhmf1o  14172