Theorem ghmf1o 18387

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 18360	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
2		ghmgrp1 18359	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
3	1, 2	jca 514	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
4	3	adantr 483	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
5		f1ocnv 6626	. . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
6	5	adantl 484	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹:𝑌–1-1-onto→𝑋)
7		f1of 6614	. . . . 5 ⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋)
8	6, 7	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹:𝑌⟶𝑋)
9		simpll 765	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
10	8	adantr 483	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ◡𝐹:𝑌⟶𝑋)
11		simprl 769	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑌)
12	10, 11	ffvelrnd 6851	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (◡𝐹‘𝑥) ∈ 𝑋)
13		simprr 771	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌)
14	10, 13	ffvelrnd 6851	. . . . . . . 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 2821	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
17		eqid 2821	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
18	15, 16, 17	ghmlin 18362	. . . . . . . 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 1367	. . . . . . 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 767	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝐹:𝑋–1-1-onto→𝑌)
21		f1ocnvfv2 7033	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
22	20, 11, 21	syl2anc 586	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
23		f1ocnvfv2 7033	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑦 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦)
24	20, 13, 23	syl2anc 586	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦)
25	22, 24	oveq12d 7173	. . . . . . 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 2856	. . . . . 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 17	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑆 ∈ Grp)
28	15, 16	grpcl 18110	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ (◡𝐹‘𝑥) ∈ 𝑋 ∧ (◡𝐹‘𝑦) ∈ 𝑋) → ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)) ∈ 𝑋)
29	27, 12, 14, 28	syl3anc 1367	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)) ∈ 𝑋)
30		f1ocnvfv 7034	. . . . . . 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 586	. . . . . 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 15	. . . . 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 3191	. . . 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 514	. . 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 18357	. . 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 585	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 GrpHom 𝑆))
38	15, 35	ghmf 18361	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌)
39	38	adantr 483	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋⟶𝑌)
40	39	ffnd 6514	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑋)
41	35, 15	ghmf 18361	. . . . 5 ⊢ (◡𝐹 ∈ (𝑇 GrpHom 𝑆) → ◡𝐹:𝑌⟶𝑋)
42	41	adantl 484	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → ◡𝐹:𝑌⟶𝑋)
43	42	ffnd 6514	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → ◡𝐹 Fn 𝑌)
44		dff1o4 6622	. . 3 ⊢ (𝐹:𝑋–1-1-onto→𝑌 ↔ (𝐹 Fn 𝑋 ∧ ◡𝐹 Fn 𝑌))
45	40, 43, 44	sylanbrc 585	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)
46	37, 45	impbida 799	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ^◡ccnv 5553   Fn wfn 6349  ⟶wf 6350  –1-1-onto→wf1o 6353  ‘cfv 6354  (class class class)co 7155  Basecbs 16482  +_gcplusg 16564  Grpcgrp 18102   GrpHom cghm 18354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-grp 18105  df-ghm 18355

This theorem is referenced by:  isgim2  18404  rhmf1o  19483  lmhmf1o  19817  rnghmf1o  44173