Theorem ghmf1o 19279

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 19250	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
2		ghmgrp1 19249	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
3	1, 2	jca 511	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
4	3	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
5		f1ocnv 6861	. . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
6	5	adantl 481	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹:𝑌–1-1-onto→𝑋)
7		f1of 6849	. . . . 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 767	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
10	8	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ◡𝐹:𝑌⟶𝑋)
11		simprl 771	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑌)
12	10, 11	ffvelcdmd 7105	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (◡𝐹‘𝑥) ∈ 𝑋)
13		simprr 773	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌)
14	10, 13	ffvelcdmd 7105	. . . . . . . 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 2735	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
17		eqid 2735	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
18	15, 16, 17	ghmlin 19252	. . . . . . . 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 1370	. . . . . . 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 769	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝐹:𝑋–1-1-onto→𝑌)
21		f1ocnvfv2 7297	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
22	20, 11, 21	syl2anc 584	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
23		f1ocnvfv2 7297	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑦 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦)
24	20, 13, 23	syl2anc 584	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦)
25	22, 24	oveq12d 7449	. . . . . . 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 2775	. . . . . 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 18972	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ (◡𝐹‘𝑥) ∈ 𝑋 ∧ (◡𝐹‘𝑦) ∈ 𝑋) → ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)) ∈ 𝑋)
29	27, 12, 14, 28	syl3anc 1370	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)) ∈ 𝑋)
30		f1ocnvfv 7298	. . . . . . 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 584	. . . . . 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 3200	. . . 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 511	. . 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 19246	. . 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 583	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 GrpHom 𝑆))
38	15, 35	ghmf 19251	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌)
39	38	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋⟶𝑌)
40	39	ffnd 6738	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑋)
41	35, 15	ghmf 19251	. . . . 5 ⊢ (◡𝐹 ∈ (𝑇 GrpHom 𝑆) → ◡𝐹:𝑌⟶𝑋)
42	41	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → ◡𝐹:𝑌⟶𝑋)
43	42	ffnd 6738	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → ◡𝐹 Fn 𝑌)
44		dff1o4 6857	. . 3 ⊢ (𝐹:𝑋–1-1-onto→𝑌 ↔ (𝐹 Fn 𝑋 ∧ ◡𝐹 Fn 𝑌))
45	40, 43, 44	sylanbrc 583	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)
46	37, 45	impbida 801	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ^◡ccnv 5688   Fn wfn 6558  ⟶wf 6559  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  Grpcgrp 18964   GrpHom cghm 19243

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-ghm 19244

This theorem is referenced by:  isgim2  19296  rnghmf1o  20469  rhmf1o  20508  lmhmf1o  21063