Theorem ghmf1o 19168

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 19139	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
2		ghmgrp1 19138	. . . . 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 6783	. . . . . 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 6771	. . . . 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 766	. . . . . . . 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 770	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑌)
12	10, 11	ffvelcdmd 7027	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (◡𝐹‘𝑥) ∈ 𝑋)
13		simprr 772	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌)
14	10, 13	ffvelcdmd 7027	. . . . . . . 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 2733	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
17		eqid 2733	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
18	15, 16, 17	ghmlin 19141	. . . . . . . 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 1373	. . . . . . 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 768	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝐹:𝑋–1-1-onto→𝑌)
21		f1ocnvfv2 7220	. . . . . . . . 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 7220	. . . . . . . . 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 7373	. . . . . . 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 2768	. . . . . 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 18862	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ (◡𝐹‘𝑥) ∈ 𝑋 ∧ (◡𝐹‘𝑦) ∈ 𝑋) → ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)) ∈ 𝑋)
29	27, 12, 14, 28	syl3anc 1373	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)) ∈ 𝑋)
30		f1ocnvfv 7221	. . . . . . 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 3176	. . . 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 19135	. . 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 19140	. . . . 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 6660	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑋)
41	35, 15	ghmf 19140	. . . . 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 6660	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → ◡𝐹 Fn 𝑌)
44		dff1o4 6779	. . 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 800	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ^◡ccnv 5620   Fn wfn 6484  ⟶wf 6485  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7355  Basecbs 17127  +_gcplusg 17168  Grpcgrp 18854   GrpHom cghm 19132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-map 8761  df-mgm 18556  df-sgrp 18635  df-mnd 18651  df-grp 18857  df-ghm 19133

This theorem is referenced by:  isgim2  19185  rnghmf1o  20379  rhmf1o  20417  lmhmf1o  20989