Theorem mhmf1o 18075

Description: A monoid homomorphism is bijective iff its converse is also a monoid homomorphism. (Contributed by AV, 22-Oct-2019.)

Hypotheses

Ref	Expression
mhmf1o.b	⊢ 𝐵 = (Base‘𝑅)
mhmf1o.c	⊢ 𝐶 = (Base‘𝑆)

Assertion

Ref	Expression
mhmf1o	⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 MndHom 𝑅)))

Proof of Theorem mhmf1o

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

Step	Hyp	Ref	Expression
1		mhmrcl2 18069	. . . . 5 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑆 ∈ Mnd)
2		mhmrcl1 18068	. . . . 5 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd)
3	1, 2	jca 515	. . . 4 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd))
4	3	adantr 484	. . 3 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd))
5		f1ocnv 6624	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐶 → ◡𝐹:𝐶–1-1-onto→𝐵)
6	5	adantl 485	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹:𝐶–1-1-onto→𝐵)
7		f1of 6612	. . . . 5 ⊢ (◡𝐹:𝐶–1-1-onto→𝐵 → ◡𝐹:𝐶⟶𝐵)
8	6, 7	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹:𝐶⟶𝐵)
9		simpll 767	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐹 ∈ (𝑅 MndHom 𝑆))
10	8	adantr 484	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ◡𝐹:𝐶⟶𝐵)
11		simprl 771	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
12	10, 11	ffvelrnd 6856	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (◡𝐹‘𝑥) ∈ 𝐵)
13		simprr 773	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
14	10, 13	ffvelrnd 6856	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (◡𝐹‘𝑦) ∈ 𝐵)
15		mhmf1o.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
16		eqid 2738	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
17		eqid 2738	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
18	15, 16, 17	mhmlin 18072	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (◡𝐹‘𝑥) ∈ 𝐵 ∧ (◡𝐹‘𝑦) ∈ 𝐵) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) = ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑆)(𝐹‘(◡𝐹‘𝑦))))
19	9, 12, 14, 18	syl3anc 1372	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) = ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑆)(𝐹‘(◡𝐹‘𝑦))))
20		simpr 488	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹:𝐵–1-1-onto→𝐶)
21	20	adantr 484	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐹:𝐵–1-1-onto→𝐶)
22		f1ocnvfv2 7039	. . . . . . . . 9 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑥 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
23	21, 11, 22	syl2anc 587	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
24		f1ocnvfv2 7039	. . . . . . . . 9 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑦 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦)
25	21, 13, 24	syl2anc 587	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦)
26	23, 25	oveq12d 7182	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑆)(𝐹‘(◡𝐹‘𝑦))) = (𝑥(+g‘𝑆)𝑦))
27	19, 26	eqtrd 2773	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑆)𝑦))
28	2	adantr 484	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝑅 ∈ Mnd)
29	28	adantr 484	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑅 ∈ Mnd)
30	15, 16	mndcl 18028	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ (◡𝐹‘𝑥) ∈ 𝐵 ∧ (◡𝐹‘𝑦) ∈ 𝐵) → ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∈ 𝐵)
31	29, 12, 14, 30	syl3anc 1372	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∈ 𝐵)
32		f1ocnvfv 7040	. . . . . . 7 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∈ 𝐵) → ((𝐹‘((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑆)𝑦) → (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))))
33	21, 31, 32	syl2anc 587	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑆)𝑦) → (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))))
34	27, 33	mpd 15	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)))
35	34	ralrimivva 3103	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)))
36		eqid 2738	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
37		eqid 2738	. . . . . . . . 9 ⊢ (0g‘𝑆) = (0g‘𝑆)
38	36, 37	mhm0 18073	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
39	38	adantr 484	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
40	39	eqcomd 2744	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (0g‘𝑆) = (𝐹‘(0g‘𝑅)))
41	40	fveq2d 6672	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹‘(0g‘𝑆)) = (◡𝐹‘(𝐹‘(0g‘𝑅))))
42	15, 36	mndidcl 18035	. . . . . . . 8 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ 𝐵)
43	2, 42	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (0g‘𝑅) ∈ 𝐵)
44	43	adantr 484	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (0g‘𝑅) ∈ 𝐵)
45		f1ocnvfv1 7038	. . . . . 6 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ (0g‘𝑅) ∈ 𝐵) → (◡𝐹‘(𝐹‘(0g‘𝑅))) = (0g‘𝑅))
46	20, 44, 45	syl2anc 587	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹‘(𝐹‘(0g‘𝑅))) = (0g‘𝑅))
47	41, 46	eqtrd 2773	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹‘(0g‘𝑆)) = (0g‘𝑅))
48	8, 35, 47	3jca 1129	. . 3 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹:𝐶⟶𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∧ (◡𝐹‘(0g‘𝑆)) = (0g‘𝑅)))
49		mhmf1o.c	. . . 4 ⊢ 𝐶 = (Base‘𝑆)
50	49, 15, 17, 16, 37, 36	ismhm 18067	. . 3 ⊢ (◡𝐹 ∈ (𝑆 MndHom 𝑅) ↔ ((𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd) ∧ (◡𝐹:𝐶⟶𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∧ (◡𝐹‘(0g‘𝑆)) = (0g‘𝑅))))
51	4, 48, 50	sylanbrc 586	. 2 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 MndHom 𝑅))
52	15, 49	mhmf 18070	. . . . 5 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹:𝐵⟶𝐶)
53	52	adantr 484	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵⟶𝐶)
54	53	ffnd 6499	. . 3 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹 Fn 𝐵)
55	49, 15	mhmf 18070	. . . . 5 ⊢ (◡𝐹 ∈ (𝑆 MndHom 𝑅) → ◡𝐹:𝐶⟶𝐵)
56	55	adantl 485	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → ◡𝐹:𝐶⟶𝐵)
57	56	ffnd 6499	. . 3 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → ◡𝐹 Fn 𝐶)
58		dff1o4 6620	. . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn 𝐶))
59	54, 57, 58	sylanbrc 586	. 2 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵–1-1-onto→𝐶)
60	51, 59	impbida 801	1 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 MndHom 𝑅)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2113  ∀wral 3053  ^◡ccnv 5518   Fn wfn 6328  ⟶wf 6329  –1-1-onto→wf1o 6332  ‘cfv 6333  (class class class)co 7164  Basecbs 16579  +_gcplusg 16661  0_gc0g 16809  Mndcmnd 18020   MndHom cmhm 18063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-map 8432  df-0g 16811  df-mgm 17961  df-sgrp 18010  df-mnd 18021  df-mhm 18065

This theorem is referenced by:  rhmf1o  19599