Theorem mhmf1o 13102

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 13096	. . . . 5 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑆 ∈ Mnd)
2		mhmrcl1 13095	. . . . 5 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd)
3	1, 2	jca 306	. . . 4 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd))
4	3	adantr 276	. . 3 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd))
5		f1ocnv 5517	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐶 → ◡𝐹:𝐶–1-1-onto→𝐵)
6	5	adantl 277	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹:𝐶–1-1-onto→𝐵)
7		f1of 5504	. . . . 5 ⊢ (◡𝐹:𝐶–1-1-onto→𝐵 → ◡𝐹:𝐶⟶𝐵)
8	6, 7	syl 14	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹:𝐶⟶𝐵)
9		simpll 527	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐹 ∈ (𝑅 MndHom 𝑆))
10	8	adantr 276	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ◡𝐹:𝐶⟶𝐵)
11		simprl 529	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
12	10, 11	ffvelcdmd 5698	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (◡𝐹‘𝑥) ∈ 𝐵)
13		simprr 531	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
14	10, 13	ffvelcdmd 5698	. . . . . . . 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 2196	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
17		eqid 2196	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
18	15, 16, 17	mhmlin 13099	. . . . . . . 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 1249	. . . . . . 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 110	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹:𝐵–1-1-onto→𝐶)
21	20	adantr 276	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐹:𝐵–1-1-onto→𝐶)
22		f1ocnvfv2 5825	. . . . . . . . 9 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑥 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
23	21, 11, 22	syl2anc 411	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
24		f1ocnvfv2 5825	. . . . . . . . 9 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑦 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦)
25	21, 13, 24	syl2anc 411	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦)
26	23, 25	oveq12d 5940	. . . . . . 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 2229	. . . . . 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 276	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝑅 ∈ Mnd)
29	28	adantr 276	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑅 ∈ Mnd)
30	15, 16	mndcl 13064	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ (◡𝐹‘𝑥) ∈ 𝐵 ∧ (◡𝐹‘𝑦) ∈ 𝐵) → ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∈ 𝐵)
31	29, 12, 14, 30	syl3anc 1249	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∈ 𝐵)
32		f1ocnvfv 5826	. . . . . . 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 411	. . . . . 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 13	. . . . 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 2579	. . . 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 2196	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
37		eqid 2196	. . . . . . . . 9 ⊢ (0g‘𝑆) = (0g‘𝑆)
38	36, 37	mhm0 13100	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
39	38	adantr 276	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
40	39	eqcomd 2202	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (0g‘𝑆) = (𝐹‘(0g‘𝑅)))
41	40	fveq2d 5562	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹‘(0g‘𝑆)) = (◡𝐹‘(𝐹‘(0g‘𝑅))))
42	15, 36	mndidcl 13071	. . . . . . . 8 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ 𝐵)
43	2, 42	syl 14	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (0g‘𝑅) ∈ 𝐵)
44	43	adantr 276	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (0g‘𝑅) ∈ 𝐵)
45		f1ocnvfv1 5824	. . . . . 6 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ (0g‘𝑅) ∈ 𝐵) → (◡𝐹‘(𝐹‘(0g‘𝑅))) = (0g‘𝑅))
46	20, 44, 45	syl2anc 411	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹‘(𝐹‘(0g‘𝑅))) = (0g‘𝑅))
47	41, 46	eqtrd 2229	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹‘(0g‘𝑆)) = (0g‘𝑅))
48	8, 35, 47	3jca 1179	. . 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 13093	. . 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 417	. 2 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 MndHom 𝑅))
52	15, 49	mhmf 13097	. . . . 5 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹:𝐵⟶𝐶)
53	52	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵⟶𝐶)
54	53	ffnd 5408	. . 3 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹 Fn 𝐵)
55	49, 15	mhmf 13097	. . . . 5 ⊢ (◡𝐹 ∈ (𝑆 MndHom 𝑅) → ◡𝐹:𝐶⟶𝐵)
56	55	adantl 277	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → ◡𝐹:𝐶⟶𝐵)
57	56	ffnd 5408	. . 3 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → ◡𝐹 Fn 𝐶)
58		dff1o4 5512	. . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn 𝐶))
59	54, 57, 58	sylanbrc 417	. 2 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵–1-1-onto→𝐶)
60	51, 59	impbida 596	1 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 MndHom 𝑅)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ^◡ccnv 4662   Fn wfn 5253  ⟶wf 5254  –1-1-onto→wf1o 5257  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  0_gc0g 12927  Mndcmnd 13057   MndHom cmhm 13089

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mhm 13091

This theorem is referenced by:  rhmf1o  13724