Theorem mhmf1o 13489

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 13483	. . . . 5 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑆 ∈ Mnd)
2		mhmrcl1 13482	. . . . 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 5581	. . . . . 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 5568	. . . . 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 5764	. . . . . . . 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 5764	. . . . . . . 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 2229	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
17		eqid 2229	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
18	15, 16, 17	mhmlin 13486	. . . . . . . 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 1271	. . . . . . 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 5895	. . . . . . . . 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 5895	. . . . . . . . 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 6012	. . . . . . 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 2262	. . . . . 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 13442	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ (◡𝐹‘𝑥) ∈ 𝐵 ∧ (◡𝐹‘𝑦) ∈ 𝐵) → ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∈ 𝐵)
31	29, 12, 14, 30	syl3anc 1271	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∈ 𝐵)
32		f1ocnvfv 5896	. . . . . . 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 2612	. . . 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 2229	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
37		eqid 2229	. . . . . . . . 9 ⊢ (0g‘𝑆) = (0g‘𝑆)
38	36, 37	mhm0 13487	. . . . . . . 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 2235	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (0g‘𝑆) = (𝐹‘(0g‘𝑅)))
41	40	fveq2d 5627	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹‘(0g‘𝑆)) = (◡𝐹‘(𝐹‘(0g‘𝑅))))
42	15, 36	mndidcl 13449	. . . . . . . 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 5894	. . . . . 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 2262	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹‘(0g‘𝑆)) = (0g‘𝑅))
48	8, 35, 47	3jca 1201	. . 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 13480	. . 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 13484	. . . . 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 5470	. . 3 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹 Fn 𝐵)
55	49, 15	mhmf 13484	. . . . 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 5470	. . 3 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → ◡𝐹 Fn 𝐶)
58		dff1o4 5576	. . 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 598	1 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 MndHom 𝑅)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ^◡ccnv 4715   Fn wfn 5309  ⟶wf 5310  –1-1-onto→wf1o 5313  ‘cfv 5314  (class class class)co 5994  Basecbs 13018  +_gcplusg 13096  0_gc0g 13275  Mndcmnd 13435   MndHom cmhm 13476

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-map 6787  df-inn 9099  df-2 9157  df-ndx 13021  df-slot 13022  df-base 13024  df-plusg 13109  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436  df-mhm 13478

This theorem is referenced by:  rhmf1o  14117