Theorem mgmhmf1o 45229

Description: A magma homomorphism is bijective iff its converse is also a magma homomorphism. (Contributed by AV, 25-Feb-2020.)

Hypotheses

Ref	Expression
mgmhmf1o.b	⊢ 𝐵 = (Base‘𝑅)
mgmhmf1o.c	⊢ 𝐶 = (Base‘𝑆)

Assertion

Ref	Expression
mgmhmf1o	⊢ (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 MgmHom 𝑅)))

Proof of Theorem mgmhmf1o

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

Step	Hyp	Ref	Expression
1		mgmhmrcl 45223	. . . . 5 ⊢ (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm))
2	1	ancomd 461	. . . 4 ⊢ (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm))
3	2	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm))
4		f1ocnv 6712	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐶 → ◡𝐹:𝐶–1-1-onto→𝐵)
5	4	adantl 481	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹:𝐶–1-1-onto→𝐵)
6		f1of 6700	. . . . 5 ⊢ (◡𝐹:𝐶–1-1-onto→𝐵 → ◡𝐹:𝐶⟶𝐵)
7	5, 6	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹:𝐶⟶𝐵)
8		simpll 763	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐹 ∈ (𝑅 MgmHom 𝑆))
9	7	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ◡𝐹:𝐶⟶𝐵)
10		simprl 767	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
11	9, 10	ffvelrnd 6944	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (◡𝐹‘𝑥) ∈ 𝐵)
12		simprr 769	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
13	9, 12	ffvelrnd 6944	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (◡𝐹‘𝑦) ∈ 𝐵)
14		mgmhmf1o.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
15		eqid 2738	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
16		eqid 2738	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
17	14, 15, 16	mgmhmlin 45228	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ (◡𝐹‘𝑥) ∈ 𝐵 ∧ (◡𝐹‘𝑦) ∈ 𝐵) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) = ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑆)(𝐹‘(◡𝐹‘𝑦))))
18	8, 11, 13, 17	syl3anc 1369	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 MgmHom 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))))
19		simplr 765	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐹:𝐵–1-1-onto→𝐶)
20		f1ocnvfv2 7130	. . . . . . . . 9 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑥 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
21	19, 10, 20	syl2anc 583	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
22		f1ocnvfv2 7130	. . . . . . . . 9 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑦 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦)
23	19, 12, 22	syl2anc 583	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦)
24	21, 23	oveq12d 7273	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑆)(𝐹‘(◡𝐹‘𝑦))) = (𝑥(+g‘𝑆)𝑦))
25	18, 24	eqtrd 2778	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑆)𝑦))
26	1	simpld 494	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑅 MgmHom 𝑆) → 𝑅 ∈ Mgm)
27	26	adantr 480	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝑅 ∈ Mgm)
28	27	adantr 480	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑅 ∈ Mgm)
29	14, 15	mgmcl 18244	. . . . . . . 8 ⊢ ((𝑅 ∈ Mgm ∧ (◡𝐹‘𝑥) ∈ 𝐵 ∧ (◡𝐹‘𝑦) ∈ 𝐵) → ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∈ 𝐵)
30	28, 11, 13, 29	syl3anc 1369	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∈ 𝐵)
31		f1ocnvfv 7131	. . . . . . 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))))
32	19, 30, 31	syl2anc 583	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 MgmHom 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))))
33	25, 32	mpd 15	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)))
34	33	ralrimivva 3114	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)))
35	7, 34	jca 511	. . 3 ⊢ ((𝐹 ∈ (𝑅 MgmHom 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))))
36		mgmhmf1o.c	. . . 4 ⊢ 𝐶 = (Base‘𝑆)
37	36, 14, 16, 15	ismgmhm 45225	. . 3 ⊢ (◡𝐹 ∈ (𝑆 MgmHom 𝑅) ↔ ((𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm) ∧ (◡𝐹:𝐶⟶𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)))))
38	3, 35, 37	sylanbrc 582	. 2 ⊢ ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 MgmHom 𝑅))
39	14, 36	mgmhmf 45226	. . . . 5 ⊢ (𝐹 ∈ (𝑅 MgmHom 𝑆) → 𝐹:𝐵⟶𝐶)
40	39	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐵⟶𝐶)
41	40	ffnd 6585	. . 3 ⊢ ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹 Fn 𝐵)
42	36, 14	mgmhmf 45226	. . . . 5 ⊢ (◡𝐹 ∈ (𝑆 MgmHom 𝑅) → ◡𝐹:𝐶⟶𝐵)
43	42	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MgmHom 𝑅)) → ◡𝐹:𝐶⟶𝐵)
44	43	ffnd 6585	. . 3 ⊢ ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MgmHom 𝑅)) → ◡𝐹 Fn 𝐶)
45		dff1o4 6708	. . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn 𝐶))
46	41, 44, 45	sylanbrc 582	. 2 ⊢ ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐵–1-1-onto→𝐶)
47	38, 46	impbida 797	1 ⊢ (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 MgmHom 𝑅)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ^◡ccnv 5579   Fn wfn 6413  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Mgmcmgm 18239   MgmHom cmgmhm 45219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-mgm 18241  df-mgmhm 45221

This theorem is referenced by:  rnghmf1o  45349