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Theorem mgmhmf1o 44044
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:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MgmHom 𝑅)))

Proof of Theorem mgmhmf1o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmhmrcl 44038 . . . . 5 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm))
21ancomd 464 . . . 4 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm))
32adantr 483 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm))
4 f1ocnv 6620 . . . . . 6 (𝐹:𝐵1-1-onto𝐶𝐹:𝐶1-1-onto𝐵)
54adantl 484 . . . . 5 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶1-1-onto𝐵)
6 f1of 6608 . . . . 5 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
75, 6syl 17 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶𝐵)
8 simpll 765 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ (𝑅 MgmHom 𝑆))
97adantr 483 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐶𝐵)
10 simprl 769 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
119, 10ffvelrnd 6845 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑥) ∈ 𝐵)
12 simprr 771 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
139, 12ffvelrnd 6845 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑦) ∈ 𝐵)
14 mgmhmf1o.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
15 eqid 2819 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
16 eqid 2819 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1714, 15, 16mgmhmlin 44043 . . . . . . . 8 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
188, 11, 13, 17syl3anc 1366 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
19 simplr 767 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐵1-1-onto𝐶)
20 f1ocnvfv2 7026 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑥𝐶) → (𝐹‘(𝐹𝑥)) = 𝑥)
2119, 10, 20syl2anc 586 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑥)) = 𝑥)
22 f1ocnvfv2 7026 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑦𝐶) → (𝐹‘(𝐹𝑦)) = 𝑦)
2319, 12, 22syl2anc 586 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2421, 23oveq12d 7166 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
2518, 24eqtrd 2854 . . . . . 6 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
261simpld 497 . . . . . . . . . 10 (𝐹 ∈ (𝑅 MgmHom 𝑆) → 𝑅 ∈ Mgm)
2726adantr 483 . . . . . . . . 9 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝑅 ∈ Mgm)
2827adantr 483 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mgm)
2914, 15mgmcl 17847 . . . . . . . 8 ((𝑅 ∈ Mgm ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
3028, 11, 13, 29syl3anc 1366 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
31 f1ocnvfv 7027 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐶 ∧ ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵) → ((𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦))))
3219, 30, 31syl2anc 586 . . . . . 6 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦))))
3325, 32mpd 15 . . . . 5 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))
3433ralrimivva 3189 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))
357, 34jca 514 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹:𝐶𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦))))
36 mgmhmf1o.c . . . 4 𝐶 = (Base‘𝑆)
3736, 14, 16, 15ismgmhm 44040 . . 3 (𝐹 ∈ (𝑆 MgmHom 𝑅) ↔ ((𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm) ∧ (𝐹:𝐶𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))))
383, 35, 37sylanbrc 585 . 2 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹 ∈ (𝑆 MgmHom 𝑅))
3914, 36mgmhmf 44041 . . . . 5 (𝐹 ∈ (𝑅 MgmHom 𝑆) → 𝐹:𝐵𝐶)
4039adantr 483 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐵𝐶)
4140ffnd 6508 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹 Fn 𝐵)
4236, 14mgmhmf 44041 . . . . 5 (𝐹 ∈ (𝑆 MgmHom 𝑅) → 𝐹:𝐶𝐵)
4342adantl 484 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐶𝐵)
4443ffnd 6508 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹 Fn 𝐶)
45 dff1o4 6616 . . 3 (𝐹:𝐵1-1-onto𝐶 ↔ (𝐹 Fn 𝐵𝐹 Fn 𝐶))
4641, 44, 45sylanbrc 585 . 2 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐵1-1-onto𝐶)
4738, 46impbida 799 1 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MgmHom 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1531  wcel 2108  wral 3136  ccnv 5547   Fn wfn 6343  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7148  Basecbs 16475  +gcplusg 16557  Mgmcmgm 17842   MgmHom cmgmhm 44034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8400  df-mgm 17844  df-mgmhm 44036
This theorem is referenced by:  rnghmf1o  44164
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