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Theorem mgmhmf1o 41577
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 41571 . . . . 5 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm))
21ancomd 466 . . . 4 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm))
32adantr 480 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm))
4 f1ocnv 6062 . . . . . 6 (𝐹:𝐵1-1-onto𝐶𝐹:𝐶1-1-onto𝐵)
54adantl 481 . . . . 5 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶1-1-onto𝐵)
6 f1of 6050 . . . . 5 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
75, 6syl 17 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶𝐵)
8 simpll 786 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ (𝑅 MgmHom 𝑆))
97adantr 480 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐶𝐵)
10 simprl 790 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
119, 10ffvelrnd 6268 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑥) ∈ 𝐵)
12 simprr 792 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
139, 12ffvelrnd 6268 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑦) ∈ 𝐵)
14 mgmhmf1o.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
15 eqid 2610 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
16 eqid 2610 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1714, 15, 16mgmhmlin 41576 . . . . . . . 8 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
188, 11, 13, 17syl3anc 1318 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
19 simplr 788 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐵1-1-onto𝐶)
20 f1ocnvfv2 6433 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑥𝐶) → (𝐹‘(𝐹𝑥)) = 𝑥)
2119, 10, 20syl2anc 691 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑥)) = 𝑥)
22 f1ocnvfv2 6433 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑦𝐶) → (𝐹‘(𝐹𝑦)) = 𝑦)
2319, 12, 22syl2anc 691 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2421, 23oveq12d 6567 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
2518, 24eqtrd 2644 . . . . . 6 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
261simpld 474 . . . . . . . . . 10 (𝐹 ∈ (𝑅 MgmHom 𝑆) → 𝑅 ∈ Mgm)
2726adantr 480 . . . . . . . . 9 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝑅 ∈ Mgm)
2827adantr 480 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mgm)
2914, 15mgmcl 17068 . . . . . . . 8 ((𝑅 ∈ Mgm ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
3028, 11, 13, 29syl3anc 1318 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
31 f1ocnvfv 6434 . . . . . . 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 691 . . . . . 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 2954 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))
357, 34jca 553 . . 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 41573 . . 3 (𝐹 ∈ (𝑆 MgmHom 𝑅) ↔ ((𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm) ∧ (𝐹:𝐶𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))))
383, 35, 37sylanbrc 695 . 2 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹 ∈ (𝑆 MgmHom 𝑅))
3914, 36mgmhmf 41574 . . . . 5 (𝐹 ∈ (𝑅 MgmHom 𝑆) → 𝐹:𝐵𝐶)
4039adantr 480 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐵𝐶)
41 ffn 5958 . . . 4 (𝐹:𝐵𝐶𝐹 Fn 𝐵)
4240, 41syl 17 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹 Fn 𝐵)
4336, 14mgmhmf 41574 . . . . 5 (𝐹 ∈ (𝑆 MgmHom 𝑅) → 𝐹:𝐶𝐵)
4443adantl 481 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐶𝐵)
45 ffn 5958 . . . 4 (𝐹:𝐶𝐵𝐹 Fn 𝐶)
4644, 45syl 17 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹 Fn 𝐶)
47 dff1o4 6058 . . 3 (𝐹:𝐵1-1-onto𝐶 ↔ (𝐹 Fn 𝐵𝐹 Fn 𝐶))
4842, 46, 47sylanbrc 695 . 2 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐵1-1-onto𝐶)
4938, 48impbida 873 1 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MgmHom 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  ccnv 5037   Fn wfn 5799  wf 5800  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Mgmcmgm 17063   MgmHom cmgmhm 41567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-mgm 17065  df-mgmhm 41569
This theorem is referenced by:  rnghmf1o  41693
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