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Theorem mgmhmf1o 46071
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 46065 . . . . 5 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm))
21ancomd 462 . . . 4 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm))
32adantr 481 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm))
4 f1ocnv 6796 . . . . . 6 (𝐹:𝐵1-1-onto𝐶𝐹:𝐶1-1-onto𝐵)
54adantl 482 . . . . 5 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶1-1-onto𝐵)
6 f1of 6784 . . . . 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 481 . . . . . . . . 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, 10ffvelcdmd 7036 . . . . . . . 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, 12ffvelcdmd 7036 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑦) ∈ 𝐵)
14 mgmhmf1o.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
15 eqid 2736 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
16 eqid 2736 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1714, 15, 16mgmhmlin 46070 . . . . . . . 8 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
188, 11, 13, 17syl3anc 1371 . . . . . . 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 7223 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑥𝐶) → (𝐹‘(𝐹𝑥)) = 𝑥)
2119, 10, 20syl2anc 584 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑥)) = 𝑥)
22 f1ocnvfv2 7223 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑦𝐶) → (𝐹‘(𝐹𝑦)) = 𝑦)
2319, 12, 22syl2anc 584 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2421, 23oveq12d 7375 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
2518, 24eqtrd 2776 . . . . . 6 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
261simpld 495 . . . . . . . . . 10 (𝐹 ∈ (𝑅 MgmHom 𝑆) → 𝑅 ∈ Mgm)
2726adantr 481 . . . . . . . . 9 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝑅 ∈ Mgm)
2827adantr 481 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mgm)
2914, 15mgmcl 18500 . . . . . . . 8 ((𝑅 ∈ Mgm ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
3028, 11, 13, 29syl3anc 1371 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
31 f1ocnvfv 7224 . . . . . . 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 584 . . . . . 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 3197 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))
357, 34jca 512 . . 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 46067 . . 3 (𝐹 ∈ (𝑆 MgmHom 𝑅) ↔ ((𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm) ∧ (𝐹:𝐶𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))))
383, 35, 37sylanbrc 583 . 2 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹 ∈ (𝑆 MgmHom 𝑅))
3914, 36mgmhmf 46068 . . . . 5 (𝐹 ∈ (𝑅 MgmHom 𝑆) → 𝐹:𝐵𝐶)
4039adantr 481 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐵𝐶)
4140ffnd 6669 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹 Fn 𝐵)
4236, 14mgmhmf 46068 . . . . 5 (𝐹 ∈ (𝑆 MgmHom 𝑅) → 𝐹:𝐶𝐵)
4342adantl 482 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐶𝐵)
4443ffnd 6669 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹 Fn 𝐶)
45 dff1o4 6792 . . 3 (𝐹:𝐵1-1-onto𝐶 ↔ (𝐹 Fn 𝐵𝐹 Fn 𝐶))
4641, 44, 45sylanbrc 583 . 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 205  wa 396   = wceq 1541  wcel 2106  wral 3064  ccnv 5632   Fn wfn 6491  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  Mgmcmgm 18495   MgmHom cmgmhm 46061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-mgm 18497  df-mgmhm 46063
This theorem is referenced by:  rnghmf1o  46191
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