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Theorem mgmhmf1o 41576
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 41570 . . . . 5 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm))
21ancomd 465 . . . 4 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm))
32adantr 479 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm))
4 f1ocnv 6044 . . . . . 6 (𝐹:𝐵1-1-onto𝐶𝐹:𝐶1-1-onto𝐵)
54adantl 480 . . . . 5 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶1-1-onto𝐵)
6 f1of 6032 . . . . 5 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
75, 6syl 17 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶𝐵)
8 simpll 785 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ (𝑅 MgmHom 𝑆))
97adantr 479 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐶𝐵)
10 simprl 789 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
119, 10ffvelrnd 6250 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑥) ∈ 𝐵)
12 simprr 791 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
139, 12ffvelrnd 6250 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑦) ∈ 𝐵)
14 mgmhmf1o.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
15 eqid 2606 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
16 eqid 2606 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1714, 15, 16mgmhmlin 41575 . . . . . . . 8 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
188, 11, 13, 17syl3anc 1317 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
19 simplr 787 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐵1-1-onto𝐶)
20 f1ocnvfv2 6408 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑥𝐶) → (𝐹‘(𝐹𝑥)) = 𝑥)
2119, 10, 20syl2anc 690 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑥)) = 𝑥)
22 f1ocnvfv2 6408 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑦𝐶) → (𝐹‘(𝐹𝑦)) = 𝑦)
2319, 12, 22syl2anc 690 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2421, 23oveq12d 6542 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
2518, 24eqtrd 2640 . . . . . 6 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
261simpld 473 . . . . . . . . . 10 (𝐹 ∈ (𝑅 MgmHom 𝑆) → 𝑅 ∈ Mgm)
2726adantr 479 . . . . . . . . 9 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝑅 ∈ Mgm)
2827adantr 479 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mgm)
2914, 15mgmcl 17011 . . . . . . . 8 ((𝑅 ∈ Mgm ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
3028, 11, 13, 29syl3anc 1317 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
31 f1ocnvfv 6409 . . . . . . 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 690 . . . . . 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 2950 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))
357, 34jca 552 . . 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 41572 . . 3 (𝐹 ∈ (𝑆 MgmHom 𝑅) ↔ ((𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm) ∧ (𝐹:𝐶𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))))
383, 35, 37sylanbrc 694 . 2 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹 ∈ (𝑆 MgmHom 𝑅))
3914, 36mgmhmf 41573 . . . . 5 (𝐹 ∈ (𝑅 MgmHom 𝑆) → 𝐹:𝐵𝐶)
4039adantr 479 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐵𝐶)
41 ffn 5941 . . . 4 (𝐹:𝐵𝐶𝐹 Fn 𝐵)
4240, 41syl 17 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹 Fn 𝐵)
4336, 14mgmhmf 41573 . . . . 5 (𝐹 ∈ (𝑆 MgmHom 𝑅) → 𝐹:𝐶𝐵)
4443adantl 480 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐶𝐵)
45 ffn 5941 . . . 4 (𝐹:𝐶𝐵𝐹 Fn 𝐶)
4644, 45syl 17 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹 Fn 𝐶)
47 dff1o4 6040 . . 3 (𝐹:𝐵1-1-onto𝐶 ↔ (𝐹 Fn 𝐵𝐹 Fn 𝐶))
4842, 46, 47sylanbrc 694 . 2 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐵1-1-onto𝐶)
4938, 48impbida 872 1 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MgmHom 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2892  ccnv 5024   Fn wfn 5782  wf 5783  1-1-ontowf1o 5786  cfv 5787  (class class class)co 6524  Basecbs 15638  +gcplusg 15711  Mgmcmgm 17006   MgmHom cmgmhm 41566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-map 7720  df-mgm 17008  df-mgmhm 41568
This theorem is referenced by:  rnghmf1o  41692
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