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Theorem mgmhmf1o 42388
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 42382 . . . . 5 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm))
21ancomd 453 . . . 4 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm))
32adantr 472 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm))
4 f1ocnv 6332 . . . . . 6 (𝐹:𝐵1-1-onto𝐶𝐹:𝐶1-1-onto𝐵)
54adantl 473 . . . . 5 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶1-1-onto𝐵)
6 f1of 6320 . . . . 5 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
75, 6syl 17 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶𝐵)
8 simpll 783 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ (𝑅 MgmHom 𝑆))
97adantr 472 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐶𝐵)
10 simprl 787 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
119, 10ffvelrnd 6550 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑥) ∈ 𝐵)
12 simprr 789 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
139, 12ffvelrnd 6550 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑦) ∈ 𝐵)
14 mgmhmf1o.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
15 eqid 2765 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
16 eqid 2765 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1714, 15, 16mgmhmlin 42387 . . . . . . . 8 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
188, 11, 13, 17syl3anc 1490 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
19 simplr 785 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐵1-1-onto𝐶)
20 f1ocnvfv2 6725 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑥𝐶) → (𝐹‘(𝐹𝑥)) = 𝑥)
2119, 10, 20syl2anc 579 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑥)) = 𝑥)
22 f1ocnvfv2 6725 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑦𝐶) → (𝐹‘(𝐹𝑦)) = 𝑦)
2319, 12, 22syl2anc 579 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2421, 23oveq12d 6860 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
2518, 24eqtrd 2799 . . . . . 6 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
261simpld 488 . . . . . . . . . 10 (𝐹 ∈ (𝑅 MgmHom 𝑆) → 𝑅 ∈ Mgm)
2726adantr 472 . . . . . . . . 9 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝑅 ∈ Mgm)
2827adantr 472 . . . . . . . 8 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mgm)
2914, 15mgmcl 17513 . . . . . . . 8 ((𝑅 ∈ Mgm ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
3028, 11, 13, 29syl3anc 1490 . . . . . . 7 (((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
31 f1ocnvfv 6726 . . . . . . 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 579 . . . . . 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 3118 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))
357, 34jca 507 . . 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 42384 . . 3 (𝐹 ∈ (𝑆 MgmHom 𝑅) ↔ ((𝑆 ∈ Mgm ∧ 𝑅 ∈ Mgm) ∧ (𝐹:𝐶𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))))
383, 35, 37sylanbrc 578 . 2 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹 ∈ (𝑆 MgmHom 𝑅))
3914, 36mgmhmf 42385 . . . . 5 (𝐹 ∈ (𝑅 MgmHom 𝑆) → 𝐹:𝐵𝐶)
4039adantr 472 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐵𝐶)
4140ffnd 6224 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹 Fn 𝐵)
4236, 14mgmhmf 42385 . . . . 5 (𝐹 ∈ (𝑆 MgmHom 𝑅) → 𝐹:𝐶𝐵)
4342adantl 473 . . . 4 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐶𝐵)
4443ffnd 6224 . . 3 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹 Fn 𝐶)
45 dff1o4 6328 . . 3 (𝐹:𝐵1-1-onto𝐶 ↔ (𝐹 Fn 𝐵𝐹 Fn 𝐶))
4641, 44, 45sylanbrc 578 . 2 ((𝐹 ∈ (𝑅 MgmHom 𝑆) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑅)) → 𝐹:𝐵1-1-onto𝐶)
4738, 46impbida 835 1 (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MgmHom 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  ccnv 5276   Fn wfn 6063  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  Basecbs 16132  +gcplusg 16216  Mgmcmgm 17508   MgmHom cmgmhm 42378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-map 8062  df-mgm 17510  df-mgmhm 42380
This theorem is referenced by:  rnghmf1o  42504
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