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Theorem mhmf1o 18075
Description: A monoid homomorphism is bijective iff its converse is also a monoid homomorphism. (Contributed by AV, 22-Oct-2019.)
Hypotheses
Ref Expression
mhmf1o.b 𝐵 = (Base‘𝑅)
mhmf1o.c 𝐶 = (Base‘𝑆)
Assertion
Ref Expression
mhmf1o (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MndHom 𝑅)))

Proof of Theorem mhmf1o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmrcl2 18069 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑆 ∈ Mnd)
2 mhmrcl1 18068 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd)
31, 2jca 515 . . . 4 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd))
43adantr 484 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd))
5 f1ocnv 6624 . . . . . 6 (𝐹:𝐵1-1-onto𝐶𝐹:𝐶1-1-onto𝐵)
65adantl 485 . . . . 5 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶1-1-onto𝐵)
7 f1of 6612 . . . . 5 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
86, 7syl 17 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶𝐵)
9 simpll 767 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ (𝑅 MndHom 𝑆))
108adantr 484 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐶𝐵)
11 simprl 771 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
1210, 11ffvelrnd 6856 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑥) ∈ 𝐵)
13 simprr 773 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
1410, 13ffvelrnd 6856 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑦) ∈ 𝐵)
15 mhmf1o.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
16 eqid 2738 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
17 eqid 2738 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1815, 16, 17mhmlin 18072 . . . . . . . 8 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
199, 12, 14, 18syl3anc 1372 . . . . . . 7 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
20 simpr 488 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐵1-1-onto𝐶)
2120adantr 484 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐵1-1-onto𝐶)
22 f1ocnvfv2 7039 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑥𝐶) → (𝐹‘(𝐹𝑥)) = 𝑥)
2321, 11, 22syl2anc 587 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑥)) = 𝑥)
24 f1ocnvfv2 7039 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑦𝐶) → (𝐹‘(𝐹𝑦)) = 𝑦)
2521, 13, 24syl2anc 587 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2623, 25oveq12d 7182 . . . . . . 7 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
2719, 26eqtrd 2773 . . . . . 6 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
282adantr 484 . . . . . . . . 9 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝑅 ∈ Mnd)
2928adantr 484 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mnd)
3015, 16mndcl 18028 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
3129, 12, 14, 30syl3anc 1372 . . . . . . 7 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
32 f1ocnvfv 7040 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐶 ∧ ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵) → ((𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦))))
3321, 31, 32syl2anc 587 . . . . . 6 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦))))
3427, 33mpd 15 . . . . 5 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))
3534ralrimivva 3103 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))
36 eqid 2738 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
37 eqid 2738 . . . . . . . . 9 (0g𝑆) = (0g𝑆)
3836, 37mhm0 18073 . . . . . . . 8 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g𝑅)) = (0g𝑆))
3938adantr 484 . . . . . . 7 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(0g𝑅)) = (0g𝑆))
4039eqcomd 2744 . . . . . 6 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (0g𝑆) = (𝐹‘(0g𝑅)))
4140fveq2d 6672 . . . . 5 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(0g𝑆)) = (𝐹‘(𝐹‘(0g𝑅))))
4215, 36mndidcl 18035 . . . . . . . 8 (𝑅 ∈ Mnd → (0g𝑅) ∈ 𝐵)
432, 42syl 17 . . . . . . 7 (𝐹 ∈ (𝑅 MndHom 𝑆) → (0g𝑅) ∈ 𝐵)
4443adantr 484 . . . . . 6 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (0g𝑅) ∈ 𝐵)
45 f1ocnvfv1 7038 . . . . . 6 ((𝐹:𝐵1-1-onto𝐶 ∧ (0g𝑅) ∈ 𝐵) → (𝐹‘(𝐹‘(0g𝑅))) = (0g𝑅))
4620, 44, 45syl2anc 587 . . . . 5 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(𝐹‘(0g𝑅))) = (0g𝑅))
4741, 46eqtrd 2773 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(0g𝑆)) = (0g𝑅))
488, 35, 473jca 1129 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹:𝐶𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑅)))
49 mhmf1o.c . . . 4 𝐶 = (Base‘𝑆)
5049, 15, 17, 16, 37, 36ismhm 18067 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑅) ↔ ((𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd) ∧ (𝐹:𝐶𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑅))))
514, 48, 50sylanbrc 586 . 2 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹 ∈ (𝑆 MndHom 𝑅))
5215, 49mhmf 18070 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹:𝐵𝐶)
5352adantr 484 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵𝐶)
5453ffnd 6499 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹 Fn 𝐵)
5549, 15mhmf 18070 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑅) → 𝐹:𝐶𝐵)
5655adantl 485 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐶𝐵)
5756ffnd 6499 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹 Fn 𝐶)
58 dff1o4 6620 . . 3 (𝐹:𝐵1-1-onto𝐶 ↔ (𝐹 Fn 𝐵𝐹 Fn 𝐶))
5954, 57, 58sylanbrc 586 . 2 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵1-1-onto𝐶)
6051, 59impbida 801 1 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MndHom 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2113  wral 3053  ccnv 5518   Fn wfn 6328  wf 6329  1-1-ontowf1o 6332  cfv 6333  (class class class)co 7164  Basecbs 16579  +gcplusg 16661  0gc0g 16809  Mndcmnd 18020   MndHom cmhm 18063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-map 8432  df-0g 16811  df-mgm 17961  df-sgrp 18010  df-mnd 18021  df-mhm 18065
This theorem is referenced by:  rhmf1o  19599
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