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Theorem mhmf1o 17266
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 17260 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑆 ∈ Mnd)
2 mhmrcl1 17259 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd)
31, 2jca 554 . . . 4 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd))
43adantr 481 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd))
5 f1ocnv 6106 . . . . . 6 (𝐹:𝐵1-1-onto𝐶𝐹:𝐶1-1-onto𝐵)
65adantl 482 . . . . 5 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶1-1-onto𝐵)
7 f1of 6094 . . . . 5 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
86, 7syl 17 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶𝐵)
9 simpll 789 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ (𝑅 MndHom 𝑆))
108adantr 481 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐶𝐵)
11 simprl 793 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
1210, 11ffvelrnd 6316 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑥) ∈ 𝐵)
13 simprr 795 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
1410, 13ffvelrnd 6316 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑦) ∈ 𝐵)
15 mhmf1o.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
16 eqid 2621 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
17 eqid 2621 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1815, 16, 17mhmlin 17263 . . . . . . . 8 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
199, 12, 14, 18syl3anc 1323 . . . . . . 7 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
20 simpr 477 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐵1-1-onto𝐶)
2120adantr 481 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐵1-1-onto𝐶)
22 f1ocnvfv2 6487 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑥𝐶) → (𝐹‘(𝐹𝑥)) = 𝑥)
2321, 11, 22syl2anc 692 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑥)) = 𝑥)
24 f1ocnvfv2 6487 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑦𝐶) → (𝐹‘(𝐹𝑦)) = 𝑦)
2521, 13, 24syl2anc 692 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2623, 25oveq12d 6622 . . . . . . 7 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
2719, 26eqtrd 2655 . . . . . 6 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
282adantr 481 . . . . . . . . 9 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝑅 ∈ Mnd)
2928adantr 481 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mnd)
3015, 16mndcl 17222 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
3129, 12, 14, 30syl3anc 1323 . . . . . . 7 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
32 f1ocnvfv 6488 . . . . . . 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 692 . . . . . 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 2965 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))
36 eqid 2621 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
37 eqid 2621 . . . . . . . . 9 (0g𝑆) = (0g𝑆)
3836, 37mhm0 17264 . . . . . . . 8 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g𝑅)) = (0g𝑆))
3938adantr 481 . . . . . . 7 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(0g𝑅)) = (0g𝑆))
4039eqcomd 2627 . . . . . 6 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (0g𝑆) = (𝐹‘(0g𝑅)))
4140fveq2d 6152 . . . . 5 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(0g𝑆)) = (𝐹‘(𝐹‘(0g𝑅))))
4215, 36mndidcl 17229 . . . . . . . 8 (𝑅 ∈ Mnd → (0g𝑅) ∈ 𝐵)
432, 42syl 17 . . . . . . 7 (𝐹 ∈ (𝑅 MndHom 𝑆) → (0g𝑅) ∈ 𝐵)
4443adantr 481 . . . . . 6 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (0g𝑅) ∈ 𝐵)
45 f1ocnvfv1 6486 . . . . . 6 ((𝐹:𝐵1-1-onto𝐶 ∧ (0g𝑅) ∈ 𝐵) → (𝐹‘(𝐹‘(0g𝑅))) = (0g𝑅))
4620, 44, 45syl2anc 692 . . . . 5 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(𝐹‘(0g𝑅))) = (0g𝑅))
4741, 46eqtrd 2655 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(0g𝑆)) = (0g𝑅))
488, 35, 473jca 1240 . . 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 17258 . . 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 697 . 2 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹 ∈ (𝑆 MndHom 𝑅))
5215, 49mhmf 17261 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹:𝐵𝐶)
5352adantr 481 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵𝐶)
54 ffn 6002 . . . 4 (𝐹:𝐵𝐶𝐹 Fn 𝐵)
5553, 54syl 17 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹 Fn 𝐵)
5649, 15mhmf 17261 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑅) → 𝐹:𝐶𝐵)
5756adantl 482 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐶𝐵)
58 ffn 6002 . . . 4 (𝐹:𝐶𝐵𝐹 Fn 𝐶)
5957, 58syl 17 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹 Fn 𝐶)
60 dff1o4 6102 . . 3 (𝐹:𝐵1-1-onto𝐶 ↔ (𝐹 Fn 𝐵𝐹 Fn 𝐶))
6155, 59, 60sylanbrc 697 . 2 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵1-1-onto𝐶)
6251, 61impbida 876 1 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MndHom 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  ccnv 5073   Fn wfn 5842  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  0gc0g 16021  Mndcmnd 17215   MndHom cmhm 17254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256
This theorem is referenced by:  rhmf1o  18653
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