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Theorem mhmf1o 17546
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 17540 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑆 ∈ Mnd)
2 mhmrcl1 17539 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd)
31, 2jca 501 . . . 4 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd))
43adantr 466 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd))
5 f1ocnv 6288 . . . . . 6 (𝐹:𝐵1-1-onto𝐶𝐹:𝐶1-1-onto𝐵)
65adantl 467 . . . . 5 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶1-1-onto𝐵)
7 f1of 6276 . . . . 5 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
86, 7syl 17 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶𝐵)
9 simpll 750 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ (𝑅 MndHom 𝑆))
108adantr 466 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐶𝐵)
11 simprl 754 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
1210, 11ffvelrnd 6501 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑥) ∈ 𝐵)
13 simprr 756 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
1410, 13ffvelrnd 6501 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑦) ∈ 𝐵)
15 mhmf1o.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
16 eqid 2771 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
17 eqid 2771 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1815, 16, 17mhmlin 17543 . . . . . . . 8 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
199, 12, 14, 18syl3anc 1476 . . . . . . 7 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
20 simpr 471 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐵1-1-onto𝐶)
2120adantr 466 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐵1-1-onto𝐶)
22 f1ocnvfv2 6674 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑥𝐶) → (𝐹‘(𝐹𝑥)) = 𝑥)
2321, 11, 22syl2anc 573 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑥)) = 𝑥)
24 f1ocnvfv2 6674 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑦𝐶) → (𝐹‘(𝐹𝑦)) = 𝑦)
2521, 13, 24syl2anc 573 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2623, 25oveq12d 6809 . . . . . . 7 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
2719, 26eqtrd 2805 . . . . . 6 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
282adantr 466 . . . . . . . . 9 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝑅 ∈ Mnd)
2928adantr 466 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mnd)
3015, 16mndcl 17502 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
3129, 12, 14, 30syl3anc 1476 . . . . . . 7 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
32 f1ocnvfv 6675 . . . . . . 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 573 . . . . . 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 3120 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))
36 eqid 2771 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
37 eqid 2771 . . . . . . . . 9 (0g𝑆) = (0g𝑆)
3836, 37mhm0 17544 . . . . . . . 8 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g𝑅)) = (0g𝑆))
3938adantr 466 . . . . . . 7 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(0g𝑅)) = (0g𝑆))
4039eqcomd 2777 . . . . . 6 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (0g𝑆) = (𝐹‘(0g𝑅)))
4140fveq2d 6334 . . . . 5 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(0g𝑆)) = (𝐹‘(𝐹‘(0g𝑅))))
4215, 36mndidcl 17509 . . . . . . . 8 (𝑅 ∈ Mnd → (0g𝑅) ∈ 𝐵)
432, 42syl 17 . . . . . . 7 (𝐹 ∈ (𝑅 MndHom 𝑆) → (0g𝑅) ∈ 𝐵)
4443adantr 466 . . . . . 6 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (0g𝑅) ∈ 𝐵)
45 f1ocnvfv1 6673 . . . . . 6 ((𝐹:𝐵1-1-onto𝐶 ∧ (0g𝑅) ∈ 𝐵) → (𝐹‘(𝐹‘(0g𝑅))) = (0g𝑅))
4620, 44, 45syl2anc 573 . . . . 5 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(𝐹‘(0g𝑅))) = (0g𝑅))
4741, 46eqtrd 2805 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(0g𝑆)) = (0g𝑅))
488, 35, 473jca 1122 . . 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 17538 . . 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 572 . 2 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹 ∈ (𝑆 MndHom 𝑅))
5215, 49mhmf 17541 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹:𝐵𝐶)
5352adantr 466 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵𝐶)
54 ffn 6183 . . . 4 (𝐹:𝐵𝐶𝐹 Fn 𝐵)
5553, 54syl 17 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹 Fn 𝐵)
5649, 15mhmf 17541 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑅) → 𝐹:𝐶𝐵)
5756adantl 467 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐶𝐵)
58 ffn 6183 . . . 4 (𝐹:𝐶𝐵𝐹 Fn 𝐶)
5957, 58syl 17 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹 Fn 𝐶)
60 dff1o4 6284 . . 3 (𝐹:𝐵1-1-onto𝐶 ↔ (𝐹 Fn 𝐵𝐹 Fn 𝐶))
6155, 59, 60sylanbrc 572 . 2 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵1-1-onto𝐶)
6251, 61impbida 802 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 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  ccnv 5248   Fn wfn 6024  wf 6025  1-1-ontowf1o 6028  cfv 6029  (class class class)co 6791  Basecbs 16057  +gcplusg 16142  0gc0g 16301  Mndcmnd 17495   MndHom cmhm 17534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-map 8009  df-0g 16303  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536
This theorem is referenced by:  rhmf1o  18935
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