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Theorem mhmf1o 17611
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 17605 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑆 ∈ Mnd)
2 mhmrcl1 17604 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd)
31, 2jca 507 . . . 4 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd))
43adantr 472 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd))
5 f1ocnv 6332 . . . . . 6 (𝐹:𝐵1-1-onto𝐶𝐹:𝐶1-1-onto𝐵)
65adantl 473 . . . . 5 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶1-1-onto𝐵)
7 f1of 6320 . . . . 5 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
86, 7syl 17 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹:𝐶𝐵)
9 simpll 783 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ (𝑅 MndHom 𝑆))
108adantr 472 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐶𝐵)
11 simprl 787 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
1210, 11ffvelrnd 6550 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑥) ∈ 𝐵)
13 simprr 789 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
1410, 13ffvelrnd 6550 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹𝑦) ∈ 𝐵)
15 mhmf1o.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
16 eqid 2765 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
17 eqid 2765 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1815, 16, 17mhmlin 17608 . . . . . . . 8 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))))
199, 12, 14, 18syl3anc 1490 . . . . . . 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 472 . . . . . . . . 9 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐵1-1-onto𝐶)
22 f1ocnvfv2 6725 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑥𝐶) → (𝐹‘(𝐹𝑥)) = 𝑥)
2321, 11, 22syl2anc 579 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑥)) = 𝑥)
24 f1ocnvfv2 6725 . . . . . . . . 9 ((𝐹:𝐵1-1-onto𝐶𝑦𝐶) → (𝐹‘(𝐹𝑦)) = 𝑦)
2521, 13, 24syl2anc 579 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2623, 25oveq12d 6860 . . . . . . 7 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹‘(𝐹𝑥))(+g𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
2719, 26eqtrd 2799 . . . . . 6 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → (𝐹‘((𝐹𝑥)(+g𝑅)(𝐹𝑦))) = (𝑥(+g𝑆)𝑦))
282adantr 472 . . . . . . . . 9 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝑅 ∈ Mnd)
2928adantr 472 . . . . . . . 8 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mnd)
3015, 16mndcl 17567 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ (𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑦) ∈ 𝐵) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
3129, 12, 14, 30syl3anc 1490 . . . . . . 7 (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥)(+g𝑅)(𝐹𝑦)) ∈ 𝐵)
32 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))))
3321, 31, 32syl2anc 579 . . . . . 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 3118 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → ∀𝑥𝐶𝑦𝐶 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑅)(𝐹𝑦)))
36 eqid 2765 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
37 eqid 2765 . . . . . . . . 9 (0g𝑆) = (0g𝑆)
3836, 37mhm0 17609 . . . . . . . 8 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g𝑅)) = (0g𝑆))
3938adantr 472 . . . . . . 7 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(0g𝑅)) = (0g𝑆))
4039eqcomd 2771 . . . . . 6 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (0g𝑆) = (𝐹‘(0g𝑅)))
4140fveq2d 6379 . . . . 5 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(0g𝑆)) = (𝐹‘(𝐹‘(0g𝑅))))
4215, 36mndidcl 17574 . . . . . . . 8 (𝑅 ∈ Mnd → (0g𝑅) ∈ 𝐵)
432, 42syl 17 . . . . . . 7 (𝐹 ∈ (𝑅 MndHom 𝑆) → (0g𝑅) ∈ 𝐵)
4443adantr 472 . . . . . 6 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (0g𝑅) ∈ 𝐵)
45 f1ocnvfv1 6724 . . . . . 6 ((𝐹:𝐵1-1-onto𝐶 ∧ (0g𝑅) ∈ 𝐵) → (𝐹‘(𝐹‘(0g𝑅))) = (0g𝑅))
4620, 44, 45syl2anc 579 . . . . 5 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(𝐹‘(0g𝑅))) = (0g𝑅))
4741, 46eqtrd 2799 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝐹‘(0g𝑆)) = (0g𝑅))
488, 35, 473jca 1158 . . 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 17603 . . 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 578 . 2 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → 𝐹 ∈ (𝑆 MndHom 𝑅))
5215, 49mhmf 17606 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹:𝐵𝐶)
5352adantr 472 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵𝐶)
5453ffnd 6224 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹 Fn 𝐵)
5549, 15mhmf 17606 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑅) → 𝐹:𝐶𝐵)
5655adantl 473 . . . 4 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐶𝐵)
5756ffnd 6224 . . 3 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹 Fn 𝐶)
58 dff1o4 6328 . . 3 (𝐹:𝐵1-1-onto𝐶 ↔ (𝐹 Fn 𝐵𝐹 Fn 𝐶))
5954, 57, 58sylanbrc 578 . 2 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵1-1-onto𝐶)
6051, 59impbida 835 1 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MndHom 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = 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 16130  +gcplusg 16214  0gc0g 16366  Mndcmnd 17560   MndHom cmhm 17599
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-reu 3062  df-rmo 3063  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-mpt 4889  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-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-map 8062  df-0g 16368  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-mhm 17601
This theorem is referenced by:  rhmf1o  19001
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