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Theorem lmhmf1o 21045
Description: A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypotheses
Ref Expression
lmhmf1o.x 𝑋 = (Base‘𝑆)
lmhmf1o.y 𝑌 = (Base‘𝑇)
Assertion
Ref Expression
lmhmf1o (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 LMHom 𝑆)))

Proof of Theorem lmhmf1o
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmf1o.y . . 3 𝑌 = (Base‘𝑇)
2 eqid 2737 . . 3 ( ·𝑠𝑇) = ( ·𝑠𝑇)
3 eqid 2737 . . 3 ( ·𝑠𝑆) = ( ·𝑠𝑆)
4 eqid 2737 . . 3 (Scalar‘𝑇) = (Scalar‘𝑇)
5 eqid 2737 . . 3 (Scalar‘𝑆) = (Scalar‘𝑆)
6 eqid 2737 . . 3 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
7 lmhmlmod2 21031 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
87adantr 480 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝑇 ∈ LMod)
9 lmhmlmod1 21032 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
109adantr 480 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝑆 ∈ LMod)
115, 4lmhmsca 21029 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
1211eqcomd 2743 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
1312adantr 480 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (Scalar‘𝑆) = (Scalar‘𝑇))
14 lmghm 21030 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
15 lmhmf1o.x . . . . . 6 𝑋 = (Base‘𝑆)
1615, 1ghmf1o 19266 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 GrpHom 𝑆)))
1714, 16syl 17 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 GrpHom 𝑆)))
1817biimpa 476 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 GrpHom 𝑆))
19 simpll 767 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
2013fveq2d 6910 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
2120eleq2d 2827 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑆)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑇))))
2221biimpar 477 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
2322adantrr 717 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
24 f1ocnv 6860 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
25 f1of 6848 . . . . . . . . . 10 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
2624, 25syl 17 . . . . . . . . 9 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌𝑋)
2726adantl 481 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌𝑋)
2827ffvelcdmda 7104 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ 𝑏𝑌) → (𝐹𝑏) ∈ 𝑋)
2928adantrl 716 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹𝑏) ∈ 𝑋)
30 eqid 2737 . . . . . . 7 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
315, 30, 15, 3, 2lmhmlin 21034 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑏) ∈ 𝑋) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))))
3219, 23, 29, 31syl3anc 1373 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))))
33 f1ocnvfv2 7297 . . . . . . 7 ((𝐹:𝑋1-1-onto𝑌𝑏𝑌) → (𝐹‘(𝐹𝑏)) = 𝑏)
3433ad2ant2l 746 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝐹𝑏)) = 𝑏)
3534oveq2d 7447 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))) = (𝑎( ·𝑠𝑇)𝑏))
3632, 35eqtrd 2777 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)𝑏))
37 simplr 769 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝐹:𝑋1-1-onto𝑌)
3810adantr 480 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝑆 ∈ LMod)
3915, 5, 3, 30lmodvscl 20876 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑏) ∈ 𝑋) → (𝑎( ·𝑠𝑆)(𝐹𝑏)) ∈ 𝑋)
4038, 23, 29, 39syl3anc 1373 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝑎( ·𝑠𝑆)(𝐹𝑏)) ∈ 𝑋)
41 f1ocnvfv 7298 . . . . 5 ((𝐹:𝑋1-1-onto𝑌 ∧ (𝑎( ·𝑠𝑆)(𝐹𝑏)) ∈ 𝑋) → ((𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)𝑏) → (𝐹‘(𝑎( ·𝑠𝑇)𝑏)) = (𝑎( ·𝑠𝑆)(𝐹𝑏))))
4237, 40, 41syl2anc 584 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → ((𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)𝑏) → (𝐹‘(𝑎( ·𝑠𝑇)𝑏)) = (𝑎( ·𝑠𝑆)(𝐹𝑏))))
4336, 42mpd 15 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝑎( ·𝑠𝑇)𝑏)) = (𝑎( ·𝑠𝑆)(𝐹𝑏)))
441, 2, 3, 4, 5, 6, 8, 10, 13, 18, 43islmhmd 21038 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 LMHom 𝑆))
4515, 1lmhmf 21033 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝑋𝑌)
4645ffnd 6737 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn 𝑋)
4746adantr 480 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑋)
481, 15lmhmf 21033 . . . . 5 (𝐹 ∈ (𝑇 LMHom 𝑆) → 𝐹:𝑌𝑋)
4948adantl 481 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑌𝑋)
5049ffnd 6737 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑌)
51 dff1o4 6856 . . 3 (𝐹:𝑋1-1-onto𝑌 ↔ (𝐹 Fn 𝑋𝐹 Fn 𝑌))
5247, 50, 51sylanbrc 583 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑋1-1-onto𝑌)
5344, 52impbida 801 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 LMHom 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  ccnv 5684   Fn wfn 6556  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Basecbs 17247  Scalarcsca 17300   ·𝑠 cvsca 17301   GrpHom cghm 19230  LModclmod 20858   LMHom clmhm 21018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-ghm 19231  df-lmod 20860  df-lmhm 21021
This theorem is referenced by:  islmim2  21065
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