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Theorem lmhmf1o 19740
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 2825 . . 3 ( ·𝑠𝑇) = ( ·𝑠𝑇)
3 eqid 2825 . . 3 ( ·𝑠𝑆) = ( ·𝑠𝑆)
4 eqid 2825 . . 3 (Scalar‘𝑇) = (Scalar‘𝑇)
5 eqid 2825 . . 3 (Scalar‘𝑆) = (Scalar‘𝑆)
6 eqid 2825 . . 3 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
7 lmhmlmod2 19726 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
87adantr 481 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝑇 ∈ LMod)
9 lmhmlmod1 19727 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
109adantr 481 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝑆 ∈ LMod)
115, 4lmhmsca 19724 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
1211eqcomd 2831 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
1312adantr 481 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (Scalar‘𝑆) = (Scalar‘𝑇))
14 lmghm 19725 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
15 lmhmf1o.x . . . . . 6 𝑋 = (Base‘𝑆)
1615, 1ghmf1o 18320 . . . . 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 477 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 GrpHom 𝑆))
19 simpll 763 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
2013fveq2d 6670 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
2120eleq2d 2902 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑆)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑇))))
2221biimpar 478 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
2322adantrr 713 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
24 f1ocnv 6623 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
25 f1of 6611 . . . . . . . . . 10 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
2624, 25syl 17 . . . . . . . . 9 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌𝑋)
2726adantl 482 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌𝑋)
2827ffvelrnda 6846 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ 𝑏𝑌) → (𝐹𝑏) ∈ 𝑋)
2928adantrl 712 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹𝑏) ∈ 𝑋)
30 eqid 2825 . . . . . . 7 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
315, 30, 15, 3, 2lmhmlin 19729 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑏) ∈ 𝑋) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))))
3219, 23, 29, 31syl3anc 1365 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))))
33 f1ocnvfv2 7031 . . . . . . 7 ((𝐹:𝑋1-1-onto𝑌𝑏𝑌) → (𝐹‘(𝐹𝑏)) = 𝑏)
3433ad2ant2l 742 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝐹𝑏)) = 𝑏)
3534oveq2d 7167 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))) = (𝑎( ·𝑠𝑇)𝑏))
3632, 35eqtrd 2860 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)𝑏))
37 simplr 765 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝐹:𝑋1-1-onto𝑌)
3810adantr 481 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝑆 ∈ LMod)
3915, 5, 3, 30lmodvscl 19573 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑏) ∈ 𝑋) → (𝑎( ·𝑠𝑆)(𝐹𝑏)) ∈ 𝑋)
4038, 23, 29, 39syl3anc 1365 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝑎( ·𝑠𝑆)(𝐹𝑏)) ∈ 𝑋)
41 f1ocnvfv 7032 . . . . 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 19733 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 LMHom 𝑆))
4515, 1lmhmf 19728 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝑋𝑌)
4645ffnd 6511 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn 𝑋)
4746adantr 481 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑋)
481, 15lmhmf 19728 . . . . 5 (𝐹 ∈ (𝑇 LMHom 𝑆) → 𝐹:𝑌𝑋)
4948adantl 482 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑌𝑋)
5049ffnd 6511 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑌)
51 dff1o4 6619 . . 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 797 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 LMHom 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  ccnv 5552   Fn wfn 6346  wf 6347  1-1-ontowf1o 6350  cfv 6351  (class class class)co 7151  Basecbs 16475  Scalarcsca 16560   ·𝑠 cvsca 16561   GrpHom cghm 18287  LModclmod 19556   LMHom clmhm 19713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-grp 18038  df-ghm 18288  df-lmod 19558  df-lmhm 19716
This theorem is referenced by:  islmim2  19760
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