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Theorem lmhmf1o 19094
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 2651 . . 3 ( ·𝑠𝑇) = ( ·𝑠𝑇)
3 eqid 2651 . . 3 ( ·𝑠𝑆) = ( ·𝑠𝑆)
4 eqid 2651 . . 3 (Scalar‘𝑇) = (Scalar‘𝑇)
5 eqid 2651 . . 3 (Scalar‘𝑆) = (Scalar‘𝑆)
6 eqid 2651 . . 3 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
7 lmhmlmod2 19080 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
87adantr 480 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝑇 ∈ LMod)
9 lmhmlmod1 19081 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
109adantr 480 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝑆 ∈ LMod)
115, 4lmhmsca 19078 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
1211eqcomd 2657 . . . 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 19079 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
15 lmhmf1o.x . . . . . 6 𝑋 = (Base‘𝑆)
1615, 1ghmf1o 17737 . . . . 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 500 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 GrpHom 𝑆))
19 simpll 805 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
2013fveq2d 6233 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
2120eleq2d 2716 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑆)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑇))))
2221biimpar 501 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
2322adantrr 753 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
24 f1ocnv 6187 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
25 f1of 6175 . . . . . . . . . 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)
2827ffvelrnda 6399 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ 𝑏𝑌) → (𝐹𝑏) ∈ 𝑋)
2928adantrl 752 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹𝑏) ∈ 𝑋)
30 eqid 2651 . . . . . . 7 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
315, 30, 15, 3, 2lmhmlin 19083 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑏) ∈ 𝑋) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))))
3219, 23, 29, 31syl3anc 1366 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))))
33 f1ocnvfv2 6573 . . . . . . 7 ((𝐹:𝑋1-1-onto𝑌𝑏𝑌) → (𝐹‘(𝐹𝑏)) = 𝑏)
3433ad2ant2l 797 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝐹𝑏)) = 𝑏)
3534oveq2d 6706 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))) = (𝑎( ·𝑠𝑇)𝑏))
3632, 35eqtrd 2685 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)𝑏))
37 simplr 807 . . . . 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 18928 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑏) ∈ 𝑋) → (𝑎( ·𝑠𝑆)(𝐹𝑏)) ∈ 𝑋)
4038, 23, 29, 39syl3anc 1366 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝑎( ·𝑠𝑆)(𝐹𝑏)) ∈ 𝑋)
41 f1ocnvfv 6574 . . . . 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 694 . . . 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 19087 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 LMHom 𝑆))
4515, 1lmhmf 19082 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝑋𝑌)
46 ffn 6083 . . . . 5 (𝐹:𝑋𝑌𝐹 Fn 𝑋)
4745, 46syl 17 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn 𝑋)
4847adantr 480 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑋)
491, 15lmhmf 19082 . . . . 5 (𝐹 ∈ (𝑇 LMHom 𝑆) → 𝐹:𝑌𝑋)
5049adantl 481 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑌𝑋)
51 ffn 6083 . . . 4 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
5250, 51syl 17 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑌)
53 dff1o4 6183 . . 3 (𝐹:𝑋1-1-onto𝑌 ↔ (𝐹 Fn 𝑋𝐹 Fn 𝑌))
5448, 52, 53sylanbrc 699 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑋1-1-onto𝑌)
5544, 54impbida 895 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 LMHom 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  ccnv 5142   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992   GrpHom cghm 17704  LModclmod 18911   LMHom clmhm 19067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-ghm 17705  df-lmod 18913  df-lmhm 19070
This theorem is referenced by:  islmim2  19114
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