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Theorem lmhmf1o 19259
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 2771 . . 3 ( ·𝑠𝑇) = ( ·𝑠𝑇)
3 eqid 2771 . . 3 ( ·𝑠𝑆) = ( ·𝑠𝑆)
4 eqid 2771 . . 3 (Scalar‘𝑇) = (Scalar‘𝑇)
5 eqid 2771 . . 3 (Scalar‘𝑆) = (Scalar‘𝑆)
6 eqid 2771 . . 3 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
7 lmhmlmod2 19245 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
87adantr 466 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝑇 ∈ LMod)
9 lmhmlmod1 19246 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
109adantr 466 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝑆 ∈ LMod)
115, 4lmhmsca 19243 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
1211eqcomd 2777 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
1312adantr 466 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (Scalar‘𝑆) = (Scalar‘𝑇))
14 lmghm 19244 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
15 lmhmf1o.x . . . . . 6 𝑋 = (Base‘𝑆)
1615, 1ghmf1o 17898 . . . . 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 462 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 GrpHom 𝑆))
19 simpll 750 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
2013fveq2d 6336 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
2120eleq2d 2836 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑆)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑇))))
2221biimpar 463 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
2322adantrr 696 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
24 f1ocnv 6290 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
25 f1of 6278 . . . . . . . . . 10 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
2624, 25syl 17 . . . . . . . . 9 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌𝑋)
2726adantl 467 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌𝑋)
2827ffvelrnda 6502 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ 𝑏𝑌) → (𝐹𝑏) ∈ 𝑋)
2928adantrl 695 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹𝑏) ∈ 𝑋)
30 eqid 2771 . . . . . . 7 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
315, 30, 15, 3, 2lmhmlin 19248 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑏) ∈ 𝑋) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))))
3219, 23, 29, 31syl3anc 1476 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))))
33 f1ocnvfv2 6676 . . . . . . 7 ((𝐹:𝑋1-1-onto𝑌𝑏𝑌) → (𝐹‘(𝐹𝑏)) = 𝑏)
3433ad2ant2l 740 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝐹𝑏)) = 𝑏)
3534oveq2d 6809 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))) = (𝑎( ·𝑠𝑇)𝑏))
3632, 35eqtrd 2805 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)𝑏))
37 simplr 752 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝐹:𝑋1-1-onto𝑌)
3810adantr 466 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝑆 ∈ LMod)
3915, 5, 3, 30lmodvscl 19090 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑏) ∈ 𝑋) → (𝑎( ·𝑠𝑆)(𝐹𝑏)) ∈ 𝑋)
4038, 23, 29, 39syl3anc 1476 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝑎( ·𝑠𝑆)(𝐹𝑏)) ∈ 𝑋)
41 f1ocnvfv 6677 . . . . 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 573 . . . 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 19252 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 LMHom 𝑆))
4515, 1lmhmf 19247 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝑋𝑌)
46 ffn 6185 . . . . 5 (𝐹:𝑋𝑌𝐹 Fn 𝑋)
4745, 46syl 17 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn 𝑋)
4847adantr 466 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑋)
491, 15lmhmf 19247 . . . . 5 (𝐹 ∈ (𝑇 LMHom 𝑆) → 𝐹:𝑌𝑋)
5049adantl 467 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑌𝑋)
51 ffn 6185 . . . 4 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
5250, 51syl 17 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑌)
53 dff1o4 6286 . . 3 (𝐹:𝑋1-1-onto𝑌 ↔ (𝐹 Fn 𝑋𝐹 Fn 𝑌))
5448, 52, 53sylanbrc 572 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑋1-1-onto𝑌)
5544, 54impbida 802 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 382   = wceq 1631  wcel 2145  ccnv 5248   Fn wfn 6026  wf 6027  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  Basecbs 16064  Scalarcsca 16152   ·𝑠 cvsca 16153   GrpHom cghm 17865  LModclmod 19073   LMHom clmhm 19232
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-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
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-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  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-iun 4656  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 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-ghm 17866  df-lmod 19075  df-lmhm 19235
This theorem is referenced by:  islmim2  19279
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