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Theorem lmhmf1o 19318
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 2765 . . 3 ( ·𝑠𝑇) = ( ·𝑠𝑇)
3 eqid 2765 . . 3 ( ·𝑠𝑆) = ( ·𝑠𝑆)
4 eqid 2765 . . 3 (Scalar‘𝑇) = (Scalar‘𝑇)
5 eqid 2765 . . 3 (Scalar‘𝑆) = (Scalar‘𝑆)
6 eqid 2765 . . 3 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
7 lmhmlmod2 19304 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
87adantr 472 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝑇 ∈ LMod)
9 lmhmlmod1 19305 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
109adantr 472 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝑆 ∈ LMod)
115, 4lmhmsca 19302 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
1211eqcomd 2771 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
1312adantr 472 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (Scalar‘𝑆) = (Scalar‘𝑇))
14 lmghm 19303 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
15 lmhmf1o.x . . . . . 6 𝑋 = (Base‘𝑆)
1615, 1ghmf1o 17954 . . . . 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 468 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 GrpHom 𝑆))
19 simpll 783 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
2013fveq2d 6379 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
2120eleq2d 2830 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑆)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑇))))
2221biimpar 469 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
2322adantrr 708 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
24 f1ocnv 6332 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
25 f1of 6320 . . . . . . . . . 10 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
2624, 25syl 17 . . . . . . . . 9 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌𝑋)
2726adantl 473 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌𝑋)
2827ffvelrnda 6549 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ 𝑏𝑌) → (𝐹𝑏) ∈ 𝑋)
2928adantrl 707 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹𝑏) ∈ 𝑋)
30 eqid 2765 . . . . . . 7 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
315, 30, 15, 3, 2lmhmlin 19307 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑏) ∈ 𝑋) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))))
3219, 23, 29, 31syl3anc 1490 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))))
33 f1ocnvfv2 6725 . . . . . . 7 ((𝐹:𝑋1-1-onto𝑌𝑏𝑌) → (𝐹‘(𝐹𝑏)) = 𝑏)
3433ad2ant2l 752 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝐹𝑏)) = 𝑏)
3534oveq2d 6858 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝑎( ·𝑠𝑇)(𝐹‘(𝐹𝑏))) = (𝑎( ·𝑠𝑇)𝑏))
3632, 35eqtrd 2799 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝐹‘(𝑎( ·𝑠𝑆)(𝐹𝑏))) = (𝑎( ·𝑠𝑇)𝑏))
37 simplr 785 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝐹:𝑋1-1-onto𝑌)
3810adantr 472 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → 𝑆 ∈ LMod)
3915, 5, 3, 30lmodvscl 19149 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑏) ∈ 𝑋) → (𝑎( ·𝑠𝑆)(𝐹𝑏)) ∈ 𝑋)
4038, 23, 29, 39syl3anc 1490 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏𝑌)) → (𝑎( ·𝑠𝑆)(𝐹𝑏)) ∈ 𝑋)
41 f1ocnvfv 6726 . . . . 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 579 . . . 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 19311 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 LMHom 𝑆))
4515, 1lmhmf 19306 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝑋𝑌)
4645ffnd 6224 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn 𝑋)
4746adantr 472 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑋)
481, 15lmhmf 19306 . . . . 5 (𝐹 ∈ (𝑇 LMHom 𝑆) → 𝐹:𝑌𝑋)
4948adantl 473 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑌𝑋)
5049ffnd 6224 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑌)
51 dff1o4 6328 . . 3 (𝐹:𝑋1-1-onto𝑌 ↔ (𝐹 Fn 𝑋𝐹 Fn 𝑌))
5247, 50, 51sylanbrc 578 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑋1-1-onto𝑌)
5344, 52impbida 835 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 LMHom 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  ccnv 5276   Fn wfn 6063  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  Basecbs 16130  Scalarcsca 16217   ·𝑠 cvsca 16218   GrpHom cghm 17921  LModclmod 19132   LMHom clmhm 19291
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-rep 4930  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-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  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-iun 4678  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-ov 6845  df-oprab 6846  df-mpt2 6847  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-grp 17692  df-ghm 17922  df-lmod 19134  df-lmhm 19294
This theorem is referenced by:  islmim2  19338
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