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Theorem lmhmf1o 16077
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  e.  ( S LMHom  T
)  ->  ( F : X -1-1-onto-> Y  <->  `' F  e.  ( 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 2404 . . 3  |-  ( .s
`  T )  =  ( .s `  T
)
3 eqid 2404 . . 3  |-  ( .s
`  S )  =  ( .s `  S
)
4 eqid 2404 . . 3  |-  (Scalar `  T )  =  (Scalar `  T )
5 eqid 2404 . . 3  |-  (Scalar `  S )  =  (Scalar `  S )
6 eqid 2404 . . 3  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
7 lmhmlmod2 16063 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
87adantr 452 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  T  e.  LMod )
9 lmhmlmod1 16064 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
109adantr 452 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  S  e.  LMod )
115, 4lmhmsca 16061 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
1211eqcomd 2409 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  S
)  =  (Scalar `  T ) )
1312adantr 452 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  (Scalar `  S )  =  (Scalar `  T ) )
14 lmghm 16062 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
15 lmhmf1o.x . . . . . 6  |-  X  =  ( Base `  S
)
1615, 1ghmf1o 14990 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F : X -1-1-onto-> Y  <->  `' F  e.  ( T  GrpHom  S ) ) )
1714, 16syl 16 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  ( F : X -1-1-onto-> Y  <->  `' F  e.  ( T  GrpHom  S ) ) )
1817biimpa 471 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  `' F  e.  ( T  GrpHom  S ) )
19 simpll 731 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  ->  F  e.  ( S LMHom  T ) )
2013fveq2d 5691 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  ( Base `  (Scalar `  S
) )  =  (
Base `  (Scalar `  T
) ) )
2120eleq2d 2471 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  (
a  e.  ( Base `  (Scalar `  S )
)  <->  a  e.  (
Base `  (Scalar `  T
) ) ) )
2221biimpar 472 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  a  e.  ( Base `  (Scalar `  T ) ) )  ->  a  e.  (
Base `  (Scalar `  S
) ) )
2322adantrr 698 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  -> 
a  e.  ( Base `  (Scalar `  S )
) )
24 f1ocnv 5646 . . . . . . . . . 10  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
25 f1of 5633 . . . . . . . . . 10  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
2624, 25syl 16 . . . . . . . . 9  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y --> X )
2726adantl 453 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  `' F : Y --> X )
2827ffvelrnda 5829 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  b  e.  Y )  ->  ( `' F `  b )  e.  X )
2928adantrl 697 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  -> 
( `' F `  b )  e.  X
)
30 eqid 2404 . . . . . . 7  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
315, 30, 15, 3, 2lmhmlin 16066 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  ( `' F `  b )  e.  X )  -> 
( F `  (
a ( .s `  S ) ( `' F `  b ) ) )  =  ( a ( .s `  T ) ( F `
 ( `' F `  b ) ) ) )
3219, 23, 29, 31syl3anc 1184 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  -> 
( F `  (
a ( .s `  S ) ( `' F `  b ) ) )  =  ( a ( .s `  T ) ( F `
 ( `' F `  b ) ) ) )
33 f1ocnvfv2 5974 . . . . . . 7  |-  ( ( F : X -1-1-onto-> Y  /\  b  e.  Y )  ->  ( F `  ( `' F `  b ) )  =  b )
3433ad2ant2l 727 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  -> 
( F `  ( `' F `  b ) )  =  b )
3534oveq2d 6056 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  -> 
( a ( .s
`  T ) ( F `  ( `' F `  b ) ) )  =  ( a ( .s `  T ) b ) )
3632, 35eqtrd 2436 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  -> 
( F `  (
a ( .s `  S ) ( `' F `  b ) ) )  =  ( a ( .s `  T ) b ) )
37 simplr 732 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  ->  F : X -1-1-onto-> Y )
3810adantr 452 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  ->  S  e.  LMod )
3915, 5, 3, 30lmodvscl 15922 . . . . . 6  |-  ( ( S  e.  LMod  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  ( `' F `  b )  e.  X )  -> 
( a ( .s
`  S ) ( `' F `  b ) )  e.  X )
4038, 23, 29, 39syl3anc 1184 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  -> 
( a ( .s
`  S ) ( `' F `  b ) )  e.  X )
41 f1ocnvfv 5975 . . . . 5  |-  ( ( F : X -1-1-onto-> Y  /\  ( a ( .s
`  S ) ( `' F `  b ) )  e.  X )  ->  ( ( F `
 ( a ( .s `  S ) ( `' F `  b ) ) )  =  ( a ( .s `  T ) b )  ->  ( `' F `  ( a ( .s `  T
) b ) )  =  ( a ( .s `  S ) ( `' F `  b ) ) ) )
4237, 40, 41syl2anc 643 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  -> 
( ( F `  ( a ( .s
`  S ) ( `' F `  b ) ) )  =  ( a ( .s `  T ) b )  ->  ( `' F `  ( a ( .s
`  T ) b ) )  =  ( a ( .s `  S ) ( `' F `  b ) ) ) )
4336, 42mpd 15 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  -> 
( `' F `  ( a ( .s
`  T ) b ) )  =  ( a ( .s `  S ) ( `' F `  b ) ) )
441, 2, 3, 4, 5, 6, 8, 10, 13, 18, 43islmhmd 16070 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  `' F  e.  ( T LMHom  S ) )
4515, 1lmhmf 16065 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F : X
--> Y )
46 ffn 5550 . . . . 5  |-  ( F : X --> Y  ->  F  Fn  X )
4745, 46syl 16 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  Fn  X )
4847adantr 452 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  F  Fn  X )
491, 15lmhmf 16065 . . . . 5  |-  ( `' F  e.  ( T LMHom 
S )  ->  `' F : Y --> X )
5049adantl 453 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  `' F : Y --> X )
51 ffn 5550 . . . 4  |-  ( `' F : Y --> X  ->  `' F  Fn  Y
)
5250, 51syl 16 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  `' F  Fn  Y )
53 dff1o4 5641 . . 3  |-  ( F : X -1-1-onto-> Y  <->  ( F  Fn  X  /\  `' F  Fn  Y ) )
5448, 52, 53sylanbrc 646 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  F : X -1-1-onto-> Y )
5544, 54impbida 806 1  |-  ( F  e.  ( S LMHom  T
)  ->  ( F : X -1-1-onto-> Y  <->  `' F  e.  ( T LMHom  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   `'ccnv 4836    Fn wfn 5408   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Basecbs 13424  Scalarcsca 13487   .scvsca 13488    GrpHom cghm 14958   LModclmod 15905   LMHom clmhm 16050
This theorem is referenced by:  islmim2  16093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-mnd 14645  df-grp 14767  df-ghm 14959  df-lmod 15907  df-lmhm 16053
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