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Theorem lmhmf1o 17049
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 2433 . . 3  |-  ( .s
`  T )  =  ( .s `  T
)
3 eqid 2433 . . 3  |-  ( .s
`  S )  =  ( .s `  S
)
4 eqid 2433 . . 3  |-  (Scalar `  T )  =  (Scalar `  T )
5 eqid 2433 . . 3  |-  (Scalar `  S )  =  (Scalar `  S )
6 eqid 2433 . . 3  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
7 lmhmlmod2 17035 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
87adantr 462 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  T  e.  LMod )
9 lmhmlmod1 17036 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
109adantr 462 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  S  e.  LMod )
115, 4lmhmsca 17033 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
1211eqcomd 2438 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  S
)  =  (Scalar `  T ) )
1312adantr 462 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  (Scalar `  S )  =  (Scalar `  T ) )
14 lmghm 17034 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
15 lmhmf1o.x . . . . . 6  |-  X  =  ( Base `  S
)
1615, 1ghmf1o 15756 . . . . 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 481 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  `' F  e.  ( T  GrpHom  S ) )
19 simpll 746 . . . . . 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 5683 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  ( Base `  (Scalar `  S
) )  =  (
Base `  (Scalar `  T
) ) )
2120eleq2d 2500 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  (
a  e.  ( Base `  (Scalar `  S )
)  <->  a  e.  (
Base `  (Scalar `  T
) ) ) )
2221biimpar 482 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  a  e.  ( Base `  (Scalar `  T ) ) )  ->  a  e.  (
Base `  (Scalar `  S
) ) )
2322adantrr 709 . . . . . 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 5641 . . . . . . . . . 10  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
25 f1of 5629 . . . . . . . . . 10  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
2624, 25syl 16 . . . . . . . . 9  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y --> X )
2726adantl 463 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  `' F : Y --> X )
2827ffvelrnda 5831 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  b  e.  Y )  ->  ( `' F `  b )  e.  X )
2928adantrl 708 . . . . . 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 2433 . . . . . . 7  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
315, 30, 15, 3, 2lmhmlin 17038 . . . . . 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 1211 . . . . 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 5971 . . . . . . 7  |-  ( ( F : X -1-1-onto-> Y  /\  b  e.  Y )  ->  ( F `  ( `' F `  b ) )  =  b )
3433ad2ant2l 738 . . . . . 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 6096 . . . . 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 2465 . . . 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 747 . . . . 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 462 . . . . . 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 16889 . . . . . 6  |-  ( ( S  e.  LMod  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  ( `' F `  b )  e.  X )  -> 
( a ( .s
`  S ) ( `' F `  b ) )  e.  X )
4038, 23, 29, 39syl3anc 1211 . . . . 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 5972 . . . . 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 654 . . . 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 17042 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  `' F  e.  ( T LMHom  S ) )
4515, 1lmhmf 17037 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F : X
--> Y )
46 ffn 5547 . . . . 5  |-  ( F : X --> Y  ->  F  Fn  X )
4745, 46syl 16 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  Fn  X )
4847adantr 462 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  F  Fn  X )
491, 15lmhmf 17037 . . . . 5  |-  ( `' F  e.  ( T LMHom 
S )  ->  `' F : Y --> X )
5049adantl 463 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  `' F : Y --> X )
51 ffn 5547 . . . 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 5637 . . 3  |-  ( F : X -1-1-onto-> Y  <->  ( F  Fn  X  /\  `' F  Fn  Y ) )
5448, 52, 53sylanbrc 657 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  F : X -1-1-onto-> Y )
5544, 54impbida 821 1  |-  ( F  e.  ( S LMHom  T
)  ->  ( F : X -1-1-onto-> Y  <->  `' F  e.  ( T LMHom  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755   `'ccnv 4826    Fn wfn 5401   -->wf 5402   -1-1-onto->wf1o 5405   ` cfv 5406  (class class class)co 6080   Basecbs 14157  Scalarcsca 14224   .scvsca 14225    GrpHom cghm 15724   LModclmod 16872   LMHom clmhm 17022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-mnd 15398  df-grp 15525  df-ghm 15725  df-lmod 16874  df-lmhm 17025
This theorem is referenced by:  islmim2  17069
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