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Theorem lmhmf1o 15803
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 2283 . . 3  |-  ( .s
`  T )  =  ( .s `  T
)
3 eqid 2283 . . 3  |-  ( .s
`  S )  =  ( .s `  S
)
4 eqid 2283 . . 3  |-  (Scalar `  T )  =  (Scalar `  T )
5 eqid 2283 . . 3  |-  (Scalar `  S )  =  (Scalar `  S )
6 eqid 2283 . . 3  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
7 lmhmlmod2 15789 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
87adantr 451 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  T  e.  LMod )
9 lmhmlmod1 15790 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
109adantr 451 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  S  e.  LMod )
115, 4lmhmsca 15787 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
1211eqcomd 2288 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  S
)  =  (Scalar `  T ) )
1312adantr 451 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  (Scalar `  S )  =  (Scalar `  T ) )
14 lmghm 15788 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
15 lmhmf1o.x . . . . . 6  |-  X  =  ( Base `  S
)
1615, 1ghmf1o 14712 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F : X -1-1-onto-> Y  <->  `' F  e.  ( T  GrpHom  S ) ) )
1714, 16syl 15 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  ( F : X -1-1-onto-> Y  <->  `' F  e.  ( T  GrpHom  S ) ) )
1817biimpa 470 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  `' F  e.  ( T  GrpHom  S ) )
19 simpll 730 . . . . . 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 5529 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  ( Base `  (Scalar `  S
) )  =  (
Base `  (Scalar `  T
) ) )
2120eleq2d 2350 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  (
a  e.  ( Base `  (Scalar `  S )
)  <->  a  e.  (
Base `  (Scalar `  T
) ) ) )
2221biimpar 471 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  a  e.  ( Base `  (Scalar `  T ) ) )  ->  a  e.  (
Base `  (Scalar `  S
) ) )
2322adantrr 697 . . . . . 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 5485 . . . . . . . . . 10  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
25 f1of 5472 . . . . . . . . . 10  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
2624, 25syl 15 . . . . . . . . 9  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y --> X )
2726adantl 452 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  `' F : Y --> X )
28 ffvelrn 5663 . . . . . . . 8  |-  ( ( `' F : Y --> X  /\  b  e.  Y )  ->  ( `' F `  b )  e.  X
)
2927, 28sylan 457 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  b  e.  Y )  ->  ( `' F `  b )  e.  X )
3029adantrl 696 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  -> 
( `' F `  b )  e.  X
)
31 eqid 2283 . . . . . . 7  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
325, 31, 15, 3, 2lmhmlin 15792 . . . . . 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 ) ) ) )
3319, 23, 30, 32syl3anc 1182 . . . . 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 ) ) ) )
34 f1ocnvfv2 5793 . . . . . . 7  |-  ( ( F : X -1-1-onto-> Y  /\  b  e.  Y )  ->  ( F `  ( `' F `  b ) )  =  b )
3534ad2ant2l 726 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  -> 
( F `  ( `' F `  b ) )  =  b )
3635oveq2d 5874 . . . . 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 ) )
3733, 36eqtrd 2315 . . . 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 ) )
38 simplr 731 . . . . 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 )
3910adantr 451 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  (
a  e.  ( Base `  (Scalar `  T )
)  /\  b  e.  Y ) )  ->  S  e.  LMod )
4015, 5, 3, 31lmodvscl 15644 . . . . . 6  |-  ( ( S  e.  LMod  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  ( `' F `  b )  e.  X )  -> 
( a ( .s
`  S ) ( `' F `  b ) )  e.  X )
4139, 23, 30, 40syl3anc 1182 . . . . 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 )
42 f1ocnvfv 5794 . . . . 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 ) ) ) )
4338, 41, 42syl2anc 642 . . . 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 ) ) ) )
4437, 43mpd 14 . . 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 ) ) )
451, 2, 3, 4, 5, 6, 8, 10, 13, 18, 44islmhmd 15796 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  `' F  e.  ( T LMHom  S ) )
4615, 1lmhmf 15791 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F : X
--> Y )
47 ffn 5389 . . . . 5  |-  ( F : X --> Y  ->  F  Fn  X )
4846, 47syl 15 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  Fn  X )
4948adantr 451 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  F  Fn  X )
501, 15lmhmf 15791 . . . . 5  |-  ( `' F  e.  ( T LMHom 
S )  ->  `' F : Y --> X )
5150adantl 452 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  `' F : Y --> X )
52 ffn 5389 . . . 4  |-  ( `' F : Y --> X  ->  `' F  Fn  Y
)
5351, 52syl 15 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  `' F  Fn  Y )
54 dff1o4 5480 . . 3  |-  ( F : X -1-1-onto-> Y  <->  ( F  Fn  X  /\  `' F  Fn  Y ) )
5549, 53, 54sylanbrc 645 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  F : X -1-1-onto-> Y )
5645, 55impbida 805 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   `'ccnv 4688    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212    GrpHom cghm 14680   LModclmod 15627   LMHom clmhm 15776
This theorem is referenced by:  islmim2  15819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-mnd 14367  df-grp 14489  df-ghm 14681  df-lmod 15629  df-lmhm 15779
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