MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmhmf1o Structured version   Unicode version

Theorem lmhmf1o 17819
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 2457 . . 3  |-  ( .s
`  T )  =  ( .s `  T
)
3 eqid 2457 . . 3  |-  ( .s
`  S )  =  ( .s `  S
)
4 eqid 2457 . . 3  |-  (Scalar `  T )  =  (Scalar `  T )
5 eqid 2457 . . 3  |-  (Scalar `  S )  =  (Scalar `  S )
6 eqid 2457 . . 3  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
7 lmhmlmod2 17805 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
87adantr 465 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  T  e.  LMod )
9 lmhmlmod1 17806 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
109adantr 465 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  S  e.  LMod )
115, 4lmhmsca 17803 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
1211eqcomd 2465 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  S
)  =  (Scalar `  T ) )
1312adantr 465 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  (Scalar `  S )  =  (Scalar `  T ) )
14 lmghm 17804 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
15 lmhmf1o.x . . . . . 6  |-  X  =  ( Base `  S
)
1615, 1ghmf1o 16423 . . . . 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 484 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  `' F  e.  ( T  GrpHom  S ) )
19 simpll 753 . . . . . 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 5876 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  ( Base `  (Scalar `  S
) )  =  (
Base `  (Scalar `  T
) ) )
2120eleq2d 2527 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  (
a  e.  ( Base `  (Scalar `  S )
)  <->  a  e.  (
Base `  (Scalar `  T
) ) ) )
2221biimpar 485 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  a  e.  ( Base `  (Scalar `  T ) ) )  ->  a  e.  (
Base `  (Scalar `  S
) ) )
2322adantrr 716 . . . . . 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 5834 . . . . . . . . . 10  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
25 f1of 5822 . . . . . . . . . 10  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
2624, 25syl 16 . . . . . . . . 9  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y --> X )
2726adantl 466 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  `' F : Y --> X )
2827ffvelrnda 6032 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  F : X -1-1-onto-> Y )  /\  b  e.  Y )  ->  ( `' F `  b )  e.  X )
2928adantrl 715 . . . . . 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 2457 . . . . . . 7  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
315, 30, 15, 3, 2lmhmlin 17808 . . . . . 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 1228 . . . . 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 6184 . . . . . . 7  |-  ( ( F : X -1-1-onto-> Y  /\  b  e.  Y )  ->  ( F `  ( `' F `  b ) )  =  b )
3433ad2ant2l 745 . . . . . 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 6312 . . . . 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 2498 . . . 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 755 . . . . 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 465 . . . . . 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 17656 . . . . . 6  |-  ( ( S  e.  LMod  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  ( `' F `  b )  e.  X )  -> 
( a ( .s
`  S ) ( `' F `  b ) )  e.  X )
4038, 23, 29, 39syl3anc 1228 . . . . 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 6185 . . . . 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 661 . . . 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 17812 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : X -1-1-onto-> Y )  ->  `' F  e.  ( T LMHom  S ) )
4515, 1lmhmf 17807 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F : X
--> Y )
46 ffn 5737 . . . . 5  |-  ( F : X --> Y  ->  F  Fn  X )
4745, 46syl 16 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  Fn  X )
4847adantr 465 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  F  Fn  X )
491, 15lmhmf 17807 . . . . 5  |-  ( `' F  e.  ( T LMHom 
S )  ->  `' F : Y --> X )
5049adantl 466 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  `' F : Y --> X )
51 ffn 5737 . . . 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 5830 . . 3  |-  ( F : X -1-1-onto-> Y  <->  ( F  Fn  X  /\  `' F  Fn  Y ) )
5448, 52, 53sylanbrc 664 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  F : X -1-1-onto-> Y )
5544, 54impbida 832 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 1395    e. wcel 1819   `'ccnv 5007    Fn wfn 5589   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   Basecbs 14644  Scalarcsca 14715   .scvsca 14716    GrpHom cghm 16391   LModclmod 17639   LMHom clmhm 17792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-ghm 16392  df-lmod 17641  df-lmhm 17795
This theorem is referenced by:  islmim2  17839
  Copyright terms: Public domain W3C validator