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Theorem tposmap 19085
Description: The transposition of an I X J -matrix is a J X I -matrix, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
Assertion
Ref Expression
tposmap  |-  ( A  e.  ( B  ^m  ( I  X.  J
) )  -> tpos  A  e.  ( B  ^m  ( J  X.  I ) ) )

Proof of Theorem tposmap
StepHypRef Expression
1 elmapi 7459 . . 3  |-  ( A  e.  ( B  ^m  ( I  X.  J
) )  ->  A : ( I  X.  J ) --> B )
2 tposf 7001 . . 3  |-  ( A : ( I  X.  J ) --> B  -> tpos  A : ( J  X.  I ) --> B )
31, 2syl 16 . 2  |-  ( A  e.  ( B  ^m  ( I  X.  J
) )  -> tpos  A :
( J  X.  I
) --> B )
4 elmapex 7458 . . 3  |-  ( A  e.  ( B  ^m  ( I  X.  J
) )  ->  ( B  e.  _V  /\  (
I  X.  J )  e.  _V ) )
5 cnvxp 5431 . . . . 5  |-  `' ( I  X.  J )  =  ( J  X.  I )
6 cnvexg 6745 . . . . 5  |-  ( ( I  X.  J )  e.  _V  ->  `' ( I  X.  J
)  e.  _V )
75, 6syl5eqelr 2550 . . . 4  |-  ( ( I  X.  J )  e.  _V  ->  ( J  X.  I )  e. 
_V )
87anim2i 569 . . 3  |-  ( ( B  e.  _V  /\  ( I  X.  J
)  e.  _V )  ->  ( B  e.  _V  /\  ( J  X.  I
)  e.  _V )
)
9 elmapg 7451 . . 3  |-  ( ( B  e.  _V  /\  ( J  X.  I
)  e.  _V )  ->  (tpos  A  e.  ( B  ^m  ( J  X.  I ) )  <-> tpos  A : ( J  X.  I ) --> B ) )
104, 8, 93syl 20 . 2  |-  ( A  e.  ( B  ^m  ( I  X.  J
) )  ->  (tpos  A  e.  ( B  ^m  ( J  X.  I
) )  <-> tpos  A : ( J  X.  I ) --> B ) )
113, 10mpbird 232 1  |-  ( A  e.  ( B  ^m  ( I  X.  J
) )  -> tpos  A  e.  ( B  ^m  ( J  X.  I ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1819   _Vcvv 3109    X. cxp 5006   `'ccnv 5007   -->wf 5590  (class class class)co 6296  tpos ctpos 6972    ^m cmap 7438
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-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-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-fo 5600  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-tpos 6973  df-map 7440
This theorem is referenced by:  mamutpos  19086
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