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

Theorem tposmap 18347
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 7239 . . 3  |-  ( A  e.  ( B  ^m  ( I  X.  J
) )  ->  A : ( I  X.  J ) --> B )
2 tposf 6778 . . 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 7238 . . 3  |-  ( A  e.  ( B  ^m  ( I  X.  J
) )  ->  ( B  e.  _V  /\  (
I  X.  J )  e.  _V ) )
5 cnvxp 5260 . . . . 5  |-  `' ( I  X.  J )  =  ( J  X.  I )
6 cnvexg 6529 . . . . 5  |-  ( ( I  X.  J )  e.  _V  ->  `' ( I  X.  J
)  e.  _V )
75, 6syl5eqelr 2528 . . . 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 7232 . . 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 1756   _Vcvv 2977    X. cxp 4843   `'ccnv 4844   -->wf 5419  (class class class)co 6096  tpos ctpos 6749    ^m cmap 7219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fo 5429  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-tpos 6750  df-map 7221
This theorem is referenced by:  mamutpos  18348
  Copyright terms: Public domain W3C validator