Theorem ovtposg 5897

Description: The transposition swaps the arguments in a two-argument function. When F is a matrix, which is to say a function from ( 1 ... m ) X. ( 1 ... n ) to the reals or some ring, tpos F is the transposition of F, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
ovtposg	|- ( ( A e. V /\ B e. W ) -> ( Atpos F B ) = ( B F A ) )

Proof of Theorem ovtposg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . . . 5 |- y e. _V
2		brtposg 5892	. . . . 5 |- ( ( A e. V /\ B e. W /\ y e. _V ) -> ( <. A , B >.tpos F y <-> <. B , A >. F y ) )
3	1, 2	mp3an3 1257	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >.tpos F y <-> <. B , A >. F y ) )
4	3	iotabidv 4908	. . 3 |- ( ( A e. V /\ B e. W ) -> ( iota y <. A , B >.tpos F y ) = ( iota y <. B , A >. F y ) )
5		df-fv 4930	. . 3 |- (tpos F ` <. A , B >. ) = ( iota y <. A , B >.tpos F y )
6		df-fv 4930	. . 3 |- ( F ` <. B , A >. ) = ( iota y <. B , A >. F y )
7	4, 5, 6	3eqtr4g 2138	. 2 |- ( ( A e. V /\ B e. W ) -> (tpos F ` <. A , B >. ) = ( F ` <. B , A >. ) )
8		df-ov 5535	. 2 |- ( Atpos F B ) = (tpos F ` <. A , B >. )
9		df-ov 5535	. 2 |- ( B F A ) = ( F ` <. B , A >. )
10	7, 8, 9	3eqtr4g 2138	1 |- ( ( A e. V /\ B e. W ) -> ( Atpos F B ) = ( B F A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   <.cop 3401   class class class wbr 3785   iotacio 4885   ` cfv 4922  (class class class)co 5532  tpos ctpos 5882

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930  df-ov 5535  df-tpos 5883

This theorem is referenced by:  tpossym  5914