Theorem ovtposg 6347

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 2775	. . . . 5 |- y e. _V
2		brtposg 6342	. . . . 5 |- ( ( A e. V /\ B e. W /\ y e. _V ) -> ( <. A , B >.tpos F y <-> <. B , A >. F y ) )
3	1, 2	mp3an3 1339	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >.tpos F y <-> <. B , A >. F y ) )
4	3	iotabidv 5255	. . 3 |- ( ( A e. V /\ B e. W ) -> ( iota y <. A , B >.tpos F y ) = ( iota y <. B , A >. F y ) )
5		df-fv 5280	. . 3 |- (tpos F ` <. A , B >. ) = ( iota y <. A , B >.tpos F y )
6		df-fv 5280	. . 3 |- ( F ` <. B , A >. ) = ( iota y <. B , A >. F y )
7	4, 5, 6	3eqtr4g 2263	. 2 |- ( ( A e. V /\ B e. W ) -> (tpos F ` <. A , B >. ) = ( F ` <. B , A >. ) )
8		df-ov 5949	. 2 |- ( Atpos F B ) = (tpos F ` <. A , B >. )
9		df-ov 5949	. 2 |- ( B F A ) = ( F ` <. B , A >. )
10	7, 8, 9	3eqtr4g 2263	1 |- ( ( A e. V /\ B e. W ) -> ( Atpos F B ) = ( B F A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   _Vcvv 2772   <.cop 3636   class class class wbr 4045   iotacio 5231   ` cfv 5272  (class class class)co 5946  tpos ctpos 6332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-ov 5949  df-tpos 6333

This theorem is referenced by:  tpossym  6364  opprmulg  13866