Theorem ovtposg 6405

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 2802	. . . . 5 |- y e. _V
2		brtposg 6400	. . . . 5 |- ( ( A e. V /\ B e. W /\ y e. _V ) -> ( <. A , B >.tpos F y <-> <. B , A >. F y ) )
3	1, 2	mp3an3 1360	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >.tpos F y <-> <. B , A >. F y ) )
4	3	iotabidv 5301	. . 3 |- ( ( A e. V /\ B e. W ) -> ( iota y <. A , B >.tpos F y ) = ( iota y <. B , A >. F y ) )
5		df-fv 5326	. . 3 |- (tpos F ` <. A , B >. ) = ( iota y <. A , B >.tpos F y )
6		df-fv 5326	. . 3 |- ( F ` <. B , A >. ) = ( iota y <. B , A >. F y )
7	4, 5, 6	3eqtr4g 2287	. 2 |- ( ( A e. V /\ B e. W ) -> (tpos F ` <. A , B >. ) = ( F ` <. B , A >. ) )
8		df-ov 6004	. 2 |- ( Atpos F B ) = (tpos F ` <. A , B >. )
9		df-ov 6004	. 2 |- ( B F A ) = ( F ` <. B , A >. )
10	7, 8, 9	3eqtr4g 2287	1 |- ( ( A e. V /\ B e. W ) -> ( Atpos F B ) = ( B F A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   <.cop 3669   class class class wbr 4083   iotacio 5276   ` cfv 5318  (class class class)co 6001  tpos ctpos 6390

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6004  df-tpos 6391

This theorem is referenced by:  tpossym  6422  opprmulg  14034