Theorem ovtpos 7367

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 RR 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
ovtpos	|- ( Atpos F B ) = ( B F A )

Proof of Theorem ovtpos

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- y e. _V
2		brtpos 7361	. . . . 5 |- ( y e. _V -> ( <. A , B >.tpos F y <-> <. B , A >. F y ) )
3	1, 2	ax-mp 5	. . . 4 |- ( <. A , B >.tpos F y <-> <. B , A >. F y )
4	3	iotabii 5873	. . 3 |- ( iota y <. A , B >.tpos F y ) = ( iota y <. B , A >. F y )
5		df-fv 5896	. . 3 |- (tpos F ` <. A , B >. ) = ( iota y <. A , B >.tpos F y )
6		df-fv 5896	. . 3 |- ( F ` <. B , A >. ) = ( iota y <. B , A >. F y )
7	4, 5, 6	3eqtr4i 2654	. 2 |- (tpos F ` <. A , B >. ) = ( F ` <. B , A >. )
8		df-ov 6653	. 2 |- ( Atpos F B ) = (tpos F ` <. A , B >. )
9		df-ov 6653	. 2 |- ( B F A ) = ( F ` <. B , A >. )
10	7, 8, 9	3eqtr4i 2654	1 |- ( Atpos F B ) = ( B F A )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   iotacio 5849   ` cfv 5888  (class class class)co 6650  tpos ctpos 7351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653  df-tpos 7352

This theorem is referenced by:  tpossym  7384  oppchom  16375  oppcco  16377  oppcmon  16398  funcoppc  16535  fulloppc  16582  fthoppc  16583  fthepi  16588  yonedalem22  16918  oppgplus  17779  oppglsm  18057  opprmul  18626  mamutpos  20264  mdettpos  20417  madutpos  20448  mdetpmtr2  29890