Theorem hmop 28781

Description: Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
hmop	|- ( ( T e. HrmOp /\ A e. ~H /\ B e. ~H ) -> ( A .ih ( T ` B ) ) = ( ( T ` A ) .ih B ) )

Proof of Theorem hmop

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elhmop 28732	. . . 4 |- ( T e. HrmOp <-> ( T : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( T ` x ) .ih y ) ) )
2	1	simprbi 480	. . 3 |- ( T e. HrmOp -> A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( T ` x ) .ih y ) )
3	2	3ad2ant1 1082	. 2 |- ( ( T e. HrmOp /\ A e. ~H /\ B e. ~H ) -> A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( T ` x ) .ih y ) )
4		oveq1 6657	. . . . 5 |- ( x = A -> ( x .ih ( T ` y ) ) = ( A .ih ( T ` y ) ) )
5		fveq2 6191	. . . . . 6 |- ( x = A -> ( T ` x ) = ( T ` A ) )
6	5	oveq1d 6665	. . . . 5 |- ( x = A -> ( ( T ` x ) .ih y ) = ( ( T ` A ) .ih y ) )
7	4, 6	eqeq12d 2637	. . . 4 |- ( x = A -> ( ( x .ih ( T ` y ) ) = ( ( T ` x ) .ih y ) <-> ( A .ih ( T ` y ) ) = ( ( T ` A ) .ih y ) ) )
8		fveq2 6191	. . . . . 6 |- ( y = B -> ( T ` y ) = ( T ` B ) )
9	8	oveq2d 6666	. . . . 5 |- ( y = B -> ( A .ih ( T ` y ) ) = ( A .ih ( T ` B ) ) )
10		oveq2 6658	. . . . 5 |- ( y = B -> ( ( T ` A ) .ih y ) = ( ( T ` A ) .ih B ) )
11	9, 10	eqeq12d 2637	. . . 4 |- ( y = B -> ( ( A .ih ( T ` y ) ) = ( ( T ` A ) .ih y ) <-> ( A .ih ( T ` B ) ) = ( ( T ` A ) .ih B ) ) )
12	7, 11	rspc2v 3322	. . 3 |- ( ( A e. ~H /\ B e. ~H ) -> ( A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( T ` x ) .ih y ) -> ( A .ih ( T ` B ) ) = ( ( T ` A ) .ih B ) ) )
13	12	3adant1 1079	. 2 |- ( ( T e. HrmOp /\ A e. ~H /\ B e. ~H ) -> ( A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( T ` x ) .ih y ) -> ( A .ih ( T ` B ) ) = ( ( T ` A ) .ih B ) ) )
14	3, 13	mpd 15	1 |- ( ( T e. HrmOp /\ A e. ~H /\ B e. ~H ) -> ( A .ih ( T ` B ) ) = ( ( T ` A ) .ih B ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   ~Hchil 27776   .ih csp 27779   HrmOpcho 27807

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  ax-hilex 27856

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-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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-hmop 28703

This theorem is referenced by:  hmopre  28782  hmopadj  28798  hmoplin  28801  eighmre  28822  eighmorth  28823  hmopbdoptHIL  28847  hmops  28879  hmopm  28880  hmopco  28882  leopsq  28988  hmopidmpji  29011