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Theorem ipcj 18542
Description: Conjugate of an inner product in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
phlsrng.f  |-  F  =  (Scalar `  W )
phllmhm.h  |-  .,  =  ( .i `  W )
phllmhm.v  |-  V  =  ( Base `  W
)
ipcj.i  |-  .*  =  ( *r `  F )
Assertion
Ref Expression
ipcj  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (  .*  `  ( A  .,  B ) )  =  ( B  .,  A
) )

Proof of Theorem ipcj
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 phllmhm.v . . . . . 6  |-  V  =  ( Base `  W
)
2 phlsrng.f . . . . . 6  |-  F  =  (Scalar `  W )
3 phllmhm.h . . . . . 6  |-  .,  =  ( .i `  W )
4 eqid 2443 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 ipcj.i . . . . . 6  |-  .*  =  ( *r `  F )
6 eqid 2443 . . . . . 6  |-  ( 0g
`  F )  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 18536 . . . . 5  |-  ( W  e.  PreHil 
<->  ( W  e.  LVec  /\  F  e.  *Ring  /\  A. x  e.  V  (
( y  e.  V  |->  ( y  .,  x
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  ( (
x  .,  x )  =  ( 0g `  F )  ->  x  =  ( 0g `  W ) )  /\  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) ) ) )
87simp3bi 1014 . . . 4  |-  ( W  e.  PreHil  ->  A. x  e.  V  ( ( y  e.  V  |->  ( y  .,  x ) )  e.  ( W LMHom  (ringLMod `  F
) )  /\  (
( x  .,  x
)  =  ( 0g
`  F )  ->  x  =  ( 0g `  W ) )  /\  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) ) )
9 simp3 999 . . . . 5  |-  ( ( ( y  e.  V  |->  ( y  .,  x
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  ( (
x  .,  x )  =  ( 0g `  F )  ->  x  =  ( 0g `  W ) )  /\  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) )  ->  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) )
109ralimi 2836 . . . 4  |-  ( A. x  e.  V  (
( y  e.  V  |->  ( y  .,  x
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  ( (
x  .,  x )  =  ( 0g `  F )  ->  x  =  ( 0g `  W ) )  /\  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) )  ->  A. x  e.  V  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) )
118, 10syl 16 . . 3  |-  ( W  e.  PreHil  ->  A. x  e.  V  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) )
12 oveq1 6288 . . . . . 6  |-  ( x  =  A  ->  (
x  .,  y )  =  ( A  .,  y ) )
1312fveq2d 5860 . . . . 5  |-  ( x  =  A  ->  (  .*  `  ( x  .,  y ) )  =  (  .*  `  ( A  .,  y ) ) )
14 oveq2 6289 . . . . 5  |-  ( x  =  A  ->  (
y  .,  x )  =  ( y  .,  A ) )
1513, 14eqeq12d 2465 . . . 4  |-  ( x  =  A  ->  (
(  .*  `  (
x  .,  y )
)  =  ( y 
.,  x )  <->  (  .*  `  ( A  .,  y
) )  =  ( y  .,  A ) ) )
16 oveq2 6289 . . . . . 6  |-  ( y  =  B  ->  ( A  .,  y )  =  ( A  .,  B
) )
1716fveq2d 5860 . . . . 5  |-  ( y  =  B  ->  (  .*  `  ( A  .,  y ) )  =  (  .*  `  ( A  .,  B ) ) )
18 oveq1 6288 . . . . 5  |-  ( y  =  B  ->  (
y  .,  A )  =  ( B  .,  A ) )
1917, 18eqeq12d 2465 . . . 4  |-  ( y  =  B  ->  (
(  .*  `  ( A  .,  y ) )  =  ( y  .,  A )  <->  (  .*  `  ( A  .,  B
) )  =  ( B  .,  A ) ) )
2015, 19rspc2v 3205 . . 3  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( A. x  e.  V  A. y  e.  V  (  .*  `  ( x  .,  y ) )  =  ( y 
.,  x )  -> 
(  .*  `  ( A  .,  B ) )  =  ( B  .,  A ) ) )
2111, 20syl5com 30 . 2  |-  ( W  e.  PreHil  ->  ( ( A  e.  V  /\  B  e.  V )  ->  (  .*  `  ( A  .,  B ) )  =  ( B  .,  A
) ) )
22213impib 1195 1  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (  .*  `  ( A  .,  B ) )  =  ( B  .,  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793    |-> cmpt 4495   ` cfv 5578  (class class class)co 6281   Basecbs 14509   *rcstv 14576  Scalarcsca 14577   .icip 14579   0gc0g 14714   *Ringcsr 17367   LMHom clmhm 17539   LVecclvec 17622  ringLModcrglmod 17689   PreHilcphl 18532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-nul 4566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-iota 5541  df-fv 5586  df-ov 6284  df-phl 18534
This theorem is referenced by:  iporthcom  18543  ip0r  18545  ipdi  18548  ipassr  18554  cphipcj  21519  tchcphlem3  21549  ipcau2  21550  tchcphlem1  21551
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