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Theorem ipcj 16855
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 2435 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 ipcj.i . . . . . 6  |-  .*  =  ( * r `  F )
6 eqid 2435 . . . . . 6  |-  ( 0g
`  F )  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 16849 . . . . 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 974 . . . 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 959 . . . . 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 2773 . . . 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 6080 . . . . . 6  |-  ( x  =  A  ->  (
x  .,  y )  =  ( A  .,  y ) )
1312fveq2d 5724 . . . . 5  |-  ( x  =  A  ->  (  .*  `  ( x  .,  y ) )  =  (  .*  `  ( A  .,  y ) ) )
14 oveq2 6081 . . . . 5  |-  ( x  =  A  ->  (
y  .,  x )  =  ( y  .,  A ) )
1513, 14eqeq12d 2449 . . . 4  |-  ( x  =  A  ->  (
(  .*  `  (
x  .,  y )
)  =  ( y 
.,  x )  <->  (  .*  `  ( A  .,  y
) )  =  ( y  .,  A ) ) )
16 oveq2 6081 . . . . . 6  |-  ( y  =  B  ->  ( A  .,  y )  =  ( A  .,  B
) )
1716fveq2d 5724 . . . . 5  |-  ( y  =  B  ->  (  .*  `  ( A  .,  y ) )  =  (  .*  `  ( A  .,  B ) ) )
18 oveq1 6080 . . . . 5  |-  ( y  =  B  ->  (
y  .,  A )  =  ( B  .,  A ) )
1917, 18eqeq12d 2449 . . . 4  |-  ( y  =  B  ->  (
(  .*  `  ( A  .,  y ) )  =  ( y  .,  A )  <->  (  .*  `  ( A  .,  B
) )  =  ( B  .,  A ) ) )
2015, 19rspc2v 3050 . . 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 28 . 2  |-  ( W  e.  PreHil  ->  ( ( A  e.  V  /\  B  e.  V )  ->  (  .*  `  ( A  .,  B ) )  =  ( B  .,  A
) ) )
22213impib 1151 1  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (  .*  `  ( A  .,  B ) )  =  ( B  .,  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   Basecbs 13459   * rcstv 13521  Scalarcsca 13522   .icip 13524   0gc0g 13713   *Ringcsr 15922   LMHom clmhm 16085   LVecclvec 16164  ringLModcrglmod 16231   PreHilcphl 16845
This theorem is referenced by:  iporthcom  16856  ip0r  16858  ipdi  16861  ipassr  16867  cphipcj  19152  tchcphlem3  19180  ipcau2  19181  tchcphlem1  19182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-iota 5410  df-fv 5454  df-ov 6076  df-phl 16847
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