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Theorem ipcj 16538
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 2283 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 ipcj.i . . . . . 6  |-  .*  =  ( * r `  F )
6 eqid 2283 . . . . . 6  |-  ( 0g
`  F )  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 16532 . . . . 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 972 . . . 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 957 . . . . 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 2618 . . . 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 15 . . 3  |-  ( W  e.  PreHil  ->  A. x  e.  V  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) )
12 oveq1 5865 . . . . . 6  |-  ( x  =  A  ->  (
x  .,  y )  =  ( A  .,  y ) )
1312fveq2d 5529 . . . . 5  |-  ( x  =  A  ->  (  .*  `  ( x  .,  y ) )  =  (  .*  `  ( A  .,  y ) ) )
14 oveq2 5866 . . . . 5  |-  ( x  =  A  ->  (
y  .,  x )  =  ( y  .,  A ) )
1513, 14eqeq12d 2297 . . . 4  |-  ( x  =  A  ->  (
(  .*  `  (
x  .,  y )
)  =  ( y 
.,  x )  <->  (  .*  `  ( A  .,  y
) )  =  ( y  .,  A ) ) )
16 oveq2 5866 . . . . . 6  |-  ( y  =  B  ->  ( A  .,  y )  =  ( A  .,  B
) )
1716fveq2d 5529 . . . . 5  |-  ( y  =  B  ->  (  .*  `  ( A  .,  y ) )  =  (  .*  `  ( A  .,  B ) ) )
18 oveq1 5865 . . . . 5  |-  ( y  =  B  ->  (
y  .,  A )  =  ( B  .,  A ) )
1917, 18eqeq12d 2297 . . . 4  |-  ( y  =  B  ->  (
(  .*  `  ( A  .,  y ) )  =  ( y  .,  A )  <->  (  .*  `  ( A  .,  B
) )  =  ( B  .,  A ) ) )
2015, 19rspc2v 2890 . . 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 26 . 2  |-  ( W  e.  PreHil  ->  ( ( A  e.  V  /\  B  e.  V )  ->  (  .*  `  ( A  .,  B ) )  =  ( B  .,  A
) ) )
22213impib 1149 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148   * rcstv 13210  Scalarcsca 13211   .icip 13213   0gc0g 13400   *Ringcsr 15609   LMHom clmhm 15776   LVecclvec 15855  ringLModcrglmod 15922   PreHilcphl 16528
This theorem is referenced by:  iporthcom  16539  ip0r  16541  ipdi  16544  ipassr  16550  cphipcj  18635  tchcphlem3  18663  ipcau2  18664  tchcphlem1  18665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-iota 5219  df-fv 5263  df-ov 5861  df-phl 16530
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