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Theorem ipcj 18181
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 2451 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 ipcj.i . . . . . 6  |-  .*  =  ( *r `  F )
6 eqid 2451 . . . . . 6  |-  ( 0g
`  F )  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 18175 . . . . 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 1005 . . . 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 990 . . . . 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 2814 . . . 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 6200 . . . . . 6  |-  ( x  =  A  ->  (
x  .,  y )  =  ( A  .,  y ) )
1312fveq2d 5796 . . . . 5  |-  ( x  =  A  ->  (  .*  `  ( x  .,  y ) )  =  (  .*  `  ( A  .,  y ) ) )
14 oveq2 6201 . . . . 5  |-  ( x  =  A  ->  (
y  .,  x )  =  ( y  .,  A ) )
1513, 14eqeq12d 2473 . . . 4  |-  ( x  =  A  ->  (
(  .*  `  (
x  .,  y )
)  =  ( y 
.,  x )  <->  (  .*  `  ( A  .,  y
) )  =  ( y  .,  A ) ) )
16 oveq2 6201 . . . . . 6  |-  ( y  =  B  ->  ( A  .,  y )  =  ( A  .,  B
) )
1716fveq2d 5796 . . . . 5  |-  ( y  =  B  ->  (  .*  `  ( A  .,  y ) )  =  (  .*  `  ( A  .,  B ) ) )
18 oveq1 6200 . . . . 5  |-  ( y  =  B  ->  (
y  .,  A )  =  ( B  .,  A ) )
1917, 18eqeq12d 2473 . . . 4  |-  ( y  =  B  ->  (
(  .*  `  ( A  .,  y ) )  =  ( y  .,  A )  <->  (  .*  `  ( A  .,  B
) )  =  ( B  .,  A ) ) )
2015, 19rspc2v 3179 . . 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 1186 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 965    = wceq 1370    e. wcel 1758   A.wral 2795    |-> cmpt 4451   ` cfv 5519  (class class class)co 6193   Basecbs 14285   *rcstv 14351  Scalarcsca 14352   .icip 14354   0gc0g 14489   *Ringcsr 17044   LMHom clmhm 17215   LVecclvec 17298  ringLModcrglmod 17365   PreHilcphl 18171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4522
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-iota 5482  df-fv 5527  df-ov 6196  df-phl 18173
This theorem is referenced by:  iporthcom  18182  ip0r  18184  ipdi  18187  ipassr  18193  cphipcj  20843  tchcphlem3  20873  ipcau2  20874  tchcphlem1  20875
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