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Theorem ipass 16549
Description: Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
phlsrng.f  |-  F  =  (Scalar `  W )
phllmhm.h  |-  .,  =  ( .i `  W )
phllmhm.v  |-  V  =  ( Base `  W
)
ipdir.f  |-  K  =  ( Base `  F
)
ipass.s  |-  .x.  =  ( .s `  W )
ipass.p  |-  .X.  =  ( .r `  F )
Assertion
Ref Expression
ipass  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .x.  B )  .,  C )  =  ( A  .X.  ( B  .,  C ) ) )

Proof of Theorem ipass
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 phlsrng.f . . . . 5  |-  F  =  (Scalar `  W )
2 phllmhm.h . . . . 5  |-  .,  =  ( .i `  W )
3 phllmhm.v . . . . 5  |-  V  =  ( Base `  W
)
4 eqid 2283 . . . . 5  |-  ( x  e.  V  |->  ( x 
.,  C ) )  =  ( x  e.  V  |->  ( x  .,  C ) )
51, 2, 3, 4phllmhm 16536 . . . 4  |-  ( ( W  e.  PreHil  /\  C  e.  V )  ->  (
x  e.  V  |->  ( x  .,  C ) )  e.  ( W LMHom 
(ringLMod `  F ) ) )
653ad2antr3 1122 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  ( x  e.  V  |->  ( x 
.,  C ) )  e.  ( W LMHom  (ringLMod `  F ) ) )
7 simpr1 961 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  K )
8 simpr2 962 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  B  e.  V )
9 ipdir.f . . . 4  |-  K  =  ( Base `  F
)
10 ipass.s . . . 4  |-  .x.  =  ( .s `  W )
11 ipass.p . . . . 5  |-  .X.  =  ( .r `  F )
12 rlmvsca 15954 . . . . 5  |-  ( .r
`  F )  =  ( .s `  (ringLMod `  F ) )
1311, 12eqtri 2303 . . . 4  |-  .X.  =  ( .s `  (ringLMod `  F
) )
141, 9, 3, 10, 13lmhmlin 15792 . . 3  |-  ( ( ( x  e.  V  |->  ( x  .,  C
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  A  e.  K  /\  B  e.  V
)  ->  ( (
x  e.  V  |->  ( x  .,  C ) ) `  ( A 
.x.  B ) )  =  ( A  .X.  ( ( x  e.  V  |->  ( x  .,  C ) ) `  B ) ) )
156, 7, 8, 14syl3anc 1182 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
x  e.  V  |->  ( x  .,  C ) ) `  ( A 
.x.  B ) )  =  ( A  .X.  ( ( x  e.  V  |->  ( x  .,  C ) ) `  B ) ) )
16 phllmod 16534 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
1716adantr 451 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  LMod )
183, 1, 10, 9lmodvscl 15644 . . . 4  |-  ( ( W  e.  LMod  /\  A  e.  K  /\  B  e.  V )  ->  ( A  .x.  B )  e.  V )
1917, 7, 8, 18syl3anc 1182 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .x.  B )  e.  V
)
20 oveq1 5865 . . . 4  |-  ( x  =  ( A  .x.  B )  ->  (
x  .,  C )  =  ( ( A 
.x.  B )  .,  C ) )
21 ovex 5883 . . . 4  |-  ( x 
.,  C )  e. 
_V
2220, 4, 21fvmpt3i 5605 . . 3  |-  ( ( A  .x.  B )  e.  V  ->  (
( x  e.  V  |->  ( x  .,  C
) ) `  ( A  .x.  B ) )  =  ( ( A 
.x.  B )  .,  C ) )
2319, 22syl 15 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
x  e.  V  |->  ( x  .,  C ) ) `  ( A 
.x.  B ) )  =  ( ( A 
.x.  B )  .,  C ) )
24 oveq1 5865 . . . . 5  |-  ( x  =  B  ->  (
x  .,  C )  =  ( B  .,  C ) )
2524, 4, 21fvmpt3i 5605 . . . 4  |-  ( B  e.  V  ->  (
( x  e.  V  |->  ( x  .,  C
) ) `  B
)  =  ( B 
.,  C ) )
268, 25syl 15 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
x  e.  V  |->  ( x  .,  C ) ) `  B )  =  ( B  .,  C ) )
2726oveq2d 5874 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .X.  ( ( x  e.  V  |->  ( x  .,  C ) ) `  B ) )  =  ( A  .X.  ( B  .,  C ) ) )
2815, 23, 273eqtr3d 2323 1  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .x.  B )  .,  C )  =  ( A  .X.  ( B  .,  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   .icip 13213   LModclmod 15627   LMHom clmhm 15776  ringLModcrglmod 15922   PreHilcphl 16528
This theorem is referenced by:  ipassr  16550  ocvlss  16572  cphass  18646  ipcau2  18664  tchcphlem2  18666
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-sets 13154  df-vsca 13225  df-lmod 15629  df-lmhm 15779  df-lvec 15856  df-sra 15925  df-rgmod 15926  df-phl 16530
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