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Theorem ipass 16835
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 2408 . . . . 5  |-  ( x  e.  V  |->  ( x 
.,  C ) )  =  ( x  e.  V  |->  ( x  .,  C ) )
51, 2, 3, 4phllmhm 16822 . . . 4  |-  ( ( W  e.  PreHil  /\  C  e.  V )  ->  (
x  e.  V  |->  ( x  .,  C ) )  e.  ( W LMHom 
(ringLMod `  F ) ) )
653ad2antr3 1124 . . 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 963 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  K )
8 simpr2 964 . . 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 16232 . . . . 5  |-  ( .r
`  F )  =  ( .s `  (ringLMod `  F ) )
1311, 12eqtri 2428 . . . 4  |-  .X.  =  ( .s `  (ringLMod `  F
) )
141, 9, 3, 10, 13lmhmlin 16070 . . 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 1184 . 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 16820 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
1716adantr 452 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  LMod )
183, 1, 10, 9lmodvscl 15926 . . . 4  |-  ( ( W  e.  LMod  /\  A  e.  K  /\  B  e.  V )  ->  ( A  .x.  B )  e.  V )
1917, 7, 8, 18syl3anc 1184 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .x.  B )  e.  V
)
20 oveq1 6051 . . . 4  |-  ( x  =  ( A  .x.  B )  ->  (
x  .,  C )  =  ( ( A 
.x.  B )  .,  C ) )
21 ovex 6069 . . . 4  |-  ( x 
.,  C )  e. 
_V
2220, 4, 21fvmpt3i 5772 . . 3  |-  ( ( A  .x.  B )  e.  V  ->  (
( x  e.  V  |->  ( x  .,  C
) ) `  ( A  .x.  B ) )  =  ( ( A 
.x.  B )  .,  C ) )
2319, 22syl 16 . 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 6051 . . . . 5  |-  ( x  =  B  ->  (
x  .,  C )  =  ( B  .,  C ) )
2524, 4, 21fvmpt3i 5772 . . . 4  |-  ( B  e.  V  ->  (
( x  e.  V  |->  ( x  .,  C
) ) `  B
)  =  ( B 
.,  C ) )
268, 25syl 16 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
x  e.  V  |->  ( x  .,  C ) ) `  B )  =  ( B  .,  C ) )
2726oveq2d 6060 . 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 2448 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    e. cmpt 4230   ` cfv 5417  (class class class)co 6044   Basecbs 13428   .rcmulr 13489  Scalarcsca 13491   .scvsca 13492   .icip 13493   LModclmod 15909   LMHom clmhm 16054  ringLModcrglmod 16200   PreHilcphl 16814
This theorem is referenced by:  ipassr  16836  ocvlss  16858  cphass  19130  ipcau2  19148  tchcphlem2  19150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-i2m1 9018  ax-1ne0 9019  ax-rrecex 9022  ax-cnre 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-recs 6596  df-rdg 6631  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-ndx 13431  df-slot 13432  df-sets 13434  df-vsca 13505  df-lmod 15911  df-lmhm 16057  df-lvec 16134  df-sra 16203  df-rgmod 16204  df-phl 16816
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