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Theorem ipass 16551
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 2285 . . . . 5  |-  ( x  e.  V  |->  ( x 
.,  C ) )  =  ( x  e.  V  |->  ( x  .,  C ) )
51, 2, 3, 4phllmhm 16538 . . . 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 15956 . . . . 5  |-  ( .r
`  F )  =  ( .s `  (ringLMod `  F ) )
1311, 12eqtri 2305 . . . 4  |-  .X.  =  ( .s `  (ringLMod `  F
) )
141, 9, 3, 10, 13lmhmlin 15794 . . 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 16536 . . . . 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 15646 . . . 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 5867 . . . 4  |-  ( x  =  ( A  .x.  B )  ->  (
x  .,  C )  =  ( ( A 
.x.  B )  .,  C ) )
21 ovex 5885 . . . 4  |-  ( x 
.,  C )  e. 
_V
2220, 4, 21fvmpt3i 5607 . . 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 5867 . . . . 5  |-  ( x  =  B  ->  (
x  .,  C )  =  ( B  .,  C ) )
2524, 4, 21fvmpt3i 5607 . . . 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 5876 . 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 2325 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 1625    e. wcel 1686    e. cmpt 4079   ` cfv 5257  (class class class)co 5860   Basecbs 13150   .rcmulr 13211  Scalarcsca 13213   .scvsca 13214   .icip 13215   LModclmod 15629   LMHom clmhm 15778  ringLModcrglmod 15924   PreHilcphl 16530
This theorem is referenced by:  ipassr  16552  ocvlss  16574  cphass  18648  ipcau2  18666  tchcphlem2  18668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-i2m1 8807  ax-1ne0 8808  ax-rrecex 8811  ax-cnre 8812
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-recs 6390  df-rdg 6425  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-ndx 13153  df-slot 13154  df-sets 13156  df-vsca 13227  df-lmod 15631  df-lmhm 15781  df-lvec 15858  df-sra 15927  df-rgmod 15928  df-phl 16532
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