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Theorem ipass 16565
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 2296 . . . . 5  |-  ( x  e.  V  |->  ( x 
.,  C ) )  =  ( x  e.  V  |->  ( x  .,  C ) )
51, 2, 3, 4phllmhm 16552 . . . 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 15970 . . . . 5  |-  ( .r
`  F )  =  ( .s `  (ringLMod `  F ) )
1311, 12eqtri 2316 . . . 4  |-  .X.  =  ( .s `  (ringLMod `  F
) )
141, 9, 3, 10, 13lmhmlin 15808 . . 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 16550 . . . . 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 15660 . . . 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 5881 . . . 4  |-  ( x  =  ( A  .x.  B )  ->  (
x  .,  C )  =  ( ( A 
.x.  B )  .,  C ) )
21 ovex 5899 . . . 4  |-  ( x 
.,  C )  e. 
_V
2220, 4, 21fvmpt3i 5621 . . 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 5881 . . . . 5  |-  ( x  =  B  ->  (
x  .,  C )  =  ( B  .,  C ) )
2524, 4, 21fvmpt3i 5621 . . . 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 5890 . 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 2336 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 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   .icip 13229   LModclmod 15643   LMHom clmhm 15792  ringLModcrglmod 15938   PreHilcphl 16544
This theorem is referenced by:  ipassr  16566  ocvlss  16588  cphass  18662  ipcau2  18680  tchcphlem2  18682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-sets 13170  df-vsca 13241  df-lmod 15645  df-lmhm 15795  df-lvec 15872  df-sra 15941  df-rgmod 15942  df-phl 16546
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