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Theorem ipdir 16793
Description: Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (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.g  |-  .+  =  ( +g  `  W )
ipdir.p  |-  .+^  =  ( +g  `  F )
Assertion
Ref Expression
ipdir  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .+  B )  .,  C )  =  ( ( A  .,  C
)  .+^  ( B  .,  C ) ) )

Proof of Theorem ipdir
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 phlsrng.f . . . . . 6  |-  F  =  (Scalar `  W )
2 phllmhm.h . . . . . 6  |-  .,  =  ( .i `  W )
3 phllmhm.v . . . . . 6  |-  V  =  ( Base `  W
)
4 eqid 2387 . . . . . 6  |-  ( x  e.  V  |->  ( x 
.,  C ) )  =  ( x  e.  V  |->  ( x  .,  C ) )
51, 2, 3, 4phllmhm 16786 . . . . 5  |-  ( ( W  e.  PreHil  /\  C  e.  V )  ->  (
x  e.  V  |->  ( x  .,  C ) )  e.  ( W LMHom 
(ringLMod `  F ) ) )
653ad2antr3 1124 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( x  e.  V  |->  ( x 
.,  C ) )  e.  ( W LMHom  (ringLMod `  F ) ) )
7 lmghm 16034 . . . 4  |-  ( ( x  e.  V  |->  ( x  .,  C ) )  e.  ( W LMHom 
(ringLMod `  F ) )  ->  ( x  e.  V  |->  ( x  .,  C ) )  e.  ( W  GrpHom  (ringLMod `  F
) ) )
86, 7syl 16 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( x  e.  V  |->  ( x 
.,  C ) )  e.  ( W  GrpHom  (ringLMod `  F ) ) )
9 simpr1 963 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
10 simpr2 964 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  B  e.  V )
11 ipdir.g . . . 4  |-  .+  =  ( +g  `  W )
12 ipdir.p . . . . 5  |-  .+^  =  ( +g  `  F )
13 rlmplusg 16195 . . . . 5  |-  ( +g  `  F )  =  ( +g  `  (ringLMod `  F
) )
1412, 13eqtri 2407 . . . 4  |-  .+^  =  ( +g  `  (ringLMod `  F
) )
153, 11, 14ghmlin 14938 . . 3  |-  ( ( ( x  e.  V  |->  ( x  .,  C
) )  e.  ( W  GrpHom  (ringLMod `  F )
)  /\  A  e.  V  /\  B  e.  V
)  ->  ( (
x  e.  V  |->  ( x  .,  C ) ) `  ( A 
.+  B ) )  =  ( ( ( x  e.  V  |->  ( x  .,  C ) ) `  A ) 
.+^  ( ( x  e.  V  |->  ( x 
.,  C ) ) `
 B ) ) )
168, 9, 10, 15syl3anc 1184 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
x  e.  V  |->  ( x  .,  C ) ) `  ( A 
.+  B ) )  =  ( ( ( x  e.  V  |->  ( x  .,  C ) ) `  A ) 
.+^  ( ( x  e.  V  |->  ( x 
.,  C ) ) `
 B ) ) )
17 phllmod 16784 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
183, 11lmodvacl 15891 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .+  B )  e.  V )
1917, 18syl3an1 1217 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .+  B )  e.  V )
20193adant3r3 1164 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .+  B )  e.  V
)
21 oveq1 6027 . . . 4  |-  ( x  =  ( A  .+  B )  ->  (
x  .,  C )  =  ( ( A 
.+  B )  .,  C ) )
22 ovex 6045 . . . 4  |-  ( x 
.,  C )  e. 
_V
2321, 4, 22fvmpt3i 5748 . . 3  |-  ( ( A  .+  B )  e.  V  ->  (
( x  e.  V  |->  ( x  .,  C
) ) `  ( A  .+  B ) )  =  ( ( A 
.+  B )  .,  C ) )
2420, 23syl 16 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
x  e.  V  |->  ( x  .,  C ) ) `  ( A 
.+  B ) )  =  ( ( A 
.+  B )  .,  C ) )
25 oveq1 6027 . . . . 5  |-  ( x  =  A  ->  (
x  .,  C )  =  ( A  .,  C ) )
2625, 4, 22fvmpt3i 5748 . . . 4  |-  ( A  e.  V  ->  (
( x  e.  V  |->  ( x  .,  C
) ) `  A
)  =  ( A 
.,  C ) )
279, 26syl 16 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
x  e.  V  |->  ( x  .,  C ) ) `  A )  =  ( A  .,  C ) )
28 oveq1 6027 . . . . 5  |-  ( x  =  B  ->  (
x  .,  C )  =  ( B  .,  C ) )
2928, 4, 22fvmpt3i 5748 . . . 4  |-  ( B  e.  V  ->  (
( x  e.  V  |->  ( x  .,  C
) ) `  B
)  =  ( B 
.,  C ) )
3010, 29syl 16 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
x  e.  V  |->  ( x  .,  C ) ) `  B )  =  ( B  .,  C ) )
3127, 30oveq12d 6038 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( x  e.  V  |->  ( x  .,  C
) ) `  A
)  .+^  ( ( x  e.  V  |->  ( x 
.,  C ) ) `
 B ) )  =  ( ( A 
.,  C )  .+^  ( B  .,  C ) ) )
3216, 24, 313eqtr3d 2427 1  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .+  B )  .,  C )  =  ( ( A  .,  C
)  .+^  ( B  .,  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456  Scalarcsca 13459   .icip 13461    GrpHom cghm 14930   LModclmod 15877   LMHom clmhm 16022  ringLModcrglmod 16168   PreHilcphl 16778
This theorem is referenced by:  ipdi  16794  ip2di  16795  ipsubdir  16796  ocvlss  16822  lsmcss  16842  cphdir  19038
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-ndx 13399  df-slot 13400  df-sets 13402  df-plusg 13469  df-sca 13472  df-vsca 13473  df-mnd 14617  df-grp 14739  df-ghm 14931  df-lmod 15879  df-lmhm 16025  df-lvec 16102  df-sra 16171  df-rgmod 16172  df-phl 16780
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