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Theorem ipdir 18043
Description: Distributive law for inner product (right-distributivity). 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 2438 . . . . . 6  |-  ( x  e.  V  |->  ( x 
.,  C ) )  =  ( x  e.  V  |->  ( x  .,  C ) )
51, 2, 3, 4phllmhm 18036 . . . . 5  |-  ( ( W  e.  PreHil  /\  C  e.  V )  ->  (
x  e.  V  |->  ( x  .,  C ) )  e.  ( W LMHom 
(ringLMod `  F ) ) )
653ad2antr3 1155 . . . 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 17089 . . . 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 994 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
10 simpr2 995 . . 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 17254 . . . . 5  |-  ( +g  `  F )  =  ( +g  `  (ringLMod `  F
) )
1412, 13eqtri 2458 . . . 4  |-  .+^  =  ( +g  `  (ringLMod `  F
) )
153, 11, 14ghmlin 15743 . . 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 1218 . 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 18034 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
183, 11lmodvacl 16940 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .+  B )  e.  V )
1917, 18syl3an1 1251 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .+  B )  e.  V )
20193adant3r3 1198 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .+  B )  e.  V
)
21 oveq1 6093 . . . 4  |-  ( x  =  ( A  .+  B )  ->  (
x  .,  C )  =  ( ( A 
.+  B )  .,  C ) )
22 ovex 6111 . . . 4  |-  ( x 
.,  C )  e. 
_V
2321, 4, 22fvmpt3i 5773 . . 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 6093 . . . . 5  |-  ( x  =  A  ->  (
x  .,  C )  =  ( A  .,  C ) )
2625, 4, 22fvmpt3i 5773 . . . 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 6093 . . . . 5  |-  ( x  =  B  ->  (
x  .,  C )  =  ( B  .,  C ) )
2928, 4, 22fvmpt3i 5773 . . . 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 6104 . 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 2478 1  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  .+  B )  .,  C )  =  ( ( A  .,  C
)  .+^  ( B  .,  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230  Scalarcsca 14233   .icip 14235    GrpHom cghm 15735   LModclmod 16926   LMHom clmhm 17077  ringLModcrglmod 17227   PreHilcphl 18028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-ndx 14169  df-slot 14170  df-sets 14172  df-plusg 14243  df-sca 14246  df-vsca 14247  df-ip 14248  df-mnd 15407  df-grp 15536  df-ghm 15736  df-lmod 16928  df-lmhm 17080  df-lvec 17161  df-sra 17230  df-rgmod 17231  df-phl 18030
This theorem is referenced by:  ipdi  18044  ip2di  18045  ipsubdir  18046  ocvlss  18072  lsmcss  18092  cphdir  20698
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