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Theorem ipdir 16470
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
StepHypRef Expression
1 phlsrng.f . . . . . 6  |-  F  =  (Scalar `  W )
2 phllmhm.h . . . . . 6  |-  .,  =  ( .i `  W )
3 phllmhm.v . . . . . 6  |-  V  =  ( Base `  W
)
4 eqid 2256 . . . . . 6  |-  ( x  e.  V  |->  ( x 
.,  C ) )  =  ( x  e.  V  |->  ( x  .,  C ) )
51, 2, 3, 4phllmhm 16463 . . . . 5  |-  ( ( W  e.  PreHil  /\  C  e.  V )  ->  (
x  e.  V  |->  ( x  .,  C ) )  e.  ( W LMHom 
(ringLMod `  F ) ) )
653ad2antr3 1127 . . . 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 15715 . . . 4  |-  ( ( x  e.  V  |->  ( x  .,  C ) )  e.  ( W LMHom 
(ringLMod `  F ) )  ->  ( x  e.  V  |->  ( x  .,  C ) )  e.  ( W  GrpHom  (ringLMod `  F
) ) )
86, 7syl 17 . . 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 966 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
10 simpr2 967 . . 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 15876 . . . . 5  |-  ( +g  `  F )  =  ( +g  `  (ringLMod `  F
) )
1412, 13eqtri 2276 . . . 4  |-  .+^  =  ( +g  `  (ringLMod `  F
) )
153, 11, 14ghmlin 14615 . . 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 1187 . 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 16461 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
183, 11lmodvacl 15568 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .+  B )  e.  V )
1917, 18syl3an1 1220 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .+  B )  e.  V )
20193adant3r3 1167 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .+  B )  e.  V
)
21 oveq1 5764 . . . 4  |-  ( x  =  ( A  .+  B )  ->  (
x  .,  C )  =  ( ( A 
.+  B )  .,  C ) )
22 ovex 5782 . . . 4  |-  ( x 
.,  C )  e. 
_V
2321, 4, 22fvmpt3i 5504 . . 3  |-  ( ( A  .+  B )  e.  V  ->  (
( x  e.  V  |->  ( x  .,  C
) ) `  ( A  .+  B ) )  =  ( ( A 
.+  B )  .,  C ) )
2420, 23syl 17 . 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 5764 . . . . 5  |-  ( x  =  A  ->  (
x  .,  C )  =  ( A  .,  C ) )
2625, 4, 22fvmpt3i 5504 . . . 4  |-  ( A  e.  V  ->  (
( x  e.  V  |->  ( x  .,  C
) ) `  A
)  =  ( A 
.,  C ) )
279, 26syl 17 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
x  e.  V  |->  ( x  .,  C ) ) `  A )  =  ( A  .,  C ) )
28 oveq1 5764 . . . . 5  |-  ( x  =  B  ->  (
x  .,  C )  =  ( B  .,  C ) )
2928, 4, 22fvmpt3i 5504 . . . 4  |-  ( B  e.  V  ->  (
( x  e.  V  |->  ( x  .,  C
) ) `  B
)  =  ( B 
.,  C ) )
3010, 29syl 17 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
x  e.  V  |->  ( x  .,  C ) ) `  B )  =  ( B  .,  C ) )
3127, 30oveq12d 5775 . 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 2296 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    e. cmpt 4017   ` cfv 4638  (class class class)co 5757   Basecbs 13075   +g cplusg 13135  Scalarcsca 13138   .icip 13140    GrpHom cghm 14607   LModclmod 15554   LMHom clmhm 15703  ringLModcrglmod 15849   PreHilcphl 16455
This theorem is referenced by:  ipdi  16471  ip2di  16472  ipsubdir  16473  ocvlss  16499  lsmcss  16519  cphdir  18567
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-ndx 13078  df-slot 13079  df-sets 13081  df-plusg 13148  df-sca 13151  df-vsca 13152  df-mnd 14294  df-grp 14416  df-ghm 14608  df-lmod 15556  df-lmhm 15706  df-lvec 15783  df-sra 15852  df-rgmod 15853  df-phl 16457
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