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Theorem ipdir 18434
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 2460 . . . . . 6  |-  ( x  e.  V  |->  ( x 
.,  C ) )  =  ( x  e.  V  |->  ( x  .,  C ) )
51, 2, 3, 4phllmhm 18427 . . . . 5  |-  ( ( W  e.  PreHil  /\  C  e.  V )  ->  (
x  e.  V  |->  ( x  .,  C ) )  e.  ( W LMHom 
(ringLMod `  F ) ) )
653ad2antr3 1158 . . . 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 17453 . . . 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 997 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
10 simpr2 998 . . 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 17618 . . . . 5  |-  ( +g  `  F )  =  ( +g  `  (ringLMod `  F
) )
1412, 13eqtri 2489 . . . 4  |-  .+^  =  ( +g  `  (ringLMod `  F
) )
153, 11, 14ghmlin 16060 . . 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 1223 . 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 18425 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
183, 11lmodvacl 17302 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .+  B )  e.  V )
1917, 18syl3an1 1256 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .+  B )  e.  V )
20193adant3r3 1202 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .+  B )  e.  V
)
21 oveq1 6282 . . . 4  |-  ( x  =  ( A  .+  B )  ->  (
x  .,  C )  =  ( ( A 
.+  B )  .,  C ) )
22 ovex 6300 . . . 4  |-  ( x 
.,  C )  e. 
_V
2321, 4, 22fvmpt3i 5945 . . 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 6282 . . . . 5  |-  ( x  =  A  ->  (
x  .,  C )  =  ( A  .,  C ) )
2625, 4, 22fvmpt3i 5945 . . . 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 6282 . . . . 5  |-  ( x  =  B  ->  (
x  .,  C )  =  ( B  .,  C ) )
2928, 4, 22fvmpt3i 5945 . . . 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 6293 . 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 2509 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 968    = wceq 1374    e. wcel 1762    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544  Scalarcsca 14547   .icip 14549    GrpHom cghm 16052   LModclmod 17288   LMHom clmhm 17441  ringLModcrglmod 17591   PreHilcphl 18419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-ndx 14482  df-slot 14483  df-sets 14485  df-plusg 14557  df-sca 14560  df-vsca 14561  df-ip 14562  df-mnd 15721  df-grp 15851  df-ghm 16053  df-lmod 17290  df-lmhm 17444  df-lvec 17525  df-sra 17594  df-rgmod 17595  df-phl 18421
This theorem is referenced by:  ipdi  18435  ip2di  18436  ipsubdir  18437  ocvlss  18463  lsmcss  18483  cphdir  21379
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