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Theorem ipdir 16506
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 2258 . . . . . 6  |-  ( x  e.  V  |->  ( x 
.,  C ) )  =  ( x  e.  V  |->  ( x  .,  C ) )
51, 2, 3, 4phllmhm 16499 . . . . 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 15751 . . . 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 15912 . . . . 5  |-  ( +g  `  F )  =  ( +g  `  (ringLMod `  F
) )
1412, 13eqtri 2278 . . . 4  |-  .+^  =  ( +g  `  (ringLMod `  F
) )
153, 11, 14ghmlin 14651 . . 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 16497 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
183, 11lmodvacl 15604 . . . . 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 5799 . . . 4  |-  ( x  =  ( A  .+  B )  ->  (
x  .,  C )  =  ( ( A 
.+  B )  .,  C ) )
22 ovex 5817 . . . 4  |-  ( x 
.,  C )  e. 
_V
2321, 4, 22fvmpt3i 5539 . . 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 5799 . . . . 5  |-  ( x  =  A  ->  (
x  .,  C )  =  ( A  .,  C ) )
2625, 4, 22fvmpt3i 5539 . . . 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 5799 . . . . 5  |-  ( x  =  B  ->  (
x  .,  C )  =  ( B  .,  C ) )
2928, 4, 22fvmpt3i 5539 . . . 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 5810 . 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 2298 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 4051   ` cfv 4673  (class class class)co 5792   Basecbs 13111   +g cplusg 13171  Scalarcsca 13174   .icip 13176    GrpHom cghm 14643   LModclmod 15590   LMHom clmhm 15739  ringLModcrglmod 15885   PreHilcphl 16491
This theorem is referenced by:  ipdi  16507  ip2di  16508  ipsubdir  16509  ocvlss  16535  lsmcss  16555  cphdir  18603
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-ndx 13114  df-slot 13115  df-sets 13117  df-plusg 13184  df-sca 13187  df-vsca 13188  df-mnd 14330  df-grp 14452  df-ghm 14644  df-lmod 15592  df-lmhm 15742  df-lvec 15819  df-sra 15888  df-rgmod 15889  df-phl 16493
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