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Theorem ipdi 18802
Description: Distributive law for inner product (left-distributivity). (Contributed by NM, 20-Nov-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
ipdi  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  ( B  .+  C
) )  =  ( ( A  .,  B
)  .+^  ( A  .,  C ) ) )

Proof of Theorem ipdi
StepHypRef Expression
1 simpl 457 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  PreHil )
2 simpr2 1003 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  B  e.  V )
3 simpr3 1004 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  C  e.  V )
4 simpr1 1002 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  A  e.  V )
5 phlsrng.f . . . . . 6  |-  F  =  (Scalar `  W )
6 phllmhm.h . . . . . 6  |-  .,  =  ( .i `  W )
7 phllmhm.v . . . . . 6  |-  V  =  ( Base `  W
)
8 ipdir.g . . . . . 6  |-  .+  =  ( +g  `  W )
9 ipdir.p . . . . . 6  |-  .+^  =  ( +g  `  F )
105, 6, 7, 8, 9ipdir 18801 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( B  e.  V  /\  C  e.  V  /\  A  e.  V )
)  ->  ( ( B  .+  C )  .,  A )  =  ( ( B  .,  A
)  .+^  ( C  .,  A ) ) )
111, 2, 3, 4, 10syl13anc 1230 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( B  .+  C )  .,  A )  =  ( ( B  .,  A
)  .+^  ( C  .,  A ) ) )
1211fveq2d 5876 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
*r `  F
) `  ( ( B  .+  C )  .,  A ) )  =  ( ( *r `  F ) `  ( ( B  .,  A )  .+^  ( C 
.,  A ) ) ) )
135phlsrng 18793 . . . . 5  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
1413adantr 465 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  F  e.  *Ring
)
15 eqid 2457 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
165, 6, 7, 15ipcl 18795 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  ( B  .,  A )  e.  ( Base `  F
) )
171, 2, 4, 16syl3anc 1228 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( B  .,  A )  e.  (
Base `  F )
)
185, 6, 7, 15ipcl 18795 . . . . 5  |-  ( ( W  e.  PreHil  /\  C  e.  V  /\  A  e.  V )  ->  ( C  .,  A )  e.  ( Base `  F
) )
191, 3, 4, 18syl3anc 1228 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( C  .,  A )  e.  (
Base `  F )
)
20 eqid 2457 . . . . 5  |-  ( *r `  F )  =  ( *r `  F )
2120, 15, 9srngadd 17633 . . . 4  |-  ( ( F  e.  *Ring  /\  ( B  .,  A )  e.  ( Base `  F
)  /\  ( C  .,  A )  e.  (
Base `  F )
)  ->  ( (
*r `  F
) `  ( ( B  .,  A )  .+^  ( C  .,  A ) ) )  =  ( ( ( *r `  F ) `  ( B  .,  A ) )  .+^  ( (
*r `  F
) `  ( C  .,  A ) ) ) )
2214, 17, 19, 21syl3anc 1228 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
*r `  F
) `  ( ( B  .,  A )  .+^  ( C  .,  A ) ) )  =  ( ( ( *r `  F ) `  ( B  .,  A ) )  .+^  ( (
*r `  F
) `  ( C  .,  A ) ) ) )
2312, 22eqtrd 2498 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
*r `  F
) `  ( ( B  .+  C )  .,  A ) )  =  ( ( ( *r `  F ) `
 ( B  .,  A ) )  .+^  ( ( *r `  F ) `  ( C  .,  A ) ) ) )
24 phllmod 18792 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
2524adantr 465 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  LMod )
267, 8lmodvacl 17653 . . . 4  |-  ( ( W  e.  LMod  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .+  C )  e.  V )
2725, 2, 3, 26syl3anc 1228 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( B  .+  C )  e.  V
)
285, 6, 7, 20ipcj 18796 . . 3  |-  ( ( W  e.  PreHil  /\  ( B  .+  C )  e.  V  /\  A  e.  V )  ->  (
( *r `  F ) `  (
( B  .+  C
)  .,  A )
)  =  ( A 
.,  ( B  .+  C ) ) )
291, 27, 4, 28syl3anc 1228 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
*r `  F
) `  ( ( B  .+  C )  .,  A ) )  =  ( A  .,  ( B  .+  C ) ) )
305, 6, 7, 20ipcj 18796 . . . 4  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  (
( *r `  F ) `  ( B  .,  A ) )  =  ( A  .,  B ) )
311, 2, 4, 30syl3anc 1228 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
*r `  F
) `  ( B  .,  A ) )  =  ( A  .,  B
) )
325, 6, 7, 20ipcj 18796 . . . 4  |-  ( ( W  e.  PreHil  /\  C  e.  V  /\  A  e.  V )  ->  (
( *r `  F ) `  ( C  .,  A ) )  =  ( A  .,  C ) )
331, 3, 4, 32syl3anc 1228 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
*r `  F
) `  ( C  .,  A ) )  =  ( A  .,  C
) )
3431, 33oveq12d 6314 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( *r `  F ) `  ( B  .,  A ) ) 
.+^  ( ( *r `  F ) `
 ( C  .,  A ) ) )  =  ( ( A 
.,  B )  .+^  ( A  .,  C ) ) )
3523, 29, 343eqtr3d 2506 1  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  .,  ( B  .+  C
) )  =  ( ( A  .,  B
)  .+^  ( A  .,  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712   *rcstv 14714  Scalarcsca 14715   .icip 14717   *Ringcsr 17620   LModclmod 17639   PreHilcphl 18786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-tpos 6973  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-ip 14730  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-grp 16184  df-ghm 16392  df-mgp 17269  df-ur 17281  df-ring 17327  df-oppr 17399  df-rnghom 17491  df-staf 17621  df-srng 17622  df-lmod 17641  df-lmhm 17795  df-lvec 17876  df-sra 17945  df-rgmod 17946  df-phl 18788
This theorem is referenced by:  ip2di  18803  ipsubdi  18805  cphdi  21778
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