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Theorem ipdi 17911
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 454 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  PreHil )
2 simpr2 988 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  B  e.  V )
3 simpr3 989 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  C  e.  V )
4 simpr1 987 . . . . 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 17910 . . . . 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 1213 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( B  .+  C )  .,  A )  =  ( ( B  .,  A
)  .+^  ( C  .,  A ) ) )
1211fveq2d 5683 . . 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 17902 . . . . 5  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
1413adantr 462 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  F  e.  *Ring
)
15 eqid 2433 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
165, 6, 7, 15ipcl 17904 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  ( B  .,  A )  e.  ( Base `  F
) )
171, 2, 4, 16syl3anc 1211 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( B  .,  A )  e.  (
Base `  F )
)
185, 6, 7, 15ipcl 17904 . . . . 5  |-  ( ( W  e.  PreHil  /\  C  e.  V  /\  A  e.  V )  ->  ( C  .,  A )  e.  ( Base `  F
) )
191, 3, 4, 18syl3anc 1211 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( C  .,  A )  e.  (
Base `  F )
)
20 eqid 2433 . . . . 5  |-  ( *r `  F )  =  ( *r `  F )
2120, 15, 9srngadd 16866 . . . 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 1211 . . 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 2465 . 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 17901 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
2524adantr 462 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  W  e.  LMod )
267, 8lmodvacl 16886 . . . 4  |-  ( ( W  e.  LMod  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .+  C )  e.  V )
2725, 2, 3, 26syl3anc 1211 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( B  .+  C )  e.  V
)
285, 6, 7, 20ipcj 17905 . . 3  |-  ( ( W  e.  PreHil  /\  ( B  .+  C )  e.  V  /\  A  e.  V )  ->  (
( *r `  F ) `  (
( B  .+  C
)  .,  A )
)  =  ( A 
.,  ( B  .+  C ) ) )
291, 27, 4, 28syl3anc 1211 . 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 17905 . . . 4  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  (
( *r `  F ) `  ( B  .,  A ) )  =  ( A  .,  B ) )
311, 2, 4, 30syl3anc 1211 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
*r `  F
) `  ( B  .,  A ) )  =  ( A  .,  B
) )
325, 6, 7, 20ipcj 17905 . . . 4  |-  ( ( W  e.  PreHil  /\  C  e.  V  /\  A  e.  V )  ->  (
( *r `  F ) `  ( C  .,  A ) )  =  ( A  .,  C ) )
331, 3, 4, 32syl3anc 1211 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
*r `  F
) `  ( C  .,  A ) )  =  ( A  .,  C
) )
3431, 33oveq12d 6098 . 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 2473 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 958    = wceq 1362    e. wcel 1755   ` cfv 5406  (class class class)co 6080   Basecbs 14157   +g cplusg 14221   *rcstv 14223  Scalarcsca 14224   .icip 14226   *Ringcsr 16853   LModclmod 16872   PreHilcphl 17895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-tpos 6734  df-recs 6818  df-rdg 6852  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-plusg 14234  df-mulr 14235  df-sca 14237  df-vsca 14238  df-ip 14239  df-0g 14363  df-mnd 15398  df-mhm 15447  df-grp 15525  df-ghm 15725  df-mgp 16566  df-rng 16580  df-ur 16582  df-oppr 16649  df-rnghom 16740  df-staf 16854  df-srng 16855  df-lmod 16874  df-lmhm 17025  df-lvec 17106  df-sra 17175  df-rgmod 17176  df-phl 17897
This theorem is referenced by:  ip2di  17912  ipsubdi  17914  cphdi  20566
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