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Theorem ipdi 18072
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 995 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  B  e.  V )
3 simpr3 996 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  C  e.  V )
4 simpr1 994 . . . . 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 18071 . . . . 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 1220 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( B  .+  C )  .,  A )  =  ( ( B  .,  A
)  .+^  ( C  .,  A ) ) )
1211fveq2d 5698 . . 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 18063 . . . . 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 2443 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
165, 6, 7, 15ipcl 18065 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  ( B  .,  A )  e.  ( Base `  F
) )
171, 2, 4, 16syl3anc 1218 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( B  .,  A )  e.  (
Base `  F )
)
185, 6, 7, 15ipcl 18065 . . . . 5  |-  ( ( W  e.  PreHil  /\  C  e.  V  /\  A  e.  V )  ->  ( C  .,  A )  e.  ( Base `  F
) )
191, 3, 4, 18syl3anc 1218 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( C  .,  A )  e.  (
Base `  F )
)
20 eqid 2443 . . . . 5  |-  ( *r `  F )  =  ( *r `  F )
2120, 15, 9srngadd 16945 . . . 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 1218 . . 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 2475 . 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 18062 . . . . 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 16965 . . . 4  |-  ( ( W  e.  LMod  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .+  C )  e.  V )
2725, 2, 3, 26syl3anc 1218 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( B  .+  C )  e.  V
)
285, 6, 7, 20ipcj 18066 . . 3  |-  ( ( W  e.  PreHil  /\  ( B  .+  C )  e.  V  /\  A  e.  V )  ->  (
( *r `  F ) `  (
( B  .+  C
)  .,  A )
)  =  ( A 
.,  ( B  .+  C ) ) )
291, 27, 4, 28syl3anc 1218 . 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 18066 . . . 4  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  (
( *r `  F ) `  ( B  .,  A ) )  =  ( A  .,  B ) )
311, 2, 4, 30syl3anc 1218 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
*r `  F
) `  ( B  .,  A ) )  =  ( A  .,  B
) )
325, 6, 7, 20ipcj 18066 . . . 4  |-  ( ( W  e.  PreHil  /\  C  e.  V  /\  A  e.  V )  ->  (
( *r `  F ) `  ( C  .,  A ) )  =  ( A  .,  C ) )
331, 3, 4, 32syl3anc 1218 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
*r `  F
) `  ( C  .,  A ) )  =  ( A  .,  C
) )
3431, 33oveq12d 6112 . 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 2483 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 965    = wceq 1369    e. wcel 1756   ` cfv 5421  (class class class)co 6094   Basecbs 14177   +g cplusg 14241   *rcstv 14243  Scalarcsca 14244   .icip 14246   *Ringcsr 16932   LModclmod 16951   PreHilcphl 18056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-tpos 6748  df-recs 6835  df-rdg 6869  df-er 7104  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-plusg 14254  df-mulr 14255  df-sca 14257  df-vsca 14258  df-ip 14259  df-0g 14383  df-mnd 15418  df-mhm 15467  df-grp 15548  df-ghm 15748  df-mgp 16595  df-ur 16607  df-rng 16650  df-oppr 16718  df-rnghom 16809  df-staf 16933  df-srng 16934  df-lmod 16953  df-lmhm 17106  df-lvec 17187  df-sra 17256  df-rgmod 17257  df-phl 18058
This theorem is referenced by:  ip2di  18073  ipsubdi  18075  cphdi  20727
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