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Theorem ipdiri 22314
Description: Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ip1i.1  |-  X  =  ( BaseSet `  U )
ip1i.2  |-  G  =  ( +v `  U
)
ip1i.4  |-  S  =  ( .s OLD `  U
)
ip1i.7  |-  P  =  ( .i OLD `  U
)
ip1i.9  |-  U  e.  CPreHil
OLD
Assertion
Ref Expression
ipdiri  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) )

Proof of Theorem ipdiri
StepHypRef Expression
1 oveq1 6074 . . . 4  |-  ( A  =  if ( A  e.  X ,  A ,  ( 0vec `  U
) )  ->  ( A G B )  =  ( if ( A  e.  X ,  A ,  ( 0vec `  U
) ) G B ) )
21oveq1d 6082 . . 3  |-  ( A  =  if ( A  e.  X ,  A ,  ( 0vec `  U
) )  ->  (
( A G B ) P C )  =  ( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) G B ) P C ) )
3 oveq1 6074 . . . 4  |-  ( A  =  if ( A  e.  X ,  A ,  ( 0vec `  U
) )  ->  ( A P C )  =  ( if ( A  e.  X ,  A ,  ( 0vec `  U
) ) P C ) )
43oveq1d 6082 . . 3  |-  ( A  =  if ( A  e.  X ,  A ,  ( 0vec `  U
) )  ->  (
( A P C )  +  ( B P C ) )  =  ( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) P C )  +  ( B P C ) ) )
52, 4eqeq12d 2444 . 2  |-  ( A  =  if ( A  e.  X ,  A ,  ( 0vec `  U
) )  ->  (
( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) )  <->  ( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) G B ) P C )  =  ( ( if ( A  e.  X ,  A , 
( 0vec `  U )
) P C )  +  ( B P C ) ) ) )
6 oveq2 6075 . . . 4  |-  ( B  =  if ( B  e.  X ,  B ,  ( 0vec `  U
) )  ->  ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) G B )  =  ( if ( A  e.  X ,  A , 
( 0vec `  U )
) G if ( B  e.  X ,  B ,  ( 0vec `  U ) ) ) )
76oveq1d 6082 . . 3  |-  ( B  =  if ( B  e.  X ,  B ,  ( 0vec `  U
) )  ->  (
( if ( A  e.  X ,  A ,  ( 0vec `  U
) ) G B ) P C )  =  ( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) G if ( B  e.  X ,  B , 
( 0vec `  U )
) ) P C ) )
8 oveq1 6074 . . . 4  |-  ( B  =  if ( B  e.  X ,  B ,  ( 0vec `  U
) )  ->  ( B P C )  =  ( if ( B  e.  X ,  B ,  ( 0vec `  U
) ) P C ) )
98oveq2d 6083 . . 3  |-  ( B  =  if ( B  e.  X ,  B ,  ( 0vec `  U
) )  ->  (
( if ( A  e.  X ,  A ,  ( 0vec `  U
) ) P C )  +  ( B P C ) )  =  ( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) P C )  +  ( if ( B  e.  X ,  B , 
( 0vec `  U )
) P C ) ) )
107, 9eqeq12d 2444 . 2  |-  ( B  =  if ( B  e.  X ,  B ,  ( 0vec `  U
) )  ->  (
( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) G B ) P C )  =  ( ( if ( A  e.  X ,  A , 
( 0vec `  U )
) P C )  +  ( B P C ) )  <->  ( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) G if ( B  e.  X ,  B , 
( 0vec `  U )
) ) P C )  =  ( ( if ( A  e.  X ,  A , 
( 0vec `  U )
) P C )  +  ( if ( B  e.  X ,  B ,  ( 0vec `  U ) ) P C ) ) ) )
11 oveq2 6075 . . 3  |-  ( C  =  if ( C  e.  X ,  C ,  ( 0vec `  U
) )  ->  (
( if ( A  e.  X ,  A ,  ( 0vec `  U
) ) G if ( B  e.  X ,  B ,  ( 0vec `  U ) ) ) P C )  =  ( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) G if ( B  e.  X ,  B , 
( 0vec `  U )
) ) P if ( C  e.  X ,  C ,  ( 0vec `  U ) ) ) )
12 oveq2 6075 . . . 4  |-  ( C  =  if ( C  e.  X ,  C ,  ( 0vec `  U
) )  ->  ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) P C )  =  ( if ( A  e.  X ,  A , 
( 0vec `  U )
) P if ( C  e.  X ,  C ,  ( 0vec `  U ) ) ) )
13 oveq2 6075 . . . 4  |-  ( C  =  if ( C  e.  X ,  C ,  ( 0vec `  U
) )  ->  ( if ( B  e.  X ,  B ,  ( 0vec `  U ) ) P C )  =  ( if ( B  e.  X ,  B , 
( 0vec `  U )
) P if ( C  e.  X ,  C ,  ( 0vec `  U ) ) ) )
1412, 13oveq12d 6085 . . 3  |-  ( C  =  if ( C  e.  X ,  C ,  ( 0vec `  U
) )  ->  (
( if ( A  e.  X ,  A ,  ( 0vec `  U
) ) P C )  +  ( if ( B  e.  X ,  B ,  ( 0vec `  U ) ) P C ) )  =  ( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) P if ( C  e.  X ,  C , 
( 0vec `  U )
) )  +  ( if ( B  e.  X ,  B , 
( 0vec `  U )
) P if ( C  e.  X ,  C ,  ( 0vec `  U ) ) ) ) )
1511, 14eqeq12d 2444 . 2  |-  ( C  =  if ( C  e.  X ,  C ,  ( 0vec `  U
) )  ->  (
( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) G if ( B  e.  X ,  B , 
( 0vec `  U )
) ) P C )  =  ( ( if ( A  e.  X ,  A , 
( 0vec `  U )
) P C )  +  ( if ( B  e.  X ,  B ,  ( 0vec `  U ) ) P C ) )  <->  ( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) G if ( B  e.  X ,  B , 
( 0vec `  U )
) ) P if ( C  e.  X ,  C ,  ( 0vec `  U ) ) )  =  ( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) P if ( C  e.  X ,  C , 
( 0vec `  U )
) )  +  ( if ( B  e.  X ,  B , 
( 0vec `  U )
) P if ( C  e.  X ,  C ,  ( 0vec `  U ) ) ) ) ) )
16 ip1i.1 . . 3  |-  X  =  ( BaseSet `  U )
17 ip1i.2 . . 3  |-  G  =  ( +v `  U
)
18 ip1i.4 . . 3  |-  S  =  ( .s OLD `  U
)
19 ip1i.7 . . 3  |-  P  =  ( .i OLD `  U
)
20 ip1i.9 . . 3  |-  U  e.  CPreHil
OLD
21 eqid 2430 . . . 4  |-  ( 0vec `  U )  =  (
0vec `  U )
2216, 21, 20elimph 22304 . . 3  |-  if ( A  e.  X ,  A ,  ( 0vec `  U ) )  e.  X
2316, 21, 20elimph 22304 . . 3  |-  if ( B  e.  X ,  B ,  ( 0vec `  U ) )  e.  X
2416, 21, 20elimph 22304 . . 3  |-  if ( C  e.  X ,  C ,  ( 0vec `  U ) )  e.  X
2516, 17, 18, 19, 20, 22, 23, 24ipdirilem 22313 . 2  |-  ( ( if ( A  e.  X ,  A , 
( 0vec `  U )
) G if ( B  e.  X ,  B ,  ( 0vec `  U ) ) ) P if ( C  e.  X ,  C ,  ( 0vec `  U
) ) )  =  ( ( if ( A  e.  X ,  A ,  ( 0vec `  U ) ) P if ( C  e.  X ,  C , 
( 0vec `  U )
) )  +  ( if ( B  e.  X ,  B , 
( 0vec `  U )
) P if ( C  e.  X ,  C ,  ( 0vec `  U ) ) ) )
265, 10, 15, 25dedth3h 3769 1  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   ifcif 3726   ` cfv 5440  (class class class)co 6067    + caddc 8977   +vcpv 22047   BaseSetcba 22048   .s
OLDcns 22049   0veccn0v 22050   .i
OLDcdip 22179   CPreHil OLDccphlo 22296
This theorem is referenced by:  ipasslem1  22315  ipasslem2  22316  ipasslem11  22324  dipdir  22326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051  ax-pre-sup 9052
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-oadd 6714  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-sup 7432  df-oi 7463  df-card 7810  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-3 10043  df-4 10044  df-n0 10206  df-z 10267  df-uz 10473  df-rp 10597  df-fz 11028  df-fzo 11119  df-seq 11307  df-exp 11366  df-hash 11602  df-cj 11887  df-re 11888  df-im 11889  df-sqr 12023  df-abs 12024  df-clim 12265  df-sum 12463  df-grpo 21762  df-gid 21763  df-ginv 21764  df-ablo 21853  df-vc 22008  df-nv 22054  df-va 22057  df-ba 22058  df-sm 22059  df-0v 22060  df-nmcv 22062  df-dip 22180  df-ph 22297
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