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Theorem dipdir 22331
Description: Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
dipdir.1  |-  X  =  ( BaseSet `  U )
dipdir.2  |-  G  =  ( +v `  U
)
dipdir.7  |-  P  =  ( .i OLD `  U
)
Assertion
Ref Expression
dipdir  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) )

Proof of Theorem dipdir
StepHypRef Expression
1 dipdir.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
2 fveq2 5719 . . . . . . 7  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( BaseSet `  U )  =  ( BaseSet `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) )
31, 2syl5eq 2479 . . . . . 6  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  X  =  ( BaseSet `  if ( U  e.  CPreHil OLD
,  U ,  <. <.  +  ,  x.  >. ,  abs >.
) ) )
43eleq2d 2502 . . . . 5  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( A  e.  X  <->  A  e.  ( BaseSet `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) ) )
53eleq2d 2502 . . . . 5  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( B  e.  X  <->  B  e.  ( BaseSet `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) ) )
63eleq2d 2502 . . . . 5  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( C  e.  X  <->  C  e.  ( BaseSet `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) ) )
74, 5, 63anbi123d 1254 . . . 4  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  <->  ( A  e.  ( BaseSet `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) )  /\  B  e.  ( BaseSet `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) )  /\  C  e.  ( BaseSet `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) ) ) )
8 dipdir.2 . . . . . . . . 9  |-  G  =  ( +v `  U
)
9 fveq2 5719 . . . . . . . . 9  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( +v `  U
)  =  ( +v
`  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )
) )
108, 9syl5eq 2479 . . . . . . . 8  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  G  =  ( +v
`  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )
) )
1110oveqd 6089 . . . . . . 7  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( A G B )  =  ( A ( +v `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) B ) )
1211oveq1d 6087 . . . . . 6  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( ( A G B ) P C )  =  ( ( A ( +v `  if ( U  e.  CPreHil OLD
,  U ,  <. <.  +  ,  x.  >. ,  abs >.
) ) B ) P C ) )
13 dipdir.7 . . . . . . . 8  |-  P  =  ( .i OLD `  U
)
14 fveq2 5719 . . . . . . . 8  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( .i OLD `  U
)  =  ( .i
OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) )
1513, 14syl5eq 2479 . . . . . . 7  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  P  =  ( .i
OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) )
1615oveqd 6089 . . . . . 6  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( ( A ( +v `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) B ) P C )  =  ( ( A ( +v
`  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )
) B ) ( .i OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C ) )
1712, 16eqtrd 2467 . . . . 5  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( ( A G B ) P C )  =  ( ( A ( +v `  if ( U  e.  CPreHil OLD
,  U ,  <. <.  +  ,  x.  >. ,  abs >.
) ) B ) ( .i OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C ) )
1815oveqd 6089 . . . . . 6  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( A P C )  =  ( A ( .i OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C ) )
1915oveqd 6089 . . . . . 6  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( B P C )  =  ( B ( .i OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C ) )
2018, 19oveq12d 6090 . . . . 5  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( ( A P C )  +  ( B P C ) )  =  ( ( A ( .i OLD `  if ( U  e.  CPreHil
OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )
) C )  +  ( B ( .i
OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C ) ) )
2117, 20eqeq12d 2449 . . . 4  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) )  <-> 
( ( A ( +v `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) B ) ( .i OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C )  =  ( ( A ( .i OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C )  +  ( B ( .i
OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C ) ) ) )
227, 21imbi12d 312 . . 3  |-  ( U  =  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  ->  ( ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  (
( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) )  <->  ( ( A  e.  ( BaseSet `  if ( U  e.  CPreHil OLD
,  U ,  <. <.  +  ,  x.  >. ,  abs >.
) )  /\  B  e.  ( BaseSet `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) )  /\  C  e.  ( BaseSet `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) )  ->  (
( A ( +v
`  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )
) B ) ( .i OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C )  =  ( ( A ( .i OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C )  +  ( B ( .i
OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C ) ) ) ) )
23 eqid 2435 . . . 4  |-  ( BaseSet `  if ( U  e.  CPreHil OLD
,  U ,  <. <.  +  ,  x.  >. ,  abs >.
) )  =  (
BaseSet `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )
)
24 eqid 2435 . . . 4  |-  ( +v
`  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )
)  =  ( +v
`  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )
)
25 eqid 2435 . . . 4  |-  ( .s
OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) )  =  ( .s
OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) )
26 eqid 2435 . . . 4  |-  ( .i
OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) )  =  ( .i
OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) )
27 elimphu 22310 . . . 4  |-  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  e.  CPreHil OLD
2823, 24, 25, 26, 27ipdiri 22319 . . 3  |-  ( ( A  e.  ( BaseSet `  if ( U  e.  CPreHil OLD
,  U ,  <. <.  +  ,  x.  >. ,  abs >.
) )  /\  B  e.  ( BaseSet `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) )  /\  C  e.  ( BaseSet `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) )  ->  (
( A ( +v
`  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )
) B ) ( .i OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C )  =  ( ( A ( .i OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C )  +  ( B ( .i
OLD `  if ( U  e.  CPreHil OLD ,  U ,  <. <.  +  ,  x.  >. ,  abs >. ) ) C ) ) )
2922, 28dedth 3772 . 2  |-  ( U  e.  CPreHil OLD  ->  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) ) )
3029imp 419 1  |-  ( ( U  e.  CPreHil OLD  /\  ( 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    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ifcif 3731   <.cop 3809   ` cfv 5445  (class class class)co 6072    + caddc 8982    x. cmul 8984   abscabs 12027   +vcpv 22052   BaseSetcba 22053   .s
OLDcns 22054   .i OLDcdip 22184   CPreHil OLDccphlo 22301
This theorem is referenced by:  dipdi  22332  ip2dii  22333  dipsubdir  22337  ipblnfi  22345  hlipdir  22402
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-sup 7437  df-oi 7468  df-card 7815  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-fz 11033  df-fzo 11124  df-seq 11312  df-exp 11371  df-hash 11607  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-clim 12270  df-sum 12468  df-grpo 21767  df-gid 21768  df-ginv 21769  df-ablo 21858  df-vc 22013  df-nv 22059  df-va 22062  df-ba 22063  df-sm 22064  df-0v 22065  df-nmcv 22067  df-dip 22185  df-ph 22302
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