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Theorem dipdi 22344
Description: Distributive law for inner product. (Contributed by NM, 20-Nov-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
dipdi  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A P ( B G C ) )  =  ( ( A P B )  +  ( A P C ) ) )

Proof of Theorem dipdi
StepHypRef Expression
1 id 20 . . 3  |-  ( ( C  e.  X  /\  B  e.  X  /\  A  e.  X )  ->  ( C  e.  X  /\  B  e.  X  /\  A  e.  X
) )
213com13 1158 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( C  e.  X  /\  B  e.  X  /\  A  e.  X
) )
3 id 20 . . . . . 6  |-  ( ( B  e.  X  /\  C  e.  X  /\  A  e.  X )  ->  ( B  e.  X  /\  C  e.  X  /\  A  e.  X
) )
433com12 1157 . . . . 5  |-  ( ( C  e.  X  /\  B  e.  X  /\  A  e.  X )  ->  ( B  e.  X  /\  C  e.  X  /\  A  e.  X
) )
5 dipdir.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
6 dipdir.2 . . . . . 6  |-  G  =  ( +v `  U
)
7 dipdir.7 . . . . . 6  |-  P  =  ( .i OLD `  U
)
85, 6, 7dipdir 22343 . . . . 5  |-  ( ( U  e.  CPreHil OLD  /\  ( B  e.  X  /\  C  e.  X  /\  A  e.  X
) )  ->  (
( B G C ) P A )  =  ( ( B P A )  +  ( C P A ) ) )
94, 8sylan2 461 . . . 4  |-  ( ( U  e.  CPreHil OLD  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X
) )  ->  (
( B G C ) P A )  =  ( ( B P A )  +  ( C P A ) ) )
109fveq2d 5732 . . 3  |-  ( ( U  e.  CPreHil OLD  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X
) )  ->  (
* `  ( ( B G C ) P A ) )  =  ( * `  (
( B P A )  +  ( C P A ) ) ) )
11 phnv 22315 . . . 4  |-  ( U  e.  CPreHil OLD  ->  U  e.  NrmCVec )
12 simpl 444 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X )
)  ->  U  e.  NrmCVec )
135, 6nvgcl 22099 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  X )  ->  ( B G C )  e.  X )
14133com23 1159 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  C  e.  X  /\  B  e.  X )  ->  ( B G C )  e.  X )
15143adant3r3 1164 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X )
)  ->  ( B G C )  e.  X
)
16 simpr3 965 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X )
)  ->  A  e.  X )
175, 7dipcj 22213 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( B G C )  e.  X  /\  A  e.  X )  ->  (
* `  ( ( B G C ) P A ) )  =  ( A P ( B G C ) ) )
1812, 15, 16, 17syl3anc 1184 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X )
)  ->  ( * `  ( ( B G C ) P A ) )  =  ( A P ( B G C ) ) )
1911, 18sylan 458 . . 3  |-  ( ( U  e.  CPreHil OLD  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X
) )  ->  (
* `  ( ( B G C ) P A ) )  =  ( A P ( B G C ) ) )
205, 7dipcl 22211 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  A  e.  X )  ->  ( B P A )  e.  CC )
21203adant3r1 1162 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X )
)  ->  ( B P A )  e.  CC )
225, 7dipcl 22211 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  C  e.  X  /\  A  e.  X )  ->  ( C P A )  e.  CC )
23223adant3r2 1163 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X )
)  ->  ( C P A )  e.  CC )
2421, 23cjaddd 12025 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X )
)  ->  ( * `  ( ( B P A )  +  ( C P A ) ) )  =  ( ( * `  ( B P A ) )  +  ( * `  ( C P A ) ) ) )
255, 7dipcj 22213 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  A  e.  X )  ->  (
* `  ( B P A ) )  =  ( A P B ) )
26253adant3r1 1162 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X )
)  ->  ( * `  ( B P A ) )  =  ( A P B ) )
275, 7dipcj 22213 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  C  e.  X  /\  A  e.  X )  ->  (
* `  ( C P A ) )  =  ( A P C ) )
28273adant3r2 1163 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X )
)  ->  ( * `  ( C P A ) )  =  ( A P C ) )
2926, 28oveq12d 6099 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X )
)  ->  ( (
* `  ( B P A ) )  +  ( * `  ( C P A ) ) )  =  ( ( A P B )  +  ( A P C ) ) )
3024, 29eqtrd 2468 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X )
)  ->  ( * `  ( ( B P A )  +  ( C P A ) ) )  =  ( ( A P B )  +  ( A P C ) ) )
3111, 30sylan 458 . . 3  |-  ( ( U  e.  CPreHil OLD  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X
) )  ->  (
* `  ( ( B P A )  +  ( C P A ) ) )  =  ( ( A P B )  +  ( A P C ) ) )
3210, 19, 313eqtr3d 2476 . 2  |-  ( ( U  e.  CPreHil OLD  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X
) )  ->  ( A P ( B G C ) )  =  ( ( A P B )  +  ( A P C ) ) )
332, 32sylan2 461 1  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A P ( B G C ) )  =  ( ( A P B )  +  ( A P C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988    + caddc 8993   *ccj 11901   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   .i
OLDcdip 22196   CPreHil OLDccphlo 22313
This theorem is referenced by:  ip2dii  22345
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-grpo 21779  df-gid 21780  df-ginv 21781  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-nmcv 22079  df-dip 22197  df-ph 22314
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