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Theorem dipsubdir 22354
Description: Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ipsubdir.1  |-  X  =  ( BaseSet `  U )
ipsubdir.3  |-  M  =  ( -v `  U
)
ipsubdir.7  |-  P  =  ( .i OLD `  U
)
Assertion
Ref Expression
dipsubdir  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A M B ) P C )  =  ( ( A P C )  -  ( B P C ) ) )

Proof of Theorem dipsubdir
StepHypRef Expression
1 idd 23 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( A  e.  X  ->  A  e.  X ) )
2 phnv 22320 . . . . . . 7  |-  ( U  e.  CPreHil OLD  ->  U  e.  NrmCVec )
3 neg1cn 10072 . . . . . . . 8  |-  -u 1  e.  CC
4 ipsubdir.1 . . . . . . . . 9  |-  X  =  ( BaseSet `  U )
5 eqid 2438 . . . . . . . . 9  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
64, 5nvscl 22112 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
73, 6mp3an2 1268 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
82, 7sylan 459 . . . . . 6  |-  ( ( U  e.  CPreHil OLD  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U
) B )  e.  X )
98ex 425 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( B  e.  X  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X ) )
10 idd 23 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( C  e.  X  ->  C  e.  X ) )
111, 9, 103anim123d 1262 . . . 4  |-  ( U  e.  CPreHil OLD  ->  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( A  e.  X  /\  ( -u 1 ( .s OLD `  U
) B )  e.  X  /\  C  e.  X ) ) )
1211imp 420 . . 3  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A  e.  X  /\  ( -u 1 ( .s
OLD `  U ) B )  e.  X  /\  C  e.  X
) )
13 eqid 2438 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
14 ipsubdir.7 . . . 4  |-  P  =  ( .i OLD `  U
)
154, 13, 14dipdir 22348 . . 3  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  ( -u 1 ( .s OLD `  U
) B )  e.  X  /\  C  e.  X ) )  -> 
( ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) B ) ) P C )  =  ( ( A P C )  +  ( ( -u 1 ( .s OLD `  U
) B ) P C ) ) )
1612, 15syldan 458 . 2  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) P C )  =  ( ( A P C )  +  ( (
-u 1 ( .s
OLD `  U ) B ) P C ) ) )
17 ipsubdir.3 . . . . . 6  |-  M  =  ( -v `  U
)
184, 13, 5, 17nvmval 22128 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) )
192, 18syl3an1 1218 . . . 4  |-  ( ( U  e.  CPreHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
20193adant3r3 1165 . . 3  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) )
2120oveq1d 6099 . 2  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A M B ) P C )  =  ( ( A ( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) P C ) )
224, 5, 14dipass 22351 . . . . . . 7  |-  ( ( U  e.  CPreHil OLD  /\  ( -u 1  e.  CC  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( -u 1 ( .s OLD `  U ) B ) P C )  =  ( -u 1  x.  ( B P C ) ) )
233, 22mp3anr1 1277 . . . . . 6  |-  ( ( U  e.  CPreHil OLD  /\  ( B  e.  X  /\  C  e.  X
) )  ->  (
( -u 1 ( .s
OLD `  U ) B ) P C )  =  ( -u
1  x.  ( B P C ) ) )
244, 14dipcl 22216 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  X )  ->  ( B P C )  e.  CC )
25243expb 1155 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B P C )  e.  CC )
262, 25sylan 459 . . . . . . 7  |-  ( ( U  e.  CPreHil OLD  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B P C )  e.  CC )
2726mulm1d 9490 . . . . . 6  |-  ( ( U  e.  CPreHil OLD  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( -u 1  x.  ( B P C ) )  =  -u ( B P C ) )
2823, 27eqtrd 2470 . . . . 5  |-  ( ( U  e.  CPreHil OLD  /\  ( B  e.  X  /\  C  e.  X
) )  ->  (
( -u 1 ( .s
OLD `  U ) B ) P C )  =  -u ( B P C ) )
29283adantr1 1117 . . . 4  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( -u 1 ( .s
OLD `  U ) B ) P C )  =  -u ( B P C ) )
3029oveq2d 6100 . . 3  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A P C )  +  ( (
-u 1 ( .s
OLD `  U ) B ) P C ) )  =  ( ( A P C )  +  -u ( B P C ) ) )
314, 14dipcl 22216 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  C  e.  X )  ->  ( A P C )  e.  CC )
32313adant3r2 1164 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A P C )  e.  CC )
33243adant3r1 1163 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B P C )  e.  CC )
3432, 33negsubd 9422 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A P C )  + 
-u ( B P C ) )  =  ( ( A P C )  -  ( B P C ) ) )
352, 34sylan 459 . . 3  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A P C )  +  -u ( B P C ) )  =  ( ( A P C )  -  ( B P C ) ) )
3630, 35eqtr2d 2471 . 2  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A P C )  -  ( B P C ) )  =  ( ( A P C )  +  ( ( -u 1
( .s OLD `  U
) B ) P C ) ) )
3716, 21, 363eqtr4d 2480 1  |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A M B ) P C )  =  ( ( A P C )  -  ( B P C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5457  (class class class)co 6084   CCcc 8993   1c1 8996    + caddc 8998    x. cmul 9000    - cmin 9296   -ucneg 9297   NrmCVeccnv 22068   +vcpv 22069   BaseSetcba 22070   .s
OLDcns 22071   -vcnsb 22073   .i OLDcdip 22201   CPreHil OLDccphlo 22318
This theorem is referenced by:  dipsubdi  22355  siilem1  22357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-icc 10928  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-cn 17296  df-cnp 17297  df-t1 17383  df-haus 17384  df-tx 17599  df-hmeo 17792  df-xms 18355  df-ms 18356  df-tms 18357  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gdiv 21787  df-ablo 21875  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-vs 22083  df-nmcv 22084  df-ims 22085  df-dip 22202  df-ph 22319
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