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Theorem dipsubdir 22310
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 22 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( A  e.  X  ->  A  e.  X ) )
2 phnv 22276 . . . . . . 7  |-  ( U  e.  CPreHil OLD  ->  U  e.  NrmCVec )
3 neg1cn 10031 . . . . . . . 8  |-  -u 1  e.  CC
4 ipsubdir.1 . . . . . . . . 9  |-  X  =  ( BaseSet `  U )
5 eqid 2412 . . . . . . . . 9  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
64, 5nvscl 22068 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
73, 6mp3an2 1267 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
82, 7sylan 458 . . . . . 6  |-  ( ( U  e.  CPreHil OLD  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U
) B )  e.  X )
98ex 424 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( B  e.  X  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X ) )
10 idd 22 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( C  e.  X  ->  C  e.  X ) )
111, 9, 103anim123d 1261 . . . 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 419 . . 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 2412 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
14 ipsubdir.7 . . . 4  |-  P  =  ( .i OLD `  U
)
154, 13, 14dipdir 22304 . . 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 457 . 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 22084 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) )
192, 18syl3an1 1217 . . . 4  |-  ( ( U  e.  CPreHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
20193adant3r3 1164 . . 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 6063 . 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 22307 . . . . . . 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 1276 . . . . . 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 22172 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  X )  ->  ( B P C )  e.  CC )
25243expb 1154 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B P C )  e.  CC )
262, 25sylan 458 . . . . . . 7  |-  ( ( U  e.  CPreHil OLD  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B P C )  e.  CC )
2726mulm1d 9449 . . . . . 6  |-  ( ( U  e.  CPreHil OLD  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( -u 1  x.  ( B P C ) )  =  -u ( B P C ) )
2823, 27eqtrd 2444 . . . . 5  |-  ( ( U  e.  CPreHil OLD  /\  ( B  e.  X  /\  C  e.  X
) )  ->  (
( -u 1 ( .s
OLD `  U ) B ) P C )  =  -u ( B P C ) )
29283adantr1 1116 . . . 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 6064 . . 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 22172 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  C  e.  X )  ->  ( A P C )  e.  CC )
32313adant3r2 1163 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A P C )  e.  CC )
33243adant3r1 1162 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B P C )  e.  CC )
3432, 33negsubd 9381 . . . 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 458 . . 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 2445 . 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 2454 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5421  (class class class)co 6048   CCcc 8952   1c1 8955    + caddc 8957    x. cmul 8959    - cmin 9255   -ucneg 9256   NrmCVeccnv 22024   +vcpv 22025   BaseSetcba 22026   .s
OLDcns 22027   -vcnsb 22029   .i OLDcdip 22157   CPreHil OLDccphlo 22274
This theorem is referenced by:  dipsubdi  22311  siilem1  22313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-icc 10887  df-fz 11008  df-fzo 11099  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-sum 12443  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-cn 17253  df-cnp 17254  df-t1 17340  df-haus 17341  df-tx 17555  df-hmeo 17748  df-xms 18311  df-ms 18312  df-tms 18313  df-grpo 21740  df-gid 21741  df-ginv 21742  df-gdiv 21743  df-ablo 21831  df-vc 21986  df-nv 22032  df-va 22035  df-ba 22036  df-sm 22037  df-0v 22038  df-vs 22039  df-nmcv 22040  df-ims 22041  df-dip 22158  df-ph 22275
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