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Theorem dipsubdir 21540
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 21 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( A  e.  X  ->  A  e.  X ) )
2 phnv 21506 . . . . . . 7  |-  ( U  e.  CPreHil OLD  ->  U  e.  NrmCVec )
3 neg1cn 9903 . . . . . . . 8  |-  -u 1  e.  CC
4 ipsubdir.1 . . . . . . . . 9  |-  X  =  ( BaseSet `  U )
5 eqid 2358 . . . . . . . . 9  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
64, 5nvscl 21298 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
73, 6mp3an2 1265 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
82, 7sylan 457 . . . . . 6  |-  ( ( U  e.  CPreHil OLD  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U
) B )  e.  X )
98ex 423 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( B  e.  X  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X ) )
10 idd 21 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( C  e.  X  ->  C  e.  X ) )
111, 9, 103anim123d 1259 . . . 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 418 . . 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 2358 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
14 ipsubdir.7 . . . 4  |-  P  =  ( .i OLD `  U
)
154, 13, 14dipdir 21534 . . 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 456 . 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 21314 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) )
192, 18syl3an1 1215 . . . 4  |-  ( ( U  e.  CPreHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
20193adant3r3 1162 . . 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 5960 . 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 21537 . . . . . . 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 1274 . . . . . 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 21402 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  X )  ->  ( B P C )  e.  CC )
25243expb 1152 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B P C )  e.  CC )
262, 25sylan 457 . . . . . . 7  |-  ( ( U  e.  CPreHil OLD  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B P C )  e.  CC )
2726mulm1d 9321 . . . . . 6  |-  ( ( U  e.  CPreHil OLD  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( -u 1  x.  ( B P C ) )  =  -u ( B P C ) )
2823, 27eqtrd 2390 . . . . 5  |-  ( ( U  e.  CPreHil OLD  /\  ( B  e.  X  /\  C  e.  X
) )  ->  (
( -u 1 ( .s
OLD `  U ) B ) P C )  =  -u ( B P C ) )
29283adantr1 1114 . . . 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 5961 . . 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 21402 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  C  e.  X )  ->  ( A P C )  e.  CC )
32313adant3r2 1161 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A P C )  e.  CC )
33243adant3r1 1160 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B P C )  e.  CC )
3432, 33negsubd 9253 . . . 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 457 . . 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 2391 . 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 2400 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710   ` cfv 5337  (class class class)co 5945   CCcc 8825   1c1 8828    + caddc 8830    x. cmul 8832    - cmin 9127   -ucneg 9128   NrmCVeccnv 21254   +vcpv 21255   BaseSetcba 21256   .s
OLDcns 21257   -vcnsb 21259   .i OLDcdip 21387   CPreHil OLDccphlo 21504
This theorem is referenced by:  dipsubdi  21541  siilem1  21543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-sup 7284  df-oi 7315  df-card 7662  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-ioo 10752  df-icc 10755  df-fz 10875  df-fzo 10963  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-sum 12256  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-hom 13329  df-cco 13330  df-rest 13426  df-topn 13427  df-topgen 13443  df-pt 13444  df-prds 13447  df-xrs 13502  df-0g 13503  df-gsum 13504  df-qtop 13509  df-imas 13510  df-xps 13512  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-submnd 14515  df-mulg 14591  df-cntz 14892  df-cmn 15190  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-cnfld 16483  df-top 16742  df-bases 16744  df-topon 16745  df-topsp 16746  df-cld 16862  df-ntr 16863  df-cls 16864  df-cn 17063  df-cnp 17064  df-t1 17148  df-haus 17149  df-tx 17363  df-hmeo 17552  df-xms 17987  df-ms 17988  df-tms 17989  df-grpo 20970  df-gid 20971  df-ginv 20972  df-gdiv 20973  df-ablo 21061  df-vc 21216  df-nv 21262  df-va 21265  df-ba 21266  df-sm 21267  df-0v 21268  df-vs 21269  df-nmcv 21270  df-ims 21271  df-dip 21388  df-ph 21505
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