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Theorem siii 24188
Description: Inference from sii 24189. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
siii.1  |-  X  =  ( BaseSet `  U )
siii.6  |-  N  =  ( normCV `  U )
siii.7  |-  P  =  ( .iOLD `  U )
siii.9  |-  U  e.  CPreHil
OLD
siii.a  |-  A  e.  X
siii.b  |-  B  e.  X
Assertion
Ref Expression
siii  |-  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) )

Proof of Theorem siii
StepHypRef Expression
1 oveq2 6098 . . . . 5  |-  ( B  =  ( 0vec `  U
)  ->  ( A P B )  =  ( A P ( 0vec `  U ) ) )
2 siii.9 . . . . . . 7  |-  U  e.  CPreHil
OLD
32phnvi 24151 . . . . . 6  |-  U  e.  NrmCVec
4 siii.a . . . . . 6  |-  A  e.  X
5 siii.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
6 eqid 2441 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
7 siii.7 . . . . . . 7  |-  P  =  ( .iOLD `  U )
85, 6, 7dip0r 24050 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P ( 0vec `  U
) )  =  0 )
93, 4, 8mp2an 667 . . . . 5  |-  ( A P ( 0vec `  U
) )  =  0
101, 9syl6eq 2489 . . . 4  |-  ( B  =  ( 0vec `  U
)  ->  ( A P B )  =  0 )
1110abs00bd 12776 . . 3  |-  ( B  =  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  =  0 )
12 siii.6 . . . . . 6  |-  N  =  ( normCV `  U )
135, 12nvge0 23997 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( N `  A
) )
143, 4, 13mp2an 667 . . . 4  |-  0  <_  ( N `  A
)
15 siii.b . . . . 5  |-  B  e.  X
165, 12nvge0 23997 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  0  <_  ( N `  B
) )
173, 15, 16mp2an 667 . . . 4  |-  0  <_  ( N `  B
)
185, 12, 3, 4nvcli 23983 . . . . 5  |-  ( N `
 A )  e.  RR
195, 12, 3, 15nvcli 23983 . . . . 5  |-  ( N `
 B )  e.  RR
2018, 19mulge0i 9883 . . . 4  |-  ( ( 0  <_  ( N `  A )  /\  0  <_  ( N `  B
) )  ->  0  <_  ( ( N `  A )  x.  ( N `  B )
) )
2114, 17, 20mp2an 667 . . 3  |-  0  <_  ( ( N `  A )  x.  ( N `  B )
)
2211, 21syl6eqbr 4326 . 2  |-  ( B  =  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) )
2319recni 9394 . . . . . . . . . . 11  |-  ( N `
 B )  e.  CC
2423sqeq0i 11943 . . . . . . . . . 10  |-  ( ( ( N `  B
) ^ 2 )  =  0  <->  ( N `  B )  =  0 )
255, 6, 12nvz 23992 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( N `  B
)  =  0  <->  B  =  ( 0vec `  U
) ) )
263, 15, 25mp2an 667 . . . . . . . . . 10  |-  ( ( N `  B )  =  0  <->  B  =  ( 0vec `  U )
)
2724, 26bitri 249 . . . . . . . . 9  |-  ( ( ( N `  B
) ^ 2 )  =  0  <->  B  =  ( 0vec `  U )
)
2827necon3bii 2638 . . . . . . . 8  |-  ( ( ( N `  B
) ^ 2 )  =/=  0  <->  B  =/=  ( 0vec `  U )
)
295, 7dipcl 24045 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  A  e.  X )  ->  ( B P A )  e.  CC )
303, 15, 4, 29mp3an 1309 . . . . . . . . 9  |-  ( B P A )  e.  CC
3119resqcli 11947 . . . . . . . . . 10  |-  ( ( N `  B ) ^ 2 )  e.  RR
3231recni 9394 . . . . . . . . 9  |-  ( ( N `  B ) ^ 2 )  e.  CC
3330, 32divcan1zi 10063 . . . . . . . 8  |-  ( ( ( N `  B
) ^ 2 )  =/=  0  ->  (
( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( ( N `  B ) ^ 2 ) )  =  ( B P A ) )
3428, 33sylbir 213 . . . . . . 7  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( ( N `
 B ) ^
2 ) )  =  ( B P A ) )
355, 7dipcj 24047 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( A P B ) )  =  ( B P A ) )
363, 4, 15, 35mp3an 1309 . . . . . . 7  |-  ( * `
 ( A P B ) )  =  ( B P A )
3734, 36syl6eqr 2491 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( ( N `
 B ) ^
2 ) )  =  ( * `  ( A P B ) ) )
3837oveq2d 6106 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  ( ( A P B )  x.  ( ( ( B P A )  / 
( ( N `  B ) ^ 2 ) )  x.  (
( N `  B
) ^ 2 ) ) )  =  ( ( A P B )  x.  ( * `
 ( A P B ) ) ) )
3938fveq2d 5692 . . . 4  |-  ( B  =/=  ( 0vec `  U
)  ->  ( sqr `  ( ( A P B )  x.  (
( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( ( N `  B ) ^ 2 ) ) ) )  =  ( sqr `  ( ( A P B )  x.  ( * `  ( A P B ) ) ) ) )
405, 7dipcl 24045 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
413, 4, 15, 40mp3an 1309 . . . . 5  |-  ( A P B )  e.  CC
42 absval 12723 . . . . 5  |-  ( ( A P B )  e.  CC  ->  ( abs `  ( A P B ) )  =  ( sqr `  (
( A P B )  x.  ( * `
 ( A P B ) ) ) ) )
4341, 42ax-mp 5 . . . 4  |-  ( abs `  ( A P B ) )  =  ( sqr `  ( ( A P B )  x.  ( * `  ( A P B ) ) ) )
4439, 43syl6reqr 2492 . . 3  |-  ( B  =/=  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  =  ( sqr `  ( ( A P B )  x.  ( ( ( B P A )  /  ( ( N `
 B ) ^
2 ) )  x.  ( ( N `  B ) ^ 2 ) ) ) ) )
4534eqcomd 2446 . . . 4  |-  ( B  =/=  ( 0vec `  U
)  ->  ( B P A )  =  ( ( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( ( N `  B ) ^ 2 ) ) )
4630, 32divclzi 10062 . . . . . 6  |-  ( ( ( N `  B
) ^ 2 )  =/=  0  ->  (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  e.  CC )
4728, 46sylbir 213 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  ( ( B P A )  / 
( ( N `  B ) ^ 2 ) )  e.  CC )
485, 7ipipcj 24048 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A P B )  x.  ( B P A ) )  =  ( ( abs `  ( A P B ) ) ^ 2 ) )
493, 4, 15, 48mp3an 1309 . . . . . . . . 9  |-  ( ( A P B )  x.  ( B P A ) )  =  ( ( abs `  ( A P B ) ) ^ 2 )
5041, 30, 49mulcomli 9389 . . . . . . . 8  |-  ( ( B P A )  x.  ( A P B ) )  =  ( ( abs `  ( A P B ) ) ^ 2 )
5150oveq1i 6100 . . . . . . 7  |-  ( ( ( B P A )  x.  ( A P B ) )  /  ( ( N `
 B ) ^
2 ) )  =  ( ( ( abs `  ( A P B ) ) ^ 2 )  /  ( ( N `  B ) ^ 2 ) )
52 div23 10009 . . . . . . . . . 10  |-  ( ( ( B P A )  e.  CC  /\  ( A P B )  e.  CC  /\  (
( ( N `  B ) ^ 2 )  e.  CC  /\  ( ( N `  B ) ^ 2 )  =/=  0 ) )  ->  ( (
( B P A )  x.  ( A P B ) )  /  ( ( N `
 B ) ^
2 ) )  =  ( ( ( B P A )  / 
( ( N `  B ) ^ 2 ) )  x.  ( A P B ) ) )
5330, 41, 52mp3an12 1299 . . . . . . . . 9  |-  ( ( ( ( N `  B ) ^ 2 )  e.  CC  /\  ( ( N `  B ) ^ 2 )  =/=  0 )  ->  ( ( ( B P A )  x.  ( A P B ) )  / 
( ( N `  B ) ^ 2 ) )  =  ( ( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( A P B ) ) )
5432, 53mpan 665 . . . . . . . 8  |-  ( ( ( N `  B
) ^ 2 )  =/=  0  ->  (
( ( B P A )  x.  ( A P B ) )  /  ( ( N `
 B ) ^
2 ) )  =  ( ( ( B P A )  / 
( ( N `  B ) ^ 2 ) )  x.  ( A P B ) ) )
5528, 54sylbir 213 . . . . . . 7  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  x.  ( A P B ) )  /  ( ( N `
 B ) ^
2 ) )  =  ( ( ( B P A )  / 
( ( N `  B ) ^ 2 ) )  x.  ( A P B ) ) )
5651, 55syl5reqr 2488 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( A P B ) )  =  ( ( ( abs `  ( A P B ) ) ^ 2 )  /  ( ( N `  B ) ^ 2 ) ) )
5741abscli 12878 . . . . . . . . 9  |-  ( abs `  ( A P B ) )  e.  RR
5857resqcli 11947 . . . . . . . 8  |-  ( ( abs `  ( A P B ) ) ^ 2 )  e.  RR
5958, 31redivclzi 10093 . . . . . . 7  |-  ( ( ( N `  B
) ^ 2 )  =/=  0  ->  (
( ( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) )  e.  RR )
6028, 59sylbir 213 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) )  e.  RR )
6156, 60eqeltrd 2515 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( A P B ) )  e.  RR )
6226necon3bii 2638 . . . . . . . 8  |-  ( ( N `  B )  =/=  0  <->  B  =/=  ( 0vec `  U )
)
6319sqgt0i 11948 . . . . . . . 8  |-  ( ( N `  B )  =/=  0  ->  0  <  ( ( N `  B ) ^ 2 ) )
6462, 63sylbir 213 . . . . . . 7  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <  ( ( N `  B
) ^ 2 ) )
6557sqge0i 11949 . . . . . . . 8  |-  0  <_  ( ( abs `  ( A P B ) ) ^ 2 )
66 divge0 10194 . . . . . . . 8  |-  ( ( ( ( ( abs `  ( A P B ) ) ^ 2 )  e.  RR  /\  0  <_  ( ( abs `  ( A P B ) ) ^ 2 ) )  /\  (
( ( N `  B ) ^ 2 )  e.  RR  /\  0  <  ( ( N `
 B ) ^
2 ) ) )  ->  0  <_  (
( ( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) ) )
6758, 65, 66mpanl12 677 . . . . . . 7  |-  ( ( ( ( N `  B ) ^ 2 )  e.  RR  /\  0  <  ( ( N `
 B ) ^
2 ) )  -> 
0  <_  ( (
( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) ) )
6831, 64, 67sylancr 658 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <_  ( ( ( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) ) )
6968, 56breqtrrd 4315 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <_  ( ( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( A P B ) ) )
70 eqid 2441 . . . . . 6  |-  ( -v
`  U )  =  ( -v `  U
)
71 eqid 2441 . . . . . 6  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
725, 12, 7, 2, 4, 15, 70, 71siilem2 24187 . . . . 5  |-  ( ( ( ( B P A )  /  (
( N `  B
) ^ 2 ) )  e.  CC  /\  ( ( ( B P A )  / 
( ( N `  B ) ^ 2 ) )  x.  ( A P B ) )  e.  RR  /\  0  <_  ( ( ( B P A )  / 
( ( N `  B ) ^ 2 ) )  x.  ( A P B ) ) )  ->  ( ( B P A )  =  ( ( ( B P A )  / 
( ( N `  B ) ^ 2 ) )  x.  (
( N `  B
) ^ 2 ) )  ->  ( sqr `  ( ( A P B )  x.  (
( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( ( N `  B ) ^ 2 ) ) ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) ) )
7347, 61, 69, 72syl3anc 1213 . . . 4  |-  ( B  =/=  ( 0vec `  U
)  ->  ( ( B P A )  =  ( ( ( B P A )  / 
( ( N `  B ) ^ 2 ) )  x.  (
( N `  B
) ^ 2 ) )  ->  ( sqr `  ( ( A P B )  x.  (
( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( ( N `  B ) ^ 2 ) ) ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) ) )
7445, 73mpd 15 . . 3  |-  ( B  =/=  ( 0vec `  U
)  ->  ( sqr `  ( ( A P B )  x.  (
( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( ( N `  B ) ^ 2 ) ) ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) )
7544, 74eqbrtrd 4309 . 2  |-  ( B  =/=  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) )
7622, 75pm2.61ine 2685 1  |-  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278    x. cmul 9283    < clt 9414    <_ cle 9415    / cdiv 9989   2c2 10367   ^cexp 11861   *ccj 12581   sqrcsqr 12718   abscabs 12719   NrmCVeccnv 23897   BaseSetcba 23899   .sOLDcns 23900   0veccn0v 23901   -vcnsb 23902   normCVcnmcv 23903   .iOLDcdip 24030   CPreHil OLDccphlo 24147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-cn 18790  df-cnp 18791  df-t1 18877  df-haus 18878  df-tx 19094  df-hmeo 19287  df-xms 19854  df-ms 19855  df-tms 19856  df-grpo 23613  df-gid 23614  df-ginv 23615  df-gdiv 23616  df-ablo 23704  df-vc 23859  df-nv 23905  df-va 23908  df-ba 23909  df-sm 23910  df-0v 23911  df-vs 23912  df-nmcv 23913  df-ims 23914  df-dip 24031  df-ph 24148
This theorem is referenced by:  sii  24189  bcsiHIL  24517
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