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Theorem siii 25430
Description: Inference from sii 25431. (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 6283 . . . . 5  |-  ( B  =  ( 0vec `  U
)  ->  ( A P B )  =  ( A P ( 0vec `  U ) ) )
2 siii.9 . . . . . . 7  |-  U  e.  CPreHil
OLD
32phnvi 25393 . . . . . 6  |-  U  e.  NrmCVec
4 siii.a . . . . . 6  |-  A  e.  X
5 siii.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
6 eqid 2460 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
7 siii.7 . . . . . . 7  |-  P  =  ( .iOLD `  U )
85, 6, 7dip0r 25292 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P ( 0vec `  U
) )  =  0 )
93, 4, 8mp2an 672 . . . . 5  |-  ( A P ( 0vec `  U
) )  =  0
101, 9syl6eq 2517 . . . 4  |-  ( B  =  ( 0vec `  U
)  ->  ( A P B )  =  0 )
1110abs00bd 13074 . . 3  |-  ( B  =  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  =  0 )
12 siii.6 . . . . . 6  |-  N  =  ( normCV `  U )
135, 12nvge0 25239 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( N `  A
) )
143, 4, 13mp2an 672 . . . 4  |-  0  <_  ( N `  A
)
15 siii.b . . . . 5  |-  B  e.  X
165, 12nvge0 25239 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  0  <_  ( N `  B
) )
173, 15, 16mp2an 672 . . . 4  |-  0  <_  ( N `  B
)
185, 12, 3, 4nvcli 25225 . . . . 5  |-  ( N `
 A )  e.  RR
195, 12, 3, 15nvcli 25225 . . . . 5  |-  ( N `
 B )  e.  RR
2018, 19mulge0i 10089 . . . 4  |-  ( ( 0  <_  ( N `  A )  /\  0  <_  ( N `  B
) )  ->  0  <_  ( ( N `  A )  x.  ( N `  B )
) )
2114, 17, 20mp2an 672 . . 3  |-  0  <_  ( ( N `  A )  x.  ( N `  B )
)
2211, 21syl6eqbr 4477 . 2  |-  ( B  =  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) )
2319recni 9597 . . . . . . . . . . 11  |-  ( N `
 B )  e.  CC
2423sqeq0i 12204 . . . . . . . . . 10  |-  ( ( ( N `  B
) ^ 2 )  =  0  <->  ( N `  B )  =  0 )
255, 6, 12nvz 25234 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( N `  B
)  =  0  <->  B  =  ( 0vec `  U
) ) )
263, 15, 25mp2an 672 . . . . . . . . . 10  |-  ( ( N `  B )  =  0  <->  B  =  ( 0vec `  U )
)
2724, 26bitri 249 . . . . . . . . 9  |-  ( ( ( N `  B
) ^ 2 )  =  0  <->  B  =  ( 0vec `  U )
)
2827necon3bii 2728 . . . . . . . 8  |-  ( ( ( N `  B
) ^ 2 )  =/=  0  <->  B  =/=  ( 0vec `  U )
)
295, 7dipcl 25287 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  A  e.  X )  ->  ( B P A )  e.  CC )
303, 15, 4, 29mp3an 1319 . . . . . . . . 9  |-  ( B P A )  e.  CC
3119resqcli 12208 . . . . . . . . . 10  |-  ( ( N `  B ) ^ 2 )  e.  RR
3231recni 9597 . . . . . . . . 9  |-  ( ( N `  B ) ^ 2 )  e.  CC
3330, 32divcan1zi 10269 . . . . . . . 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 25289 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( A P B ) )  =  ( B P A ) )
363, 4, 15, 35mp3an 1319 . . . . . . 7  |-  ( * `
 ( A P B ) )  =  ( B P A )
3734, 36syl6eqr 2519 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( ( N `
 B ) ^
2 ) )  =  ( * `  ( A P B ) ) )
3837oveq2d 6291 . . . . 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 5861 . . . 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 25287 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
413, 4, 15, 40mp3an 1319 . . . . 5  |-  ( A P B )  e.  CC
42 absval 13021 . . . . 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 2520 . . 3  |-  ( B  =/=  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  =  ( sqr `  ( ( A P B )  x.  ( ( ( B P A )  /  ( ( N `
 B ) ^
2 ) )  x.  ( ( N `  B ) ^ 2 ) ) ) ) )
4534eqcomd 2468 . . . 4  |-  ( B  =/=  ( 0vec `  U
)  ->  ( B P A )  =  ( ( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( ( N `  B ) ^ 2 ) ) )
4630, 32divclzi 10268 . . . . . 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 25290 . . . . . . . . . 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 1319 . . . . . . . . 9  |-  ( ( A P B )  x.  ( B P A ) )  =  ( ( abs `  ( A P B ) ) ^ 2 )
5041, 30, 49mulcomli 9592 . . . . . . . 8  |-  ( ( B P A )  x.  ( A P B ) )  =  ( ( abs `  ( A P B ) ) ^ 2 )
5150oveq1i 6285 . . . . . . 7  |-  ( ( ( B P A )  x.  ( A P B ) )  /  ( ( N `
 B ) ^
2 ) )  =  ( ( ( abs `  ( A P B ) ) ^ 2 )  /  ( ( N `  B ) ^ 2 ) )
52 div23 10215 . . . . . . . . . 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 1309 . . . . . . . . 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 670 . . . . . . . 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 2516 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( A P B ) )  =  ( ( ( abs `  ( A P B ) ) ^ 2 )  /  ( ( N `  B ) ^ 2 ) ) )
5741abscli 13176 . . . . . . . . 9  |-  ( abs `  ( A P B ) )  e.  RR
5857resqcli 12208 . . . . . . . 8  |-  ( ( abs `  ( A P B ) ) ^ 2 )  e.  RR
5958, 31redivclzi 10299 . . . . . . 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 2548 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( A P B ) )  e.  RR )
6226necon3bii 2728 . . . . . . . 8  |-  ( ( N `  B )  =/=  0  <->  B  =/=  ( 0vec `  U )
)
6319sqgt0i 12209 . . . . . . . 8  |-  ( ( N `  B )  =/=  0  ->  0  <  ( ( N `  B ) ^ 2 ) )
6462, 63sylbir 213 . . . . . . 7  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <  ( ( N `  B
) ^ 2 ) )
6557sqge0i 12210 . . . . . . . 8  |-  0  <_  ( ( abs `  ( A P B ) ) ^ 2 )
66 divge0 10400 . . . . . . . 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 682 . . . . . . 7  |-  ( ( ( ( N `  B ) ^ 2 )  e.  RR  /\  0  <  ( ( N `
 B ) ^
2 ) )  -> 
0  <_  ( (
( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) ) )
6831, 64, 67sylancr 663 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <_  ( ( ( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) ) )
6968, 56breqtrrd 4466 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <_  ( ( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( A P B ) ) )
70 eqid 2460 . . . . . 6  |-  ( -v
`  U )  =  ( -v `  U
)
71 eqid 2460 . . . . . 6  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
725, 12, 7, 2, 4, 15, 70, 71siilem2 25429 . . . . 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 1223 . . . 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 4460 . 2  |-  ( B  =/=  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) )
7622, 75pm2.61ine 2773 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 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486    < clt 9617    <_ cle 9618    / cdiv 10195   2c2 10574   ^cexp 12122   *ccj 12879   sqrcsqr 13016   abscabs 13017   NrmCVeccnv 25139   BaseSetcba 25141   .sOLDcns 25142   0veccn0v 25143   -vcnsb 25144   normCVcnmcv 25145   .iOLDcdip 25272   CPreHil OLDccphlo 25389
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-cn 19487  df-cnp 19488  df-t1 19574  df-haus 19575  df-tx 19791  df-hmeo 19984  df-xms 20551  df-ms 20552  df-tms 20553  df-grpo 24855  df-gid 24856  df-ginv 24857  df-gdiv 24858  df-ablo 24946  df-vc 25101  df-nv 25147  df-va 25150  df-ba 25151  df-sm 25152  df-0v 25153  df-vs 25154  df-nmcv 25155  df-ims 25156  df-dip 25273  df-ph 25390
This theorem is referenced by:  sii  25431  bcsiHIL  25759
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