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Theorem siii 25966
Description: Inference from sii 25967. (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 6278 . . . . 5  |-  ( B  =  ( 0vec `  U
)  ->  ( A P B )  =  ( A P ( 0vec `  U ) ) )
2 siii.9 . . . . . . 7  |-  U  e.  CPreHil
OLD
32phnvi 25929 . . . . . 6  |-  U  e.  NrmCVec
4 siii.a . . . . . 6  |-  A  e.  X
5 siii.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
6 eqid 2454 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
7 siii.7 . . . . . . 7  |-  P  =  ( .iOLD `  U )
85, 6, 7dip0r 25828 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P ( 0vec `  U
) )  =  0 )
93, 4, 8mp2an 670 . . . . 5  |-  ( A P ( 0vec `  U
) )  =  0
101, 9syl6eq 2511 . . . 4  |-  ( B  =  ( 0vec `  U
)  ->  ( A P B )  =  0 )
1110abs00bd 13206 . . 3  |-  ( B  =  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  =  0 )
12 siii.6 . . . . . 6  |-  N  =  ( normCV `  U )
135, 12nvge0 25775 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( N `  A
) )
143, 4, 13mp2an 670 . . . 4  |-  0  <_  ( N `  A
)
15 siii.b . . . . 5  |-  B  e.  X
165, 12nvge0 25775 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  0  <_  ( N `  B
) )
173, 15, 16mp2an 670 . . . 4  |-  0  <_  ( N `  B
)
185, 12, 3, 4nvcli 25761 . . . . 5  |-  ( N `
 A )  e.  RR
195, 12, 3, 15nvcli 25761 . . . . 5  |-  ( N `
 B )  e.  RR
2018, 19mulge0i 10096 . . . 4  |-  ( ( 0  <_  ( N `  A )  /\  0  <_  ( N `  B
) )  ->  0  <_  ( ( N `  A )  x.  ( N `  B )
) )
2114, 17, 20mp2an 670 . . 3  |-  0  <_  ( ( N `  A )  x.  ( N `  B )
)
2211, 21syl6eqbr 4476 . 2  |-  ( B  =  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) )
2319recni 9597 . . . . . . . . . . 11  |-  ( N `
 B )  e.  CC
2423sqeq0i 12231 . . . . . . . . . 10  |-  ( ( ( N `  B
) ^ 2 )  =  0  <->  ( N `  B )  =  0 )
255, 6, 12nvz 25770 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( N `  B
)  =  0  <->  B  =  ( 0vec `  U
) ) )
263, 15, 25mp2an 670 . . . . . . . . . 10  |-  ( ( N `  B )  =  0  <->  B  =  ( 0vec `  U )
)
2724, 26bitri 249 . . . . . . . . 9  |-  ( ( ( N `  B
) ^ 2 )  =  0  <->  B  =  ( 0vec `  U )
)
2827necon3bii 2722 . . . . . . . 8  |-  ( ( ( N `  B
) ^ 2 )  =/=  0  <->  B  =/=  ( 0vec `  U )
)
295, 7dipcl 25823 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  A  e.  X )  ->  ( B P A )  e.  CC )
303, 15, 4, 29mp3an 1322 . . . . . . . . 9  |-  ( B P A )  e.  CC
3119resqcli 12235 . . . . . . . . . 10  |-  ( ( N `  B ) ^ 2 )  e.  RR
3231recni 9597 . . . . . . . . 9  |-  ( ( N `  B ) ^ 2 )  e.  CC
3330, 32divcan1zi 10276 . . . . . . . 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 25825 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( A P B ) )  =  ( B P A ) )
363, 4, 15, 35mp3an 1322 . . . . . . 7  |-  ( * `
 ( A P B ) )  =  ( B P A )
3734, 36syl6eqr 2513 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( ( N `
 B ) ^
2 ) )  =  ( * `  ( A P B ) ) )
3837oveq2d 6286 . . . . 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 5852 . . . 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 25823 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
413, 4, 15, 40mp3an 1322 . . . . 5  |-  ( A P B )  e.  CC
42 absval 13153 . . . . 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 2514 . . 3  |-  ( B  =/=  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  =  ( sqr `  ( ( A P B )  x.  ( ( ( B P A )  /  ( ( N `
 B ) ^
2 ) )  x.  ( ( N `  B ) ^ 2 ) ) ) ) )
4534eqcomd 2462 . . . 4  |-  ( B  =/=  ( 0vec `  U
)  ->  ( B P A )  =  ( ( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( ( N `  B ) ^ 2 ) ) )
4630, 32divclzi 10275 . . . . . 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 25826 . . . . . . . . . 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 1322 . . . . . . . . 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 6280 . . . . . . 7  |-  ( ( ( B P A )  x.  ( A P B ) )  /  ( ( N `
 B ) ^
2 ) )  =  ( ( ( abs `  ( A P B ) ) ^ 2 )  /  ( ( N `  B ) ^ 2 ) )
52 div23 10222 . . . . . . . . . 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 1312 . . . . . . . . 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 668 . . . . . . . 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 2510 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( A P B ) )  =  ( ( ( abs `  ( A P B ) ) ^ 2 )  /  ( ( N `  B ) ^ 2 ) ) )
5741abscli 13309 . . . . . . . . 9  |-  ( abs `  ( A P B ) )  e.  RR
5857resqcli 12235 . . . . . . . 8  |-  ( ( abs `  ( A P B ) ) ^ 2 )  e.  RR
5958, 31redivclzi 10306 . . . . . . 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 2542 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( A P B ) )  e.  RR )
6226necon3bii 2722 . . . . . . . 8  |-  ( ( N `  B )  =/=  0  <->  B  =/=  ( 0vec `  U )
)
6319sqgt0i 12236 . . . . . . . 8  |-  ( ( N `  B )  =/=  0  ->  0  <  ( ( N `  B ) ^ 2 ) )
6462, 63sylbir 213 . . . . . . 7  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <  ( ( N `  B
) ^ 2 ) )
6557sqge0i 12237 . . . . . . . 8  |-  0  <_  ( ( abs `  ( A P B ) ) ^ 2 )
66 divge0 10407 . . . . . . . 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 680 . . . . . . 7  |-  ( ( ( ( N `  B ) ^ 2 )  e.  RR  /\  0  <  ( ( N `
 B ) ^
2 ) )  -> 
0  <_  ( (
( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) ) )
6831, 64, 67sylancr 661 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <_  ( ( ( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) ) )
6968, 56breqtrrd 4465 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <_  ( ( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( A P B ) ) )
70 eqid 2454 . . . . . 6  |-  ( -v
`  U )  =  ( -v `  U
)
71 eqid 2454 . . . . . 6  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
725, 12, 7, 2, 4, 15, 70, 71siilem2 25965 . . . . 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 1226 . . . 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 4459 . 2  |-  ( B  =/=  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) )
7622, 75pm2.61ine 2767 1  |-  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486    < clt 9617    <_ cle 9618    / cdiv 10202   2c2 10581   ^cexp 12148   *ccj 13011   sqrcsqrt 13148   abscabs 13149   NrmCVeccnv 25675   BaseSetcba 25677   .sOLDcns 25678   0veccn0v 25679   -vcnsb 25680   normCVcnmcv 25681   .iOLDcdip 25808   CPreHil OLDccphlo 25925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-cn 19895  df-cnp 19896  df-t1 19982  df-haus 19983  df-tx 20229  df-hmeo 20422  df-xms 20989  df-ms 20990  df-tms 20991  df-grpo 25391  df-gid 25392  df-ginv 25393  df-gdiv 25394  df-ablo 25482  df-vc 25637  df-nv 25683  df-va 25686  df-ba 25687  df-sm 25688  df-0v 25689  df-vs 25690  df-nmcv 25691  df-ims 25692  df-dip 25809  df-ph 25926
This theorem is referenced by:  sii  25967  bcsiHIL  26295
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