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Theorem siii 24256
Description: Inference from sii 24257. (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 6102 . . . . 5  |-  ( B  =  ( 0vec `  U
)  ->  ( A P B )  =  ( A P ( 0vec `  U ) ) )
2 siii.9 . . . . . . 7  |-  U  e.  CPreHil
OLD
32phnvi 24219 . . . . . 6  |-  U  e.  NrmCVec
4 siii.a . . . . . 6  |-  A  e.  X
5 siii.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
6 eqid 2443 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
7 siii.7 . . . . . . 7  |-  P  =  ( .iOLD `  U )
85, 6, 7dip0r 24118 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P ( 0vec `  U
) )  =  0 )
93, 4, 8mp2an 672 . . . . 5  |-  ( A P ( 0vec `  U
) )  =  0
101, 9syl6eq 2491 . . . 4  |-  ( B  =  ( 0vec `  U
)  ->  ( A P B )  =  0 )
1110abs00bd 12783 . . 3  |-  ( B  =  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  =  0 )
12 siii.6 . . . . . 6  |-  N  =  ( normCV `  U )
135, 12nvge0 24065 . . . . 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 24065 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  0  <_  ( N `  B
) )
173, 15, 16mp2an 672 . . . 4  |-  0  <_  ( N `  B
)
185, 12, 3, 4nvcli 24051 . . . . 5  |-  ( N `
 A )  e.  RR
195, 12, 3, 15nvcli 24051 . . . . 5  |-  ( N `
 B )  e.  RR
2018, 19mulge0i 9890 . . . 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 4332 . 2  |-  ( B  =  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) )
2319recni 9401 . . . . . . . . . . 11  |-  ( N `
 B )  e.  CC
2423sqeq0i 11950 . . . . . . . . . 10  |-  ( ( ( N `  B
) ^ 2 )  =  0  <->  ( N `  B )  =  0 )
255, 6, 12nvz 24060 . . . . . . . . . . 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 2643 . . . . . . . 8  |-  ( ( ( N `  B
) ^ 2 )  =/=  0  <->  B  =/=  ( 0vec `  U )
)
295, 7dipcl 24113 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  A  e.  X )  ->  ( B P A )  e.  CC )
303, 15, 4, 29mp3an 1314 . . . . . . . . 9  |-  ( B P A )  e.  CC
3119resqcli 11954 . . . . . . . . . 10  |-  ( ( N `  B ) ^ 2 )  e.  RR
3231recni 9401 . . . . . . . . 9  |-  ( ( N `  B ) ^ 2 )  e.  CC
3330, 32divcan1zi 10070 . . . . . . . 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 24115 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( A P B ) )  =  ( B P A ) )
363, 4, 15, 35mp3an 1314 . . . . . . 7  |-  ( * `
 ( A P B ) )  =  ( B P A )
3734, 36syl6eqr 2493 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( ( N `
 B ) ^
2 ) )  =  ( * `  ( A P B ) ) )
3837oveq2d 6110 . . . . 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 5698 . . . 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 24113 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
413, 4, 15, 40mp3an 1314 . . . . 5  |-  ( A P B )  e.  CC
42 absval 12730 . . . . 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 2494 . . 3  |-  ( B  =/=  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  =  ( sqr `  ( ( A P B )  x.  ( ( ( B P A )  /  ( ( N `
 B ) ^
2 ) )  x.  ( ( N `  B ) ^ 2 ) ) ) ) )
4534eqcomd 2448 . . . 4  |-  ( B  =/=  ( 0vec `  U
)  ->  ( B P A )  =  ( ( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( ( N `  B ) ^ 2 ) ) )
4630, 32divclzi 10069 . . . . . 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 24116 . . . . . . . . . 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 1314 . . . . . . . . 9  |-  ( ( A P B )  x.  ( B P A ) )  =  ( ( abs `  ( A P B ) ) ^ 2 )
5041, 30, 49mulcomli 9396 . . . . . . . 8  |-  ( ( B P A )  x.  ( A P B ) )  =  ( ( abs `  ( A P B ) ) ^ 2 )
5150oveq1i 6104 . . . . . . 7  |-  ( ( ( B P A )  x.  ( A P B ) )  /  ( ( N `
 B ) ^
2 ) )  =  ( ( ( abs `  ( A P B ) ) ^ 2 )  /  ( ( N `  B ) ^ 2 ) )
52 div23 10016 . . . . . . . . . 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 1304 . . . . . . . . 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 2490 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( A P B ) )  =  ( ( ( abs `  ( A P B ) ) ^ 2 )  /  ( ( N `  B ) ^ 2 ) ) )
5741abscli 12885 . . . . . . . . 9  |-  ( abs `  ( A P B ) )  e.  RR
5857resqcli 11954 . . . . . . . 8  |-  ( ( abs `  ( A P B ) ) ^ 2 )  e.  RR
5958, 31redivclzi 10100 . . . . . . 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 2517 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( A P B ) )  e.  RR )
6226necon3bii 2643 . . . . . . . 8  |-  ( ( N `  B )  =/=  0  <->  B  =/=  ( 0vec `  U )
)
6319sqgt0i 11955 . . . . . . . 8  |-  ( ( N `  B )  =/=  0  ->  0  <  ( ( N `  B ) ^ 2 ) )
6462, 63sylbir 213 . . . . . . 7  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <  ( ( N `  B
) ^ 2 ) )
6557sqge0i 11956 . . . . . . . 8  |-  0  <_  ( ( abs `  ( A P B ) ) ^ 2 )
66 divge0 10201 . . . . . . . 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 4321 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <_  ( ( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( A P B ) ) )
70 eqid 2443 . . . . . 6  |-  ( -v
`  U )  =  ( -v `  U
)
71 eqid 2443 . . . . . 6  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
725, 12, 7, 2, 4, 15, 70, 71siilem2 24255 . . . . 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 1218 . . . 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 4315 . 2  |-  ( B  =/=  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) )
7622, 75pm2.61ine 2690 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 1369    e. wcel 1756    =/= wne 2609   class class class wbr 4295   ` cfv 5421  (class class class)co 6094   CCcc 9283   RRcr 9284   0cc0 9285    x. cmul 9290    < clt 9421    <_ cle 9422    / cdiv 9996   2c2 10374   ^cexp 11868   *ccj 12588   sqrcsqr 12725   abscabs 12726   NrmCVeccnv 23965   BaseSetcba 23967   .sOLDcns 23968   0veccn0v 23969   -vcnsb 23970   normCVcnmcv 23971   .iOLDcdip 24098   CPreHil OLDccphlo 24215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364  ax-mulf 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-iin 4177  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  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-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-supp 6694  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-map 7219  df-ixp 7267  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-fsupp 7624  df-fi 7664  df-sup 7694  df-oi 7727  df-card 8112  df-cda 8340  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-q 10957  df-rp 10995  df-xneg 11092  df-xadd 11093  df-xmul 11094  df-ioo 11307  df-icc 11310  df-fz 11441  df-fzo 11552  df-seq 11810  df-exp 11869  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-sum 13167  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-starv 14256  df-sca 14257  df-vsca 14258  df-ip 14259  df-tset 14260  df-ple 14261  df-ds 14263  df-unif 14264  df-hom 14265  df-cco 14266  df-rest 14364  df-topn 14365  df-0g 14383  df-gsum 14384  df-topgen 14385  df-pt 14386  df-prds 14389  df-xrs 14443  df-qtop 14448  df-imas 14449  df-xps 14451  df-mre 14527  df-mrc 14528  df-acs 14530  df-mnd 15418  df-submnd 15468  df-mulg 15551  df-cntz 15838  df-cmn 16282  df-psmet 17812  df-xmet 17813  df-met 17814  df-bl 17815  df-mopn 17816  df-cnfld 17822  df-top 18506  df-bases 18508  df-topon 18509  df-topsp 18510  df-cld 18626  df-ntr 18627  df-cls 18628  df-cn 18834  df-cnp 18835  df-t1 18921  df-haus 18922  df-tx 19138  df-hmeo 19331  df-xms 19898  df-ms 19899  df-tms 19900  df-grpo 23681  df-gid 23682  df-ginv 23683  df-gdiv 23684  df-ablo 23772  df-vc 23927  df-nv 23973  df-va 23976  df-ba 23977  df-sm 23978  df-0v 23979  df-vs 23980  df-nmcv 23981  df-ims 23982  df-dip 24099  df-ph 24216
This theorem is referenced by:  sii  24257  bcsiHIL  24585
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