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Theorem siii 21545
Description: Inference from sii 21546. (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  =  ( .i OLD `  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 5953 . . . . . 6  |-  ( B  =  ( 0vec `  U
)  ->  ( A P B )  =  ( A P ( 0vec `  U ) ) )
2 siii.9 . . . . . . . 8  |-  U  e.  CPreHil
OLD
32phnvi 21508 . . . . . . 7  |-  U  e.  NrmCVec
4 siii.a . . . . . . 7  |-  A  e.  X
5 siii.1 . . . . . . . 8  |-  X  =  ( BaseSet `  U )
6 eqid 2358 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
7 siii.7 . . . . . . . 8  |-  P  =  ( .i OLD `  U
)
85, 6, 7dip0r 21407 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P ( 0vec `  U
) )  =  0 )
93, 4, 8mp2an 653 . . . . . 6  |-  ( A P ( 0vec `  U
) )  =  0
101, 9syl6eq 2406 . . . . 5  |-  ( B  =  ( 0vec `  U
)  ->  ( A P B )  =  0 )
1110fveq2d 5612 . . . 4  |-  ( B  =  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  =  ( abs `  0 ) )
12 abs0 11866 . . . 4  |-  ( abs `  0 )  =  0
1311, 12syl6eq 2406 . . 3  |-  ( B  =  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  =  0 )
14 siii.6 . . . . . 6  |-  N  =  ( normCV `  U )
155, 14nvge0 21354 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( N `  A
) )
163, 4, 15mp2an 653 . . . 4  |-  0  <_  ( N `  A
)
17 siii.b . . . . 5  |-  B  e.  X
185, 14nvge0 21354 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  0  <_  ( N `  B
) )
193, 17, 18mp2an 653 . . . 4  |-  0  <_  ( N `  B
)
205, 14, 3, 4nvcli 21340 . . . . 5  |-  ( N `
 A )  e.  RR
215, 14, 3, 17nvcli 21340 . . . . 5  |-  ( N `
 B )  e.  RR
2220, 21mulge0i 9410 . . . 4  |-  ( ( 0  <_  ( N `  A )  /\  0  <_  ( N `  B
) )  ->  0  <_  ( ( N `  A )  x.  ( N `  B )
) )
2316, 19, 22mp2an 653 . . 3  |-  0  <_  ( ( N `  A )  x.  ( N `  B )
)
2413, 23syl6eqbr 4141 . 2  |-  ( B  =  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) )
2521recni 8939 . . . . . . . . . . 11  |-  ( N `
 B )  e.  CC
2625sqeq0i 11278 . . . . . . . . . 10  |-  ( ( ( N `  B
) ^ 2 )  =  0  <->  ( N `  B )  =  0 )
275, 6, 14nvz 21349 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( N `  B
)  =  0  <->  B  =  ( 0vec `  U
) ) )
283, 17, 27mp2an 653 . . . . . . . . . 10  |-  ( ( N `  B )  =  0  <->  B  =  ( 0vec `  U )
)
2926, 28bitri 240 . . . . . . . . 9  |-  ( ( ( N `  B
) ^ 2 )  =  0  <->  B  =  ( 0vec `  U )
)
3029necon3bii 2553 . . . . . . . 8  |-  ( ( ( N `  B
) ^ 2 )  =/=  0  <->  B  =/=  ( 0vec `  U )
)
315, 7dipcl 21402 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  A  e.  X )  ->  ( B P A )  e.  CC )
323, 17, 4, 31mp3an 1277 . . . . . . . . 9  |-  ( B P A )  e.  CC
3321resqcli 11282 . . . . . . . . . 10  |-  ( ( N `  B ) ^ 2 )  e.  RR
3433recni 8939 . . . . . . . . 9  |-  ( ( N `  B ) ^ 2 )  e.  CC
3532, 34divcan1zi 9586 . . . . . . . 8  |-  ( ( ( N `  B
) ^ 2 )  =/=  0  ->  (
( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( ( N `  B ) ^ 2 ) )  =  ( B P A ) )
3630, 35sylbir 204 . . . . . . 7  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( ( N `
 B ) ^
2 ) )  =  ( B P A ) )
375, 7dipcj 21404 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( A P B ) )  =  ( B P A ) )
383, 4, 17, 37mp3an 1277 . . . . . . 7  |-  ( * `
 ( A P B ) )  =  ( B P A )
3936, 38syl6eqr 2408 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( ( N `
 B ) ^
2 ) )  =  ( * `  ( A P B ) ) )
4039oveq2d 5961 . . . . 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 ) ) ) )
4140fveq2d 5612 . . . 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 ) ) ) ) )
425, 7dipcl 21402 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
433, 4, 17, 42mp3an 1277 . . . . 5  |-  ( A P B )  e.  CC
44 absval 11819 . . . . 5  |-  ( ( A P B )  e.  CC  ->  ( abs `  ( A P B ) )  =  ( sqr `  (
( A P B )  x.  ( * `
 ( A P B ) ) ) ) )
4543, 44ax-mp 8 . . . 4  |-  ( abs `  ( A P B ) )  =  ( sqr `  ( ( A P B )  x.  ( * `  ( A P B ) ) ) )
4641, 45syl6reqr 2409 . . 3  |-  ( B  =/=  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  =  ( sqr `  ( ( A P B )  x.  ( ( ( B P A )  /  ( ( N `
 B ) ^
2 ) )  x.  ( ( N `  B ) ^ 2 ) ) ) ) )
4736eqcomd 2363 . . . 4  |-  ( B  =/=  ( 0vec `  U
)  ->  ( B P A )  =  ( ( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( ( N `  B ) ^ 2 ) ) )
4832, 34divclzi 9585 . . . . . 6  |-  ( ( ( N `  B
) ^ 2 )  =/=  0  ->  (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  e.  CC )
4930, 48sylbir 204 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  ( ( B P A )  / 
( ( N `  B ) ^ 2 ) )  e.  CC )
505, 7ipipcj 21405 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A P B )  x.  ( B P A ) )  =  ( ( abs `  ( A P B ) ) ^ 2 ) )
513, 4, 17, 50mp3an 1277 . . . . . . . . 9  |-  ( ( A P B )  x.  ( B P A ) )  =  ( ( abs `  ( A P B ) ) ^ 2 )
5243, 32, 51mulcomli 8934 . . . . . . . 8  |-  ( ( B P A )  x.  ( A P B ) )  =  ( ( abs `  ( A P B ) ) ^ 2 )
5352oveq1i 5955 . . . . . . 7  |-  ( ( ( B P A )  x.  ( A P B ) )  /  ( ( N `
 B ) ^
2 ) )  =  ( ( ( abs `  ( A P B ) ) ^ 2 )  /  ( ( N `  B ) ^ 2 ) )
54 div23 9533 . . . . . . . . . 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 ) ) )
5532, 43, 54mp3an12 1267 . . . . . . . . 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 ) ) )
5634, 55mpan 651 . . . . . . . 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 ) ) )
5730, 56sylbir 204 . . . . . . 7  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  x.  ( A P B ) )  /  ( ( N `
 B ) ^
2 ) )  =  ( ( ( B P A )  / 
( ( N `  B ) ^ 2 ) )  x.  ( A P B ) ) )
5853, 57syl5reqr 2405 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( A P B ) )  =  ( ( ( abs `  ( A P B ) ) ^ 2 )  /  ( ( N `  B ) ^ 2 ) ) )
5943abscli 11974 . . . . . . . . 9  |-  ( abs `  ( A P B ) )  e.  RR
6059resqcli 11282 . . . . . . . 8  |-  ( ( abs `  ( A P B ) ) ^ 2 )  e.  RR
6160, 33redivclzi 9616 . . . . . . 7  |-  ( ( ( N `  B
) ^ 2 )  =/=  0  ->  (
( ( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) )  e.  RR )
6230, 61sylbir 204 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) )  e.  RR )
6358, 62eqeltrd 2432 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  ( (
( B P A )  /  ( ( N `  B ) ^ 2 ) )  x.  ( A P B ) )  e.  RR )
6428necon3bii 2553 . . . . . . . 8  |-  ( ( N `  B )  =/=  0  <->  B  =/=  ( 0vec `  U )
)
6521sqgt0i 11283 . . . . . . . 8  |-  ( ( N `  B )  =/=  0  ->  0  <  ( ( N `  B ) ^ 2 ) )
6664, 65sylbir 204 . . . . . . 7  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <  ( ( N `  B
) ^ 2 ) )
6759sqge0i 11284 . . . . . . . 8  |-  0  <_  ( ( abs `  ( A P B ) ) ^ 2 )
68 divge0 9715 . . . . . . . 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 ) ) )
6960, 67, 68mpanl12 663 . . . . . . 7  |-  ( ( ( ( N `  B ) ^ 2 )  e.  RR  /\  0  <  ( ( N `
 B ) ^
2 ) )  -> 
0  <_  ( (
( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) ) )
7033, 66, 69sylancr 644 . . . . . 6  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <_  ( ( ( abs `  ( A P B ) ) ^ 2 )  / 
( ( N `  B ) ^ 2 ) ) )
7170, 58breqtrrd 4130 . . . . 5  |-  ( B  =/=  ( 0vec `  U
)  ->  0  <_  ( ( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( A P B ) ) )
72 eqid 2358 . . . . . 6  |-  ( -v
`  U )  =  ( -v `  U
)
73 eqid 2358 . . . . . 6  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
745, 14, 7, 2, 4, 17, 72, 73siilem2 21544 . . . . 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 ) ) ) )
7549, 63, 71, 74syl3anc 1182 . . . 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 ) ) ) )
7647, 75mpd 14 . . 3  |-  ( B  =/=  ( 0vec `  U
)  ->  ( sqr `  ( ( A P B )  x.  (
( ( B P A )  /  (
( N `  B
) ^ 2 ) )  x.  ( ( N `  B ) ^ 2 ) ) ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) )
7746, 76eqbrtrd 4124 . 2  |-  ( B  =/=  ( 0vec `  U
)  ->  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) )
7824, 77pm2.61ine 2597 1  |-  ( abs `  ( A P B ) )  <_  (
( N `  A
)  x.  ( N `
 B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   CCcc 8825   RRcr 8826   0cc0 8827    x. cmul 8832    < clt 8957    <_ cle 8958    / cdiv 9513   2c2 9885   ^cexp 11197   *ccj 11677   sqrcsqr 11814   abscabs 11815   NrmCVeccnv 21254   BaseSetcba 21256   .s OLDcns 21257   0veccn0v 21258   -vcnsb 21259   normCVcnmcv 21260   .i OLDcdip 21387   CPreHil OLDccphlo 21504
This theorem is referenced by:  sii  21546  bcsiHIL  21873
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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