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Theorem siilem2 21430
Description: Lemma for sii 21432. (Contributed by NM, 24-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
siii2.3  |-  M  =  ( -v `  U
)
siii2.4  |-  S  =  ( .s OLD `  U
)
Assertion
Ref Expression
siilem2  |-  ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) )  ->  ( ( B P A )  =  ( C  x.  (
( N `  B
) ^ 2 ) )  ->  ( sqr `  ( ( A P B )  x.  ( C  x.  ( ( N `  B ) ^ 2 ) ) ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) ) )

Proof of Theorem siilem2
StepHypRef Expression
1 oveq1 5865 . . . 4  |-  ( C  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  ( C  x.  ( ( N `  B ) ^ 2 ) )  =  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  (
( N `  B
) ^ 2 ) ) )
21eqeq2d 2294 . . 3  |-  ( C  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  (
( B P A )  =  ( C  x.  ( ( N `
 B ) ^
2 ) )  <->  ( B P A )  =  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_ 
( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  (
( N `  B
) ^ 2 ) ) ) )
31oveq2d 5874 . . . . 5  |-  ( C  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  (
( A P B )  x.  ( C  x.  ( ( N `
 B ) ^
2 ) ) )  =  ( ( A P B )  x.  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  (
( N `  B
) ^ 2 ) ) ) )
43fveq2d 5529 . . . 4  |-  ( C  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  ( sqr `  ( ( A P B )  x.  ( C  x.  (
( N `  B
) ^ 2 ) ) ) )  =  ( sqr `  (
( A P B )  x.  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_ 
( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  (
( N `  B
) ^ 2 ) ) ) ) )
54breq1d 4033 . . 3  |-  ( C  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  (
( sqr `  (
( A P B )  x.  ( C  x.  ( ( N `
 B ) ^
2 ) ) ) )  <_  ( ( N `  A )  x.  ( N `  B
) )  <->  ( sqr `  ( ( A P B )  x.  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_ 
( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  (
( N `  B
) ^ 2 ) ) ) )  <_ 
( ( N `  A )  x.  ( N `  B )
) ) )
62, 5imbi12d 311 . 2  |-  ( C  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  (
( ( B P A )  =  ( C  x.  ( ( N `  B ) ^ 2 ) )  ->  ( sqr `  (
( A P B )  x.  ( C  x.  ( ( N `
 B ) ^
2 ) ) ) )  <_  ( ( N `  A )  x.  ( N `  B
) ) )  <->  ( ( B P A )  =  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  (
( N `  B
) ^ 2 ) )  ->  ( sqr `  ( ( A P B )  x.  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_ 
( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  (
( N `  B
) ^ 2 ) ) ) )  <_ 
( ( N `  A )  x.  ( N `  B )
) ) ) )
7 siii.1 . . 3  |-  X  =  ( BaseSet `  U )
8 siii.6 . . 3  |-  N  =  ( normCV `  U )
9 siii.7 . . 3  |-  P  =  ( .i OLD `  U
)
10 siii.9 . . 3  |-  U  e.  CPreHil
OLD
11 siii.a . . 3  |-  A  e.  X
12 siii.b . . 3  |-  B  e.  X
13 siii2.3 . . 3  |-  M  =  ( -v `  U
)
14 siii2.4 . . 3  |-  S  =  ( .s OLD `  U
)
15 eleq1 2343 . . . . . 6  |-  ( C  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  ( C  e.  CC  <->  if (
( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  e.  CC ) )
16 oveq1 5865 . . . . . . 7  |-  ( C  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  ( C  x.  ( A P B ) )  =  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) ) )
1716eleq1d 2349 . . . . . 6  |-  ( C  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  (
( C  x.  ( A P B ) )  e.  RR  <->  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_ 
( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) )  e.  RR ) )
1816breq2d 4035 . . . . . 6  |-  ( C  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  (
0  <_  ( C  x.  ( A P B ) )  <->  0  <_  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_ 
( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) ) ) )
1915, 17, 183anbi123d 1252 . . . . 5  |-  ( C  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  (
( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) )  <->  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  e.  CC  /\  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) )  e.  RR  /\  0  <_  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) ) ) ) )
20 eleq1 2343 . . . . . 6  |-  ( 0  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  (
0  e.  CC  <->  if (
( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  e.  CC ) )
21 oveq1 5865 . . . . . . 7  |-  ( 0  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  (
0  x.  ( A P B ) )  =  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) ) )
2221eleq1d 2349 . . . . . 6  |-  ( 0  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  (
( 0  x.  ( A P B ) )  e.  RR  <->  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_ 
( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) )  e.  RR ) )
2321breq2d 4035 . . . . . 6  |-  ( 0  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  (
0  <_  ( 0  x.  ( A P B ) )  <->  0  <_  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_ 
( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) ) ) )
2420, 22, 233anbi123d 1252 . . . . 5  |-  ( 0  =  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  ->  (
( 0  e.  CC  /\  ( 0  x.  ( A P B ) )  e.  RR  /\  0  <_  ( 0  x.  ( A P B ) ) )  <->  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  e.  CC  /\  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) )  e.  RR  /\  0  <_  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) ) ) ) )
25 0cn 8831 . . . . . 6  |-  0  e.  CC
2610phnvi 21394 . . . . . . . . 9  |-  U  e.  NrmCVec
277, 9dipcl 21288 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
2826, 11, 12, 27mp3an 1277 . . . . . . . 8  |-  ( A P B )  e.  CC
2928mul02i 9001 . . . . . . 7  |-  ( 0  x.  ( A P B ) )  =  0
30 0re 8838 . . . . . . 7  |-  0  e.  RR
3129, 30eqeltri 2353 . . . . . 6  |-  ( 0  x.  ( A P B ) )  e.  RR
32 0le0 9827 . . . . . . 7  |-  0  <_  0
3332, 29breqtrri 4048 . . . . . 6  |-  0  <_  ( 0  x.  ( A P B ) )
3425, 31, 333pm3.2i 1130 . . . . 5  |-  ( 0  e.  CC  /\  (
0  x.  ( A P B ) )  e.  RR  /\  0  <_  ( 0  x.  ( A P B ) ) )
3519, 24, 34elimhyp 3613 . . . 4  |-  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_ 
( C  x.  ( A P B ) ) ) ,  C , 
0 )  e.  CC  /\  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) )  e.  RR  /\  0  <_  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) ) )
3635simp1i 964 . . 3  |-  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  e.  CC
3735simp2i 965 . . 3  |-  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_ 
( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) )  e.  RR
3835simp3i 966 . . 3  |-  0  <_  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  ( A P B ) )
397, 8, 9, 10, 11, 12, 13, 14, 36, 37, 38siilem1 21429 . 2  |-  ( ( B P A )  =  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  (
( N `  B
) ^ 2 ) )  ->  ( sqr `  ( ( A P B )  x.  ( if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_ 
( C  x.  ( A P B ) ) ) ,  C , 
0 )  x.  (
( N `  B
) ^ 2 ) ) ) )  <_ 
( ( N `  A )  x.  ( N `  B )
) )
406, 39dedth 3606 1  |-  ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) )  ->  ( ( B P A )  =  ( C  x.  (
( N `  B
) ^ 2 ) )  ->  ( sqr `  ( ( A P B )  x.  ( C  x.  ( ( N `  B ) ^ 2 ) ) ) )  <_  (
( N `  A
)  x.  ( N `
 B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   ifcif 3565   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742    <_ cle 8868   2c2 9795   ^cexp 11104   sqrcsqr 11718   NrmCVeccnv 21140   BaseSetcba 21142   .s
OLDcns 21143   -vcnsb 21145   normCVcnmcv 21146   .i OLDcdip 21273   CPreHil OLDccphlo 21390
This theorem is referenced by:  siii  21431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-cn 16957  df-cnp 16958  df-t1 17042  df-haus 17043  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-dip 21274  df-ph 21391
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