MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  siilem2 Structured version   Unicode version

Theorem siilem2 25590
Description: Lemma for sii 25592. (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  =  ( .iOLD `  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  =  ( .sOLD `  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 6302 . . . 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 2481 . . 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 6311 . . . . 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 5876 . . . 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 4463 . . 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 320 . 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  =  ( .iOLD `  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  =  ( .sOLD `  U )
15 eleq1 2539 . . . . . 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 6302 . . . . . . 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 2536 . . . . . 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 4465 . . . . . 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 1299 . . . . 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 2539 . . . . . 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 6302 . . . . . . 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 2536 . . . . . 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 4465 . . . . . 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 1299 . . . . 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 9600 . . . . . 6  |-  0  e.  CC
2610phnvi 25554 . . . . . . . . 9  |-  U  e.  NrmCVec
277, 9dipcl 25448 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
2826, 11, 12, 27mp3an 1324 . . . . . . . 8  |-  ( A P B )  e.  CC
2928mul02i 9780 . . . . . . 7  |-  ( 0  x.  ( A P B ) )  =  0
30 0re 9608 . . . . . . 7  |-  0  e.  RR
3129, 30eqeltri 2551 . . . . . 6  |-  ( 0  x.  ( A P B ) )  e.  RR
32 0le0 10637 . . . . . . 7  |-  0  <_  0
3332, 29breqtrri 4478 . . . . . 6  |-  0  <_  ( 0  x.  ( A P B ) )
3425, 31, 333pm3.2i 1174 . . . . 5  |-  ( 0  e.  CC  /\  (
0  x.  ( A P B ) )  e.  RR  /\  0  <_  ( 0  x.  ( A P B ) ) )
3519, 24, 34elimhyp 4004 . . . 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 1005 . . 3  |-  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  e.  CC
3735simp2i 1006 . . 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 1007 . . 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 25589 . 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 3997 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   ifcif 3945   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504    x. cmul 9509    <_ cle 9641   2c2 10597   ^cexp 12146   sqrcsqrt 13046   NrmCVeccnv 25300   BaseSetcba 25302   .sOLDcns 25303   -vcnsb 25305   normCVcnmcv 25306   .iOLDcdip 25433   CPreHil OLDccphlo 25550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-icc 11548  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-cn 19596  df-cnp 19597  df-t1 19683  df-haus 19684  df-tx 19931  df-hmeo 20124  df-xms 20691  df-ms 20692  df-tms 20693  df-grpo 25016  df-gid 25017  df-ginv 25018  df-gdiv 25019  df-ablo 25107  df-vc 25262  df-nv 25308  df-va 25311  df-ba 25312  df-sm 25313  df-0v 25314  df-vs 25315  df-nmcv 25316  df-ims 25317  df-dip 25434  df-ph 25551
This theorem is referenced by:  siii  25591
  Copyright terms: Public domain W3C validator