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Theorem siilem2 24257
Description: Lemma for sii 24259. (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 6103 . . . 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 2454 . . 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 6112 . . . . 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 5700 . . . 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 4307 . . 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 2503 . . . . . 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 6103 . . . . . . 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 2509 . . . . . 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 4309 . . . . . 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 1289 . . . . 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 2503 . . . . . 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 6103 . . . . . . 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 2509 . . . . . 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 4309 . . . . . 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 1289 . . . . 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 9383 . . . . . 6  |-  0  e.  CC
2610phnvi 24221 . . . . . . . . 9  |-  U  e.  NrmCVec
277, 9dipcl 24115 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
2826, 11, 12, 27mp3an 1314 . . . . . . . 8  |-  ( A P B )  e.  CC
2928mul02i 9563 . . . . . . 7  |-  ( 0  x.  ( A P B ) )  =  0
30 0re 9391 . . . . . . 7  |-  0  e.  RR
3129, 30eqeltri 2513 . . . . . 6  |-  ( 0  x.  ( A P B ) )  e.  RR
32 0le0 10416 . . . . . . 7  |-  0  <_  0
3332, 29breqtrri 4322 . . . . . 6  |-  0  <_  ( 0  x.  ( A P B ) )
3425, 31, 333pm3.2i 1166 . . . . 5  |-  ( 0  e.  CC  /\  (
0  x.  ( A P B ) )  e.  RR  /\  0  <_  ( 0  x.  ( A P B ) ) )
3519, 24, 34elimhyp 3853 . . . 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 997 . . 3  |-  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  e.  CC
3735simp2i 998 . . 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 999 . . 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 24256 . 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 3846 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 965    = wceq 1369    e. wcel 1756   ifcif 3796   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287    x. cmul 9292    <_ cle 9424   2c2 10376   ^cexp 11870   sqrcsqr 12727   NrmCVeccnv 23967   BaseSetcba 23969   .sOLDcns 23970   -vcnsb 23972   normCVcnmcv 23973   .iOLDcdip 24100   CPreHil OLDccphlo 24217
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-icc 11312  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-cn 18836  df-cnp 18837  df-t1 18923  df-haus 18924  df-tx 19140  df-hmeo 19333  df-xms 19900  df-ms 19901  df-tms 19902  df-grpo 23683  df-gid 23684  df-ginv 23685  df-gdiv 23686  df-ablo 23774  df-vc 23929  df-nv 23975  df-va 23978  df-ba 23979  df-sm 23980  df-0v 23981  df-vs 23982  df-nmcv 23983  df-ims 23984  df-dip 24101  df-ph 24218
This theorem is referenced by:  siii  24258
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