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Theorem siilem2 22345
Description: Lemma for sii 22347. (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 6080 . . . 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 2446 . . 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 6089 . . . . 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 5724 . . . 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 4214 . . 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 312 . 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 2495 . . . . . 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 6080 . . . . . . 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 2501 . . . . . 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 4216 . . . . . 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 1254 . . . . 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 2495 . . . . . 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 6080 . . . . . . 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 2501 . . . . . 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 4216 . . . . . 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 1254 . . . . 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 9076 . . . . . 6  |-  0  e.  CC
2610phnvi 22309 . . . . . . . . 9  |-  U  e.  NrmCVec
277, 9dipcl 22203 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
2826, 11, 12, 27mp3an 1279 . . . . . . . 8  |-  ( A P B )  e.  CC
2928mul02i 9247 . . . . . . 7  |-  ( 0  x.  ( A P B ) )  =  0
30 0re 9083 . . . . . . 7  |-  0  e.  RR
3129, 30eqeltri 2505 . . . . . 6  |-  ( 0  x.  ( A P B ) )  e.  RR
32 0le0 10073 . . . . . . 7  |-  0  <_  0
3332, 29breqtrri 4229 . . . . . 6  |-  0  <_  ( 0  x.  ( A P B ) )
3425, 31, 333pm3.2i 1132 . . . . 5  |-  ( 0  e.  CC  /\  (
0  x.  ( A P B ) )  e.  RR  /\  0  <_  ( 0  x.  ( A P B ) ) )
3519, 24, 34elimhyp 3779 . . . 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 966 . . 3  |-  if ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e.  RR  /\  0  <_  ( C  x.  ( A P B ) ) ) ,  C , 
0 )  e.  CC
3735simp2i 967 . . 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 968 . . 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 22344 . 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 3772 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 936    = wceq 1652    e. wcel 1725   ifcif 3731   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982    x. cmul 8987    <_ cle 9113   2c2 10041   ^cexp 11374   sqrcsqr 12030   NrmCVeccnv 22055   BaseSetcba 22057   .s
OLDcns 22058   -vcnsb 22060   normCVcnmcv 22061   .i OLDcdip 22188   CPreHil OLDccphlo 22305
This theorem is referenced by:  siii  22346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-cn 17283  df-cnp 17284  df-t1 17370  df-haus 17371  df-tx 17586  df-hmeo 17779  df-xms 18342  df-ms 18343  df-tms 18344  df-grpo 21771  df-gid 21772  df-ginv 21773  df-gdiv 21774  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-vs 22070  df-nmcv 22071  df-ims 22072  df-dip 22189  df-ph 22306
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