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Theorem siilem2 22203
Description: Lemma for sii 22205. (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 6029 . . . 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 2400 . . 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 6038 . . . . 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 5674 . . . 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 4165 . . 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 2449 . . . . . 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 6029 . . . . . . 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 2455 . . . . . 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 4167 . . . . . 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 2449 . . . . . 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 6029 . . . . . . 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 2455 . . . . . 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 4167 . . . . . 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 9019 . . . . . 6  |-  0  e.  CC
2610phnvi 22167 . . . . . . . . 9  |-  U  e.  NrmCVec
277, 9dipcl 22061 . . . . . . . . 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 9189 . . . . . . 7  |-  ( 0  x.  ( A P B ) )  =  0
30 0re 9026 . . . . . . 7  |-  0  e.  RR
3129, 30eqeltri 2459 . . . . . 6  |-  ( 0  x.  ( A P B ) )  e.  RR
32 0le0 10015 . . . . . . 7  |-  0  <_  0
3332, 29breqtrri 4180 . . . . . 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 3732 . . . 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 22202 . 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 3725 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 1649    e. wcel 1717   ifcif 3684   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   CCcc 8923   RRcr 8924   0cc0 8925    x. cmul 8930    <_ cle 9056   2c2 9983   ^cexp 11311   sqrcsqr 11967   NrmCVeccnv 21913   BaseSetcba 21915   .s
OLDcns 21916   -vcnsb 21918   normCVcnmcv 21919   .i OLDcdip 22046   CPreHil OLDccphlo 22163
This theorem is referenced by:  siii  22204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-addf 9004  ax-mulf 9005
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-sup 7383  df-oi 7414  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-ioo 10854  df-icc 10857  df-fz 10978  df-fzo 11068  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211  df-sum 12409  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-starv 13473  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-hom 13482  df-cco 13483  df-rest 13579  df-topn 13580  df-topgen 13596  df-pt 13597  df-prds 13600  df-xrs 13655  df-0g 13656  df-gsum 13657  df-qtop 13662  df-imas 13663  df-xps 13665  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-submnd 14668  df-mulg 14744  df-cntz 15045  df-cmn 15343  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-cnfld 16629  df-top 16888  df-bases 16890  df-topon 16891  df-topsp 16892  df-cld 17008  df-ntr 17009  df-cls 17010  df-cn 17215  df-cnp 17216  df-t1 17302  df-haus 17303  df-tx 17517  df-hmeo 17710  df-xms 18261  df-ms 18262  df-tms 18263  df-grpo 21629  df-gid 21630  df-ginv 21631  df-gdiv 21632  df-ablo 21720  df-vc 21875  df-nv 21921  df-va 21924  df-ba 21925  df-sm 21926  df-0v 21927  df-vs 21928  df-nmcv 21929  df-ims 21930  df-dip 22047  df-ph 22164
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