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Theorem normlem6 22607
Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
normlem1.1  |-  S  e.  CC
normlem1.2  |-  F  e. 
~H
normlem1.3  |-  G  e. 
~H
normlem2.4  |-  B  = 
-u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) )
normlem3.5  |-  A  =  ( G  .ih  G
)
normlem3.6  |-  C  =  ( F  .ih  F
)
normlem6.7  |-  ( abs `  S )  =  1
Assertion
Ref Expression
normlem6  |-  ( abs `  B )  <_  (
2  x.  ( ( sqr `  A )  x.  ( sqr `  C
) ) )

Proof of Theorem normlem6
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 normlem3.5 . . . . . . . . 9  |-  A  =  ( G  .ih  G
)
2 normlem1.3 . . . . . . . . . 10  |-  G  e. 
~H
3 hiidrcl 22587 . . . . . . . . . 10  |-  ( G  e.  ~H  ->  ( G  .ih  G )  e.  RR )
42, 3ax-mp 8 . . . . . . . . 9  |-  ( G 
.ih  G )  e.  RR
51, 4eqeltri 2505 . . . . . . . 8  |-  A  e.  RR
65a1i 11 . . . . . . 7  |-  (  T. 
->  A  e.  RR )
7 normlem1.1 . . . . . . . . 9  |-  S  e.  CC
8 normlem1.2 . . . . . . . . 9  |-  F  e. 
~H
9 normlem2.4 . . . . . . . . 9  |-  B  = 
-u ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G 
.ih  F ) ) )
107, 8, 2, 9normlem2 22603 . . . . . . . 8  |-  B  e.  RR
1110a1i 11 . . . . . . 7  |-  (  T. 
->  B  e.  RR )
12 normlem3.6 . . . . . . . . 9  |-  C  =  ( F  .ih  F
)
13 hiidrcl 22587 . . . . . . . . . 10  |-  ( F  e.  ~H  ->  ( F  .ih  F )  e.  RR )
148, 13ax-mp 8 . . . . . . . . 9  |-  ( F 
.ih  F )  e.  RR
1512, 14eqeltri 2505 . . . . . . . 8  |-  C  e.  RR
1615a1i 11 . . . . . . 7  |-  (  T. 
->  C  e.  RR )
17 oveq1 6080 . . . . . . . . . . . . 13  |-  ( x  =  if ( x  e.  RR ,  x ,  0 )  -> 
( x ^ 2 )  =  ( if ( x  e.  RR ,  x ,  0 ) ^ 2 ) )
1817oveq2d 6089 . . . . . . . . . . . 12  |-  ( x  =  if ( x  e.  RR ,  x ,  0 )  -> 
( A  x.  (
x ^ 2 ) )  =  ( A  x.  ( if ( x  e.  RR ,  x ,  0 ) ^ 2 ) ) )
19 oveq2 6081 . . . . . . . . . . . 12  |-  ( x  =  if ( x  e.  RR ,  x ,  0 )  -> 
( B  x.  x
)  =  ( B  x.  if ( x  e.  RR ,  x ,  0 ) ) )
2018, 19oveq12d 6091 . . . . . . . . . . 11  |-  ( x  =  if ( x  e.  RR ,  x ,  0 )  -> 
( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  =  ( ( A  x.  ( if ( x  e.  RR ,  x ,  0 ) ^ 2 ) )  +  ( B  x.  if ( x  e.  RR ,  x ,  0 ) ) ) )
2120oveq1d 6088 . . . . . . . . . 10  |-  ( x  =  if ( x  e.  RR ,  x ,  0 )  -> 
( ( ( A  x.  ( x ^
2 ) )  +  ( B  x.  x
) )  +  C
)  =  ( ( ( A  x.  ( if ( x  e.  RR ,  x ,  0 ) ^ 2 ) )  +  ( B  x.  if ( x  e.  RR ,  x ,  0 ) ) )  +  C
) )
2221breq2d 4216 . . . . . . . . 9  |-  ( x  =  if ( x  e.  RR ,  x ,  0 )  -> 
( 0  <_  (
( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C )  <->  0  <_  ( (
( A  x.  ( if ( x  e.  RR ,  x ,  0 ) ^ 2 ) )  +  ( B  x.  if ( x  e.  RR ,  x ,  0 ) ) )  +  C
) ) )
23 0re 9081 . . . . . . . . . . 11  |-  0  e.  RR
2423elimel 3783 . . . . . . . . . 10  |-  if ( x  e.  RR ,  x ,  0 )  e.  RR
25 normlem6.7 . . . . . . . . . 10  |-  ( abs `  S )  =  1
267, 8, 2, 9, 1, 12, 24, 25normlem5 22606 . . . . . . . . 9  |-  0  <_  ( ( ( A  x.  ( if ( x  e.  RR ,  x ,  0 ) ^ 2 ) )  +  ( B  x.  if ( x  e.  RR ,  x ,  0 ) ) )  +  C
)
2722, 26dedth 3772 . . . . . . . 8  |-  ( x  e.  RR  ->  0  <_  ( ( ( A  x.  ( x ^
2 ) )  +  ( B  x.  x
) )  +  C
) )
2827adantl 453 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR )  ->  0  <_  ( ( ( A  x.  ( x ^
2 ) )  +  ( B  x.  x
) )  +  C
) )
296, 11, 16, 28discr 11506 . . . . . 6  |-  (  T. 
->  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )  <_  0 )
3029trud 1332 . . . . 5  |-  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) )  <_ 
0
3110resqcli 11457 . . . . . 6  |-  ( B ^ 2 )  e.  RR
32 4re 10063 . . . . . . 7  |-  4  e.  RR
335, 15remulcli 9094 . . . . . . 7  |-  ( A  x.  C )  e.  RR
3432, 33remulcli 9094 . . . . . 6  |-  ( 4  x.  ( A  x.  C ) )  e.  RR
3531, 34, 23lesubadd2i 9577 . . . . 5  |-  ( ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  <_  0  <->  ( B ^ 2 )  <_ 
( ( 4  x.  ( A  x.  C
) )  +  0 ) )
3630, 35mpbi 200 . . . 4  |-  ( B ^ 2 )  <_ 
( ( 4  x.  ( A  x.  C
) )  +  0 )
3734recni 9092 . . . . 5  |-  ( 4  x.  ( A  x.  C ) )  e.  CC
3837addid1i 9243 . . . 4  |-  ( ( 4  x.  ( A  x.  C ) )  +  0 )  =  ( 4  x.  ( A  x.  C )
)
3936, 38breqtri 4227 . . 3  |-  ( B ^ 2 )  <_ 
( 4  x.  ( A  x.  C )
)
4010sqge0i 11459 . . . 4  |-  0  <_  ( B ^ 2 )
41 4pos 10076 . . . . . 6  |-  0  <  4
4223, 32, 41ltleii 9186 . . . . 5  |-  0  <_  4
43 hiidge0 22590 . . . . . . . 8  |-  ( G  e.  ~H  ->  0  <_  ( G  .ih  G
) )
442, 43ax-mp 8 . . . . . . 7  |-  0  <_  ( G  .ih  G
)
4544, 1breqtrri 4229 . . . . . 6  |-  0  <_  A
46 hiidge0 22590 . . . . . . . 8  |-  ( F  e.  ~H  ->  0  <_  ( F  .ih  F
) )
478, 46ax-mp 8 . . . . . . 7  |-  0  <_  ( F  .ih  F
)
4847, 12breqtrri 4229 . . . . . 6  |-  0  <_  C
495, 15mulge0i 9564 . . . . . 6  |-  ( ( 0  <_  A  /\  0  <_  C )  -> 
0  <_  ( A  x.  C ) )
5045, 48, 49mp2an 654 . . . . 5  |-  0  <_  ( A  x.  C
)
5132, 33mulge0i 9564 . . . . 5  |-  ( ( 0  <_  4  /\  0  <_  ( A  x.  C ) )  -> 
0  <_  ( 4  x.  ( A  x.  C ) ) )
5242, 50, 51mp2an 654 . . . 4  |-  0  <_  ( 4  x.  ( A  x.  C )
)
5331, 34sqrlei 12182 . . . 4  |-  ( ( 0  <_  ( B ^ 2 )  /\  0  <_  ( 4  x.  ( A  x.  C
) ) )  -> 
( ( B ^
2 )  <_  (
4  x.  ( A  x.  C ) )  <-> 
( sqr `  ( B ^ 2 ) )  <_  ( sqr `  (
4  x.  ( A  x.  C ) ) ) ) )
5440, 52, 53mp2an 654 . . 3  |-  ( ( B ^ 2 )  <_  ( 4  x.  ( A  x.  C
) )  <->  ( sqr `  ( B ^ 2 ) )  <_  ( sqr `  ( 4  x.  ( A  x.  C
) ) ) )
5539, 54mpbi 200 . 2  |-  ( sqr `  ( B ^ 2 ) )  <_  ( sqr `  ( 4  x.  ( A  x.  C
) ) )
5610absrei 12175 . 2  |-  ( abs `  B )  =  ( sqr `  ( B ^ 2 ) )
5732, 33, 42, 50sqrmulii 12180 . . 3  |-  ( sqr `  ( 4  x.  ( A  x.  C )
) )  =  ( ( sqr `  4
)  x.  ( sqr `  ( A  x.  C
) ) )
58 sqr4 12068 . . . 4  |-  ( sqr `  4 )  =  2
595, 15, 45, 48sqrmulii 12180 . . . 4  |-  ( sqr `  ( A  x.  C
) )  =  ( ( sqr `  A
)  x.  ( sqr `  C ) )
6058, 59oveq12i 6085 . . 3  |-  ( ( sqr `  4 )  x.  ( sqr `  ( A  x.  C )
) )  =  ( 2  x.  ( ( sqr `  A )  x.  ( sqr `  C
) ) )
6157, 60eqtr2i 2456 . 2  |-  ( 2  x.  ( ( sqr `  A )  x.  ( sqr `  C ) ) )  =  ( sqr `  ( 4  x.  ( A  x.  C )
) )
6255, 56, 613brtr4i 4232 1  |-  ( abs `  B )  <_  (
2  x.  ( ( sqr `  A )  x.  ( sqr `  C
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    T. wtru 1325    = wceq 1652    e. wcel 1725   ifcif 3731   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8978   RRcr 8979   0cc0 8980   1c1 8981    + caddc 8983    x. cmul 8985    <_ cle 9111    - cmin 9281   -ucneg 9282   2c2 10039   4c4 10041   ^cexp 11372   *ccj 11891   sqrcsqr 12028   abscabs 12029   ~Hchil 22412    .ih csp 22415
This theorem is referenced by:  normlem7  22608
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058  ax-hfvadd 22493  ax-hv0cl 22496  ax-hfvmul 22498  ax-hvmulass 22500  ax-hvmul0 22503  ax-hfi 22571  ax-his1 22574  ax-his2 22575  ax-his3 22576  ax-his4 22577
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-iun 4087  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-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-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-hvsub 22464
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