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Theorem normlem6 21710
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 21690 . . . . . . . . . 10  |-  ( G  e.  ~H  ->  ( G  .ih  G )  e.  RR )
42, 3ax-mp 8 . . . . . . . . 9  |-  ( G 
.ih  G )  e.  RR
51, 4eqeltri 2366 . . . . . . . 8  |-  A  e.  RR
65a1i 10 . . . . . . 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 21706 . . . . . . . 8  |-  B  e.  RR
1110a1i 10 . . . . . . 7  |-  (  T. 
->  B  e.  RR )
12 normlem3.6 . . . . . . . . 9  |-  C  =  ( F  .ih  F
)
13 hiidrcl 21690 . . . . . . . . . 10  |-  ( F  e.  ~H  ->  ( F  .ih  F )  e.  RR )
148, 13ax-mp 8 . . . . . . . . 9  |-  ( F 
.ih  F )  e.  RR
1512, 14eqeltri 2366 . . . . . . . 8  |-  C  e.  RR
1615a1i 10 . . . . . . 7  |-  (  T. 
->  C  e.  RR )
17 oveq1 5881 . . . . . . . . . . . . 13  |-  ( x  =  if ( x  e.  RR ,  x ,  0 )  -> 
( x ^ 2 )  =  ( if ( x  e.  RR ,  x ,  0 ) ^ 2 ) )
1817oveq2d 5890 . . . . . . . . . . . 12  |-  ( x  =  if ( x  e.  RR ,  x ,  0 )  -> 
( A  x.  (
x ^ 2 ) )  =  ( A  x.  ( if ( x  e.  RR ,  x ,  0 ) ^ 2 ) ) )
19 oveq2 5882 . . . . . . . . . . . 12  |-  ( x  =  if ( x  e.  RR ,  x ,  0 )  -> 
( B  x.  x
)  =  ( B  x.  if ( x  e.  RR ,  x ,  0 ) ) )
2018, 19oveq12d 5892 . . . . . . . . . . 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 5889 . . . . . . . . . 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 4051 . . . . . . . . 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 8854 . . . . . . . . . . 11  |-  0  e.  RR
2423elimel 3630 . . . . . . . . . 10  |-  if ( x  e.  RR ,  x ,  0 )  e.  RR
25 normlem6.7 . . . . . . . . . 10  |-  ( abs `  S )  =  1
267, 8, 2, 9, 1, 12, 24, 25normlem5 21709 . . . . . . . . 9  |-  0  <_  ( ( ( A  x.  ( if ( x  e.  RR ,  x ,  0 ) ^ 2 ) )  +  ( B  x.  if ( x  e.  RR ,  x ,  0 ) ) )  +  C
)
2722, 26dedth 3619 . . . . . . . 8  |-  ( x  e.  RR  ->  0  <_  ( ( ( A  x.  ( x ^
2 ) )  +  ( B  x.  x
) )  +  C
) )
2827adantl 452 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR )  ->  0  <_  ( ( ( A  x.  ( x ^
2 ) )  +  ( B  x.  x
) )  +  C
) )
296, 11, 16, 28discr 11254 . . . . . 6  |-  (  T. 
->  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )  <_  0 )
3029trud 1314 . . . . 5  |-  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) )  <_ 
0
3110resqcli 11205 . . . . . 6  |-  ( B ^ 2 )  e.  RR
32 4re 9835 . . . . . . 7  |-  4  e.  RR
335, 15remulcli 8867 . . . . . . 7  |-  ( A  x.  C )  e.  RR
3432, 33remulcli 8867 . . . . . 6  |-  ( 4  x.  ( A  x.  C ) )  e.  RR
3531, 34, 23lesubadd2i 9349 . . . . 5  |-  ( ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  <_  0  <->  ( B ^ 2 )  <_ 
( ( 4  x.  ( A  x.  C
) )  +  0 ) )
3630, 35mpbi 199 . . . 4  |-  ( B ^ 2 )  <_ 
( ( 4  x.  ( A  x.  C
) )  +  0 )
3734recni 8865 . . . . 5  |-  ( 4  x.  ( A  x.  C ) )  e.  CC
3837addid1i 9015 . . . 4  |-  ( ( 4  x.  ( A  x.  C ) )  +  0 )  =  ( 4  x.  ( A  x.  C )
)
3936, 38breqtri 4062 . . 3  |-  ( B ^ 2 )  <_ 
( 4  x.  ( A  x.  C )
)
4010sqge0i 11207 . . . 4  |-  0  <_  ( B ^ 2 )
41 4pos 9848 . . . . . 6  |-  0  <  4
4223, 32, 41ltleii 8957 . . . . 5  |-  0  <_  4
43 hiidge0 21693 . . . . . . . 8  |-  ( G  e.  ~H  ->  0  <_  ( G  .ih  G
) )
442, 43ax-mp 8 . . . . . . 7  |-  0  <_  ( G  .ih  G
)
4544, 1breqtrri 4064 . . . . . 6  |-  0  <_  A
46 hiidge0 21693 . . . . . . . 8  |-  ( F  e.  ~H  ->  0  <_  ( F  .ih  F
) )
478, 46ax-mp 8 . . . . . . 7  |-  0  <_  ( F  .ih  F
)
4847, 12breqtrri 4064 . . . . . 6  |-  0  <_  C
495, 15mulge0i 9336 . . . . . 6  |-  ( ( 0  <_  A  /\  0  <_  C )  -> 
0  <_  ( A  x.  C ) )
5045, 48, 49mp2an 653 . . . . 5  |-  0  <_  ( A  x.  C
)
5132, 33mulge0i 9336 . . . . 5  |-  ( ( 0  <_  4  /\  0  <_  ( A  x.  C ) )  -> 
0  <_  ( 4  x.  ( A  x.  C ) ) )
5242, 50, 51mp2an 653 . . . 4  |-  0  <_  ( 4  x.  ( A  x.  C )
)
5331, 34sqrlei 11888 . . . 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 653 . . 3  |-  ( ( B ^ 2 )  <_  ( 4  x.  ( A  x.  C
) )  <->  ( sqr `  ( B ^ 2 ) )  <_  ( sqr `  ( 4  x.  ( A  x.  C
) ) ) )
5539, 54mpbi 199 . 2  |-  ( sqr `  ( B ^ 2 ) )  <_  ( sqr `  ( 4  x.  ( A  x.  C
) ) )
5610absrei 11881 . 2  |-  ( abs `  B )  =  ( sqr `  ( B ^ 2 ) )
5732, 33, 42, 50sqrmulii 11886 . . 3  |-  ( sqr `  ( 4  x.  ( A  x.  C )
) )  =  ( ( sqr `  4
)  x.  ( sqr `  ( A  x.  C
) ) )
58 sqr4 11774 . . . 4  |-  ( sqr `  4 )  =  2
595, 15, 45, 48sqrmulii 11886 . . . 4  |-  ( sqr `  ( A  x.  C
) )  =  ( ( sqr `  A
)  x.  ( sqr `  C ) )
6058, 59oveq12i 5886 . . 3  |-  ( ( sqr `  4 )  x.  ( sqr `  ( A  x.  C )
) )  =  ( 2  x.  ( ( sqr `  A )  x.  ( sqr `  C
) ) )
6157, 60eqtr2i 2317 . 2  |-  ( 2  x.  ( ( sqr `  A )  x.  ( sqr `  C ) ) )  =  ( sqr `  ( 4  x.  ( A  x.  C )
) )
6255, 56, 613brtr4i 4067 1  |-  ( abs `  B )  <_  (
2  x.  ( ( sqr `  A )  x.  ( sqr `  C
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    T. wtru 1307    = wceq 1632    e. wcel 1696   ifcif 3578   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053   -ucneg 9054   2c2 9811   4c4 9813   ^cexp 11120   *ccj 11597   sqrcsqr 11734   abscabs 11735   ~Hchil 21515    .ih csp 21518
This theorem is referenced by:  normlem7  21711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-hfvadd 21596  ax-hv0cl 21599  ax-hfvmul 21601  ax-hvmulass 21603  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-hvsub 21567
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