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Theorem normlem6 21696
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 21676 . . . . . . . . . 10  |-  ( G  e.  ~H  ->  ( G  .ih  G )  e.  RR )
42, 3ax-mp 8 . . . . . . . . 9  |-  ( G 
.ih  G )  e.  RR
51, 4eqeltri 2355 . . . . . . . 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 21692 . . . . . . . 8  |-  B  e.  RR
1110a1i 10 . . . . . . 7  |-  (  T. 
->  B  e.  RR )
12 normlem3.6 . . . . . . . . 9  |-  C  =  ( F  .ih  F
)
13 hiidrcl 21676 . . . . . . . . . 10  |-  ( F  e.  ~H  ->  ( F  .ih  F )  e.  RR )
148, 13ax-mp 8 . . . . . . . . 9  |-  ( F 
.ih  F )  e.  RR
1512, 14eqeltri 2355 . . . . . . . 8  |-  C  e.  RR
1615a1i 10 . . . . . . 7  |-  (  T. 
->  C  e.  RR )
17 oveq1 5867 . . . . . . . . . . . . 13  |-  ( x  =  if ( x  e.  RR ,  x ,  0 )  -> 
( x ^ 2 )  =  ( if ( x  e.  RR ,  x ,  0 ) ^ 2 ) )
1817oveq2d 5876 . . . . . . . . . . . 12  |-  ( x  =  if ( x  e.  RR ,  x ,  0 )  -> 
( A  x.  (
x ^ 2 ) )  =  ( A  x.  ( if ( x  e.  RR ,  x ,  0 ) ^ 2 ) ) )
19 oveq2 5868 . . . . . . . . . . . 12  |-  ( x  =  if ( x  e.  RR ,  x ,  0 )  -> 
( B  x.  x
)  =  ( B  x.  if ( x  e.  RR ,  x ,  0 ) ) )
2018, 19oveq12d 5878 . . . . . . . . . . 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 5875 . . . . . . . . . 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 4037 . . . . . . . . 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 8840 . . . . . . . . . . 11  |-  0  e.  RR
2423elimel 3619 . . . . . . . . . 10  |-  if ( x  e.  RR ,  x ,  0 )  e.  RR
25 normlem6.7 . . . . . . . . . 10  |-  ( abs `  S )  =  1
267, 8, 2, 9, 1, 12, 24, 25normlem5 21695 . . . . . . . . 9  |-  0  <_  ( ( ( A  x.  ( if ( x  e.  RR ,  x ,  0 ) ^ 2 ) )  +  ( B  x.  if ( x  e.  RR ,  x ,  0 ) ) )  +  C
)
2722, 26dedth 3608 . . . . . . . 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 11240 . . . . . 6  |-  (  T. 
->  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )  <_  0 )
3029trud 1314 . . . . 5  |-  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) )  <_ 
0
3110resqcli 11191 . . . . . 6  |-  ( B ^ 2 )  e.  RR
32 4re 9821 . . . . . . 7  |-  4  e.  RR
335, 15remulcli 8853 . . . . . . 7  |-  ( A  x.  C )  e.  RR
3432, 33remulcli 8853 . . . . . 6  |-  ( 4  x.  ( A  x.  C ) )  e.  RR
3531, 34, 23lesubadd2i 9335 . . . . 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 8851 . . . . 5  |-  ( 4  x.  ( A  x.  C ) )  e.  CC
3837addid1i 9001 . . . 4  |-  ( ( 4  x.  ( A  x.  C ) )  +  0 )  =  ( 4  x.  ( A  x.  C )
)
3936, 38breqtri 4048 . . 3  |-  ( B ^ 2 )  <_ 
( 4  x.  ( A  x.  C )
)
4010sqge0i 11193 . . . 4  |-  0  <_  ( B ^ 2 )
41 4pos 9834 . . . . . 6  |-  0  <  4
4223, 32, 41ltleii 8943 . . . . 5  |-  0  <_  4
43 hiidge0 21679 . . . . . . . 8  |-  ( G  e.  ~H  ->  0  <_  ( G  .ih  G
) )
442, 43ax-mp 8 . . . . . . 7  |-  0  <_  ( G  .ih  G
)
4544, 1breqtrri 4050 . . . . . 6  |-  0  <_  A
46 hiidge0 21679 . . . . . . . 8  |-  ( F  e.  ~H  ->  0  <_  ( F  .ih  F
) )
478, 46ax-mp 8 . . . . . . 7  |-  0  <_  ( F  .ih  F
)
4847, 12breqtrri 4050 . . . . . 6  |-  0  <_  C
495, 15mulge0i 9322 . . . . . 6  |-  ( ( 0  <_  A  /\  0  <_  C )  -> 
0  <_  ( A  x.  C ) )
5045, 48, 49mp2an 653 . . . . 5  |-  0  <_  ( A  x.  C
)
5132, 33mulge0i 9322 . . . . 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 11874 . . . 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 11867 . 2  |-  ( abs `  B )  =  ( sqr `  ( B ^ 2 ) )
5732, 33, 42, 50sqrmulii 11872 . . 3  |-  ( sqr `  ( 4  x.  ( A  x.  C )
) )  =  ( ( sqr `  4
)  x.  ( sqr `  ( A  x.  C
) ) )
58 sqr4 11760 . . . 4  |-  ( sqr `  4 )  =  2
595, 15, 45, 48sqrmulii 11872 . . . 4  |-  ( sqr `  ( A  x.  C
) )  =  ( ( sqr `  A
)  x.  ( sqr `  C ) )
6058, 59oveq12i 5872 . . 3  |-  ( ( sqr `  4 )  x.  ( sqr `  ( A  x.  C )
) )  =  ( 2  x.  ( ( sqr `  A )  x.  ( sqr `  C
) ) )
6157, 60eqtr2i 2306 . 2  |-  ( 2  x.  ( ( sqr `  A )  x.  ( sqr `  C ) ) )  =  ( sqr `  ( 4  x.  ( A  x.  C )
) )
6255, 56, 613brtr4i 4053 1  |-  ( abs `  B )  <_  (
2  x.  ( ( sqr `  A )  x.  ( sqr `  C
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    T. wtru 1307    = wceq 1625    e. wcel 1686   ifcif 3567   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    + caddc 8742    x. cmul 8744    <_ cle 8870    - cmin 9039   -ucneg 9040   2c2 9797   4c4 9799   ^cexp 11106   *ccj 11583   sqrcsqr 11720   abscabs 11721   ~Hchil 21501    .ih csp 21504
This theorem is referenced by:  normlem7  21697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-hfvadd 21582  ax-hv0cl 21585  ax-hfvmul 21587  ax-hvmulass 21589  ax-hvmul0 21592  ax-hfi 21660  ax-his1 21663  ax-his2 21664  ax-his3 21665  ax-his4 21666
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-sup 7196  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-n0 9968  df-z 10027  df-uz 10233  df-rp 10357  df-seq 11049  df-exp 11107  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-hvsub 21553
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