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Theorem normlem7tALT 22609
Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
normlem7t.1  |-  A  e. 
~H
normlem7t.2  |-  B  e. 
~H
Assertion
Ref Expression
normlem7tALT  |-  ( ( S  e.  CC  /\  ( abs `  S )  =  1 )  -> 
( ( ( * `
 S )  x.  ( A  .ih  B
) )  +  ( S  x.  ( B 
.ih  A ) ) )  <_  ( 2  x.  ( ( sqr `  ( B  .ih  B
) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )

Proof of Theorem normlem7tALT
StepHypRef Expression
1 fveq2 5719 . . . . 5  |-  ( S  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( * `  S )  =  ( * `  if ( ( S  e.  CC  /\  ( abs `  S
)  =  1 ) ,  S ,  1 ) ) )
21oveq1d 6087 . . . 4  |-  ( S  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( ( * `
 S )  x.  ( A  .ih  B
) )  =  ( ( * `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 ) )  x.  ( A  .ih  B
) ) )
3 oveq1 6079 . . . 4  |-  ( S  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( S  x.  ( B  .ih  A ) )  =  ( if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 )  x.  ( B  .ih  A ) ) )
42, 3oveq12d 6090 . . 3  |-  ( S  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( ( ( * `  S )  x.  ( A  .ih  B ) )  +  ( S  x.  ( B 
.ih  A ) ) )  =  ( ( ( * `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 ) )  x.  ( A  .ih  B
) )  +  ( if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  x.  ( B  .ih  A
) ) ) )
54breq1d 4214 . 2  |-  ( S  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( ( ( ( * `  S
)  x.  ( A 
.ih  B ) )  +  ( S  x.  ( B  .ih  A ) ) )  <_  (
2  x.  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) ) )  <->  ( (
( * `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 ) )  x.  ( A  .ih  B
) )  +  ( if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  x.  ( B  .ih  A
) ) )  <_ 
( 2  x.  (
( sqr `  ( B  .ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) ) )
6 eleq1 2495 . . . . . 6  |-  ( S  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( S  e.  CC  <->  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  e.  CC ) )
7 fveq2 5719 . . . . . . 7  |-  ( S  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( abs `  S
)  =  ( abs `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 ) ) )
87eqeq1d 2443 . . . . . 6  |-  ( S  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( ( abs `  S )  =  1  <-> 
( abs `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 ) )  =  1 ) )
96, 8anbi12d 692 . . . . 5  |-  ( S  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( ( S  e.  CC  /\  ( abs `  S )  =  1 )  <->  ( if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 )  e.  CC  /\  ( abs `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 ) )  =  1 ) ) )
10 eleq1 2495 . . . . . 6  |-  ( 1  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( 1  e.  CC  <->  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  e.  CC ) )
11 fveq2 5719 . . . . . . 7  |-  ( 1  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( abs `  1
)  =  ( abs `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 ) ) )
1211eqeq1d 2443 . . . . . 6  |-  ( 1  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( ( abs `  1 )  =  1  <->  ( abs `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 ) )  =  1 ) )
1310, 12anbi12d 692 . . . . 5  |-  ( 1  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( ( 1  e.  CC  /\  ( abs `  1 )  =  1 )  <->  ( if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 )  e.  CC  /\  ( abs `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 ) )  =  1 ) ) )
14 ax-1cn 9037 . . . . . 6  |-  1  e.  CC
15 abs1 12090 . . . . . 6  |-  ( abs `  1 )  =  1
1614, 15pm3.2i 442 . . . . 5  |-  ( 1  e.  CC  /\  ( abs `  1 )  =  1 )
179, 13, 16elimhyp 3779 . . . 4  |-  ( if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 )  e.  CC  /\  ( abs `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 ) )  =  1 )
1817simpli 445 . . 3  |-  if ( ( S  e.  CC  /\  ( abs `  S
)  =  1 ) ,  S ,  1 )  e.  CC
19 normlem7t.1 . . 3  |-  A  e. 
~H
20 normlem7t.2 . . 3  |-  B  e. 
~H
2117simpri 449 . . 3  |-  ( abs `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 ) )  =  1
2218, 19, 20, 21normlem7 22606 . 2  |-  ( ( ( * `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 ) )  x.  ( A  .ih  B
) )  +  ( if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  x.  ( B  .ih  A
) ) )  <_ 
( 2  x.  (
( sqr `  ( B  .ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) ) )
235, 22dedth 3772 1  |-  ( ( S  e.  CC  /\  ( abs `  S )  =  1 )  -> 
( ( ( * `
 S )  x.  ( A  .ih  B
) )  +  ( S  x.  ( B 
.ih  A ) ) )  <_  ( 2  x.  ( ( sqr `  ( B  .ih  B
) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ifcif 3731   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   CCcc 8977   1c1 8980    + caddc 8982    x. cmul 8984    <_ cle 9110   2c2 10038   *ccj 11889   sqrcsqr 12026   abscabs 12027   ~Hchil 22410    .ih csp 22413
This theorem is referenced by:  bcsiALT  22669
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 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-hfvadd 22491  ax-hv0cl 22494  ax-hfvmul 22496  ax-hvmulass 22498  ax-hvmul0 22501  ax-hfi 22569  ax-his1 22572  ax-his2 22573  ax-his3 22574  ax-his4 22575
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 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-seq 11312  df-exp 11371  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-hvsub 22462
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