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Theorem normlem7tALT 21690
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 5485 . . . . 5  |-  ( S  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( * `  S )  =  ( * `  if ( ( S  e.  CC  /\  ( abs `  S
)  =  1 ) ,  S ,  1 ) ) )
21oveq1d 5834 . . . 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 5826 . . . 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 5837 . . 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 4034 . 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 2344 . . . . . 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 5485 . . . . . . 7  |-  ( S  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( abs `  S
)  =  ( abs `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 ) ) )
87eqeq1d 2292 . . . . . 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 693 . . . . 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 2344 . . . . . 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 5485 . . . . . . 7  |-  ( 1  =  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 )  ->  ( abs `  1
)  =  ( abs `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 ) ) )
1211eqeq1d 2292 . . . . . 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 693 . . . . 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 8790 . . . . . 6  |-  1  e.  CC
15 abs1 11776 . . . . . 6  |-  ( abs `  1 )  =  1
1614, 15pm3.2i 443 . . . . 5  |-  ( 1  e.  CC  /\  ( abs `  1 )  =  1 )
179, 13, 16elimhyp 3614 . . . 4  |-  ( if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 )  e.  CC  /\  ( abs `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S , 
1 ) )  =  1 )
1817simpli 446 . . 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 450 . . 3  |-  ( abs `  if ( ( S  e.  CC  /\  ( abs `  S )  =  1 ) ,  S ,  1 ) )  =  1
2218, 19, 20, 21normlem7 21687 . 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 3607 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 6    /\ wa 360    = wceq 1624    e. wcel 1685   ifcif 3566   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   CCcc 8730   1c1 8733    + caddc 8735    x. cmul 8737    <_ cle 8863   2c2 9790   *ccj 11575   sqrcsqr 11712   abscabs 11713   ~Hchil 21491    .ih csp 21494
This theorem is referenced by:  bcsiALT  21750
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-hfvadd 21572  ax-hv0cl 21575  ax-hfvmul 21577  ax-hvmulass 21579  ax-hvmul0 21582  ax-hfi 21650  ax-his1 21653  ax-his2 21654  ax-his3 21655  ax-his4 21656
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-sup 7189  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-seq 11041  df-exp 11099  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-hvsub 21543
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