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Theorem abslem2 12144
Description: Lemma involving absolute values. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
Assertion
Ref Expression
abslem2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( * `
 ( A  / 
( abs `  A
) ) )  x.  A )  +  ( ( A  /  ( abs `  A ) )  x.  ( * `  A ) ) )  =  ( 2  x.  ( abs `  A
) ) )

Proof of Theorem abslem2
StepHypRef Expression
1 absvalsq 12086 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
21adantr 453 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
3 abscl 12084 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
43adantr 453 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  RR )
54recnd 9115 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  CC )
65sqvald 11521 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
) ^ 2 )  =  ( ( abs `  A )  x.  ( abs `  A ) ) )
72, 6eqtr3d 2471 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  x.  (
* `  A )
)  =  ( ( abs `  A )  x.  ( abs `  A
) ) )
87oveq1d 6097 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  x.  ( * `  A
) )  /  ( abs `  A ) )  =  ( ( ( abs `  A )  x.  ( abs `  A
) )  /  ( abs `  A ) ) )
9 simpl 445 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  e.  CC )
109cjcld 12002 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( * `  A
)  e.  CC )
11 abs00 12095 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( abs `  A
)  =  0  <->  A  =  0 ) )
1211necon3bid 2637 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( abs `  A
)  =/=  0  <->  A  =/=  0 ) )
1312biimpar 473 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =/=  0 )
149, 10, 5, 13div23d 9828 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  x.  ( * `  A
) )  /  ( abs `  A ) )  =  ( ( A  /  ( abs `  A
) )  x.  (
* `  A )
) )
155, 5, 13divcan3d 9796 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( abs `  A )  x.  ( abs `  A ) )  /  ( abs `  A
) )  =  ( abs `  A ) )
168, 14, 153eqtr3d 2477 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  / 
( abs `  A
) )  x.  (
* `  A )
)  =  ( abs `  A ) )
1716fveq2d 5733 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( * `  (
( A  /  ( abs `  A ) )  x.  ( * `  A ) ) )  =  ( * `  ( abs `  A ) ) )
189, 5, 13divcld 9791 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  /  ( abs `  A ) )  e.  CC )
1918, 10cjmuld 12027 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( * `  (
( A  /  ( abs `  A ) )  x.  ( * `  A ) ) )  =  ( ( * `
 ( A  / 
( abs `  A
) ) )  x.  ( * `  (
* `  A )
) ) )
209cjcjd 12005 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( * `  (
* `  A )
)  =  A )
2120oveq2d 6098 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( * `  ( A  /  ( abs `  A ) ) )  x.  ( * `
 ( * `  A ) ) )  =  ( ( * `
 ( A  / 
( abs `  A
) ) )  x.  A ) )
2219, 21eqtrd 2469 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( * `  (
( A  /  ( abs `  A ) )  x.  ( * `  A ) ) )  =  ( ( * `
 ( A  / 
( abs `  A
) ) )  x.  A ) )
234cjred 12032 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( * `  ( abs `  A ) )  =  ( abs `  A
) )
2417, 22, 233eqtr3d 2477 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( * `  ( A  /  ( abs `  A ) ) )  x.  A )  =  ( abs `  A
) )
2524, 16oveq12d 6100 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( * `
 ( A  / 
( abs `  A
) ) )  x.  A )  +  ( ( A  /  ( abs `  A ) )  x.  ( * `  A ) ) )  =  ( ( abs `  A )  +  ( abs `  A ) ) )
2652timesd 10211 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 2  x.  ( abs `  A ) )  =  ( ( abs `  A )  +  ( abs `  A ) ) )
2725, 26eqtr4d 2472 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( * `
 ( A  / 
( abs `  A
) ) )  x.  A )  +  ( ( A  /  ( abs `  A ) )  x.  ( * `  A ) ) )  =  ( 2  x.  ( abs `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   ` cfv 5455  (class class class)co 6082   CCcc 8989   RRcr 8990   0cc0 8991    + caddc 8994    x. cmul 8996    / cdiv 9678   2c2 10050   ^cexp 11383   *ccj 11902   abscabs 12040
This theorem is referenced by:  bcsiALT  22682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042
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