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Theorem abs1m 12835
Description: For any complex number, there exists a unit-magnitude multiplier that produces its absolute value. Part of proof of Theorem 13-2.12 of [Gleason] p. 195. (Contributed by NM, 26-Mar-2005.)
Assertion
Ref Expression
abs1m  |-  ( A  e.  CC  ->  E. x  e.  CC  ( ( abs `  x )  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) ) )
Distinct variable group:    x, A

Proof of Theorem abs1m
StepHypRef Expression
1 fveq2 5703 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
2 abs0 12786 . . . . . 6  |-  ( abs `  0 )  =  0
31, 2syl6eq 2491 . . . . 5  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
4 oveq2 6111 . . . . 5  |-  ( A  =  0  ->  (
x  x.  A )  =  ( x  x.  0 ) )
53, 4eqeq12d 2457 . . . 4  |-  ( A  =  0  ->  (
( abs `  A
)  =  ( x  x.  A )  <->  0  =  ( x  x.  0
) ) )
65anbi2d 703 . . 3  |-  ( A  =  0  ->  (
( ( abs `  x
)  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) )  <-> 
( ( abs `  x
)  =  1  /\  0  =  ( x  x.  0 ) ) ) )
76rexbidv 2748 . 2  |-  ( A  =  0  ->  ( E. x  e.  CC  ( ( abs `  x
)  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) )  <->  E. x  e.  CC  ( ( abs `  x
)  =  1  /\  0  =  ( x  x.  0 ) ) ) )
8 simpl 457 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  e.  CC )
98cjcld 12697 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( * `  A
)  e.  CC )
10 abscl 12779 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
1110adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  RR )
1211recnd 9424 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  CC )
13 abs00 12790 . . . . . 6  |-  ( A  e.  CC  ->  (
( abs `  A
)  =  0  <->  A  =  0 ) )
1413necon3bid 2655 . . . . 5  |-  ( A  e.  CC  ->  (
( abs `  A
)  =/=  0  <->  A  =/=  0 ) )
1514biimpar 485 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =/=  0 )
169, 12, 15divcld 10119 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( * `  A )  /  ( abs `  A ) )  e.  CC )
17 absdiv 12796 . . . . 5  |-  ( ( ( * `  A
)  e.  CC  /\  ( abs `  A )  e.  CC  /\  ( abs `  A )  =/=  0 )  ->  ( abs `  ( ( * `
 A )  / 
( abs `  A
) ) )  =  ( ( abs `  (
* `  A )
)  /  ( abs `  ( abs `  A
) ) ) )
189, 12, 15, 17syl3anc 1218 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
( * `  A
)  /  ( abs `  A ) ) )  =  ( ( abs `  ( * `  A
) )  /  ( abs `  ( abs `  A
) ) ) )
19 abscj 12780 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( * `  A ) )  =  ( abs `  A
) )
2019adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
* `  A )
)  =  ( abs `  A ) )
21 absidm 12823 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( abs `  A
) )  =  ( abs `  A ) )
2221adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  ( abs `  A ) )  =  ( abs `  A
) )
2320, 22oveq12d 6121 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  (
* `  A )
)  /  ( abs `  ( abs `  A
) ) )  =  ( ( abs `  A
)  /  ( abs `  A ) ) )
2412, 15dividd 10117 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
)  /  ( abs `  A ) )  =  1 )
2518, 23, 243eqtrd 2479 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
( * `  A
)  /  ( abs `  A ) ) )  =  1 )
268, 9, 12, 15divassd 10154 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  x.  ( * `  A
) )  /  ( abs `  A ) )  =  ( A  x.  ( ( * `  A )  /  ( abs `  A ) ) ) )
2712, 12, 15divcan3d 10124 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( abs `  A )  x.  ( abs `  A ) )  /  ( abs `  A
) )  =  ( abs `  A ) )
2812sqvald 12017 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
) ^ 2 )  =  ( ( abs `  A )  x.  ( abs `  A ) ) )
29 absvalsq 12781 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
3029adantr 465 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
3128, 30eqtr3d 2477 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
)  x.  ( abs `  A ) )  =  ( A  x.  (
* `  A )
) )
3231oveq1d 6118 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( abs `  A )  x.  ( abs `  A ) )  /  ( abs `  A
) )  =  ( ( A  x.  (
* `  A )
)  /  ( abs `  A ) ) )
3327, 32eqtr3d 2477 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =  ( ( A  x.  ( * `
 A ) )  /  ( abs `  A
) ) )
3416, 8mulcomd 9419 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( * `
 A )  / 
( abs `  A
) )  x.  A
)  =  ( A  x.  ( ( * `
 A )  / 
( abs `  A
) ) ) )
3526, 33, 343eqtr4d 2485 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =  ( ( ( * `  A
)  /  ( abs `  A ) )  x.  A ) )
36 fveq2 5703 . . . . . 6  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  ( abs `  x )  =  ( abs `  (
( * `  A
)  /  ( abs `  A ) ) ) )
3736eqeq1d 2451 . . . . 5  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
( abs `  x
)  =  1  <->  ( abs `  ( ( * `
 A )  / 
( abs `  A
) ) )  =  1 ) )
38 oveq1 6110 . . . . . 6  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
x  x.  A )  =  ( ( ( * `  A )  /  ( abs `  A
) )  x.  A
) )
3938eqeq2d 2454 . . . . 5  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
( abs `  A
)  =  ( x  x.  A )  <->  ( abs `  A )  =  ( ( ( * `  A )  /  ( abs `  A ) )  x.  A ) ) )
4037, 39anbi12d 710 . . . 4  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
( ( abs `  x
)  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) )  <-> 
( ( abs `  (
( * `  A
)  /  ( abs `  A ) ) )  =  1  /\  ( abs `  A )  =  ( ( ( * `
 A )  / 
( abs `  A
) )  x.  A
) ) ) )
4140rspcev 3085 . . 3  |-  ( ( ( ( * `  A )  /  ( abs `  A ) )  e.  CC  /\  (
( abs `  (
( * `  A
)  /  ( abs `  A ) ) )  =  1  /\  ( abs `  A )  =  ( ( ( * `
 A )  / 
( abs `  A
) )  x.  A
) ) )  ->  E. x  e.  CC  ( ( abs `  x
)  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) ) )
4216, 25, 35, 41syl12anc 1216 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  E. x  e.  CC  ( ( abs `  x
)  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) ) )
43 ax-icn 9353 . . . 4  |-  _i  e.  CC
44 absi 12787 . . . . 5  |-  ( abs `  _i )  =  1
45 it0e0 10559 . . . . . 6  |-  ( _i  x.  0 )  =  0
4645eqcomi 2447 . . . . 5  |-  0  =  ( _i  x.  0 )
4744, 46pm3.2i 455 . . . 4  |-  ( ( abs `  _i )  =  1  /\  0  =  ( _i  x.  0 ) )
48 fveq2 5703 . . . . . . 7  |-  ( x  =  _i  ->  ( abs `  x )  =  ( abs `  _i ) )
4948eqeq1d 2451 . . . . . 6  |-  ( x  =  _i  ->  (
( abs `  x
)  =  1  <->  ( abs `  _i )  =  1 ) )
50 oveq1 6110 . . . . . . 7  |-  ( x  =  _i  ->  (
x  x.  0 )  =  ( _i  x.  0 ) )
5150eqeq2d 2454 . . . . . 6  |-  ( x  =  _i  ->  (
0  =  ( x  x.  0 )  <->  0  =  ( _i  x.  0
) ) )
5249, 51anbi12d 710 . . . . 5  |-  ( x  =  _i  ->  (
( ( abs `  x
)  =  1  /\  0  =  ( x  x.  0 ) )  <-> 
( ( abs `  _i )  =  1  /\  0  =  ( _i  x.  0 ) ) ) )
5352rspcev 3085 . . . 4  |-  ( ( _i  e.  CC  /\  ( ( abs `  _i )  =  1  /\  0  =  ( _i  x.  0 ) ) )  ->  E. x  e.  CC  ( ( abs `  x
)  =  1  /\  0  =  ( x  x.  0 ) ) )
5443, 47, 53mp2an 672 . . 3  |-  E. x  e.  CC  ( ( abs `  x )  =  1  /\  0  =  ( x  x.  0 ) )
5554a1i 11 . 2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( ( abs `  x )  =  1  /\  0  =  ( x  x.  0 ) ) )
567, 42, 55pm2.61ne 2698 1  |-  ( A  e.  CC  ->  E. x  e.  CC  ( ( abs `  x )  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2618   E.wrex 2728   ` cfv 5430  (class class class)co 6103   CCcc 9292   RRcr 9293   0cc0 9294   1c1 9295   _ici 9296    x. cmul 9299    / cdiv 10005   2c2 10383   ^cexp 11877   *ccj 12597   abscabs 12735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-seq 11819  df-exp 11878  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737
This theorem is referenced by: (None)
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