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Theorem abs1m 12129
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 5720 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
2 abs0 12080 . . . . . 6  |-  ( abs `  0 )  =  0
31, 2syl6eq 2483 . . . . 5  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
4 oveq2 6081 . . . . 5  |-  ( A  =  0  ->  (
x  x.  A )  =  ( x  x.  0 ) )
53, 4eqeq12d 2449 . . . 4  |-  ( A  =  0  ->  (
( abs `  A
)  =  ( x  x.  A )  <->  0  =  ( x  x.  0
) ) )
65anbi2d 685 . . 3  |-  ( A  =  0  ->  (
( ( abs `  x
)  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) )  <-> 
( ( abs `  x
)  =  1  /\  0  =  ( x  x.  0 ) ) ) )
76rexbidv 2718 . 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 444 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  e.  CC )
98cjcld 11991 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( * `  A
)  e.  CC )
10 abscl 12073 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
1110adantr 452 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  RR )
1211recnd 9104 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  CC )
13 abs00 12084 . . . . . 6  |-  ( A  e.  CC  ->  (
( abs `  A
)  =  0  <->  A  =  0 ) )
1413necon3bid 2633 . . . . 5  |-  ( A  e.  CC  ->  (
( abs `  A
)  =/=  0  <->  A  =/=  0 ) )
1514biimpar 472 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =/=  0 )
169, 12, 15divcld 9780 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( * `  A )  /  ( abs `  A ) )  e.  CC )
17 absdiv 12090 . . . . 5  |-  ( ( ( * `  A
)  e.  CC  /\  ( abs `  A )  e.  CC  /\  ( abs `  A )  =/=  0 )  ->  ( abs `  ( ( * `
 A )  / 
( abs `  A
) ) )  =  ( ( abs `  (
* `  A )
)  /  ( abs `  ( abs `  A
) ) ) )
189, 12, 15, 17syl3anc 1184 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
( * `  A
)  /  ( abs `  A ) ) )  =  ( ( abs `  ( * `  A
) )  /  ( abs `  ( abs `  A
) ) ) )
19 abscj 12074 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( * `  A ) )  =  ( abs `  A
) )
2019adantr 452 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
* `  A )
)  =  ( abs `  A ) )
21 absidm 12117 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( abs `  A
) )  =  ( abs `  A ) )
2221adantr 452 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  ( abs `  A ) )  =  ( abs `  A
) )
2320, 22oveq12d 6091 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  (
* `  A )
)  /  ( abs `  ( abs `  A
) ) )  =  ( ( abs `  A
)  /  ( abs `  A ) ) )
2412, 15dividd 9778 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
)  /  ( abs `  A ) )  =  1 )
2518, 23, 243eqtrd 2471 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
( * `  A
)  /  ( abs `  A ) ) )  =  1 )
268, 9, 12, 15divassd 9815 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  x.  ( * `  A
) )  /  ( abs `  A ) )  =  ( A  x.  ( ( * `  A )  /  ( abs `  A ) ) ) )
2712, 12, 15divcan3d 9785 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( abs `  A )  x.  ( abs `  A ) )  /  ( abs `  A
) )  =  ( abs `  A ) )
2812sqvald 11510 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
) ^ 2 )  =  ( ( abs `  A )  x.  ( abs `  A ) ) )
29 absvalsq 12075 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
3029adantr 452 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
3128, 30eqtr3d 2469 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
)  x.  ( abs `  A ) )  =  ( A  x.  (
* `  A )
) )
3231oveq1d 6088 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( abs `  A )  x.  ( abs `  A ) )  /  ( abs `  A
) )  =  ( ( A  x.  (
* `  A )
)  /  ( abs `  A ) ) )
3327, 32eqtr3d 2469 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =  ( ( A  x.  ( * `
 A ) )  /  ( abs `  A
) ) )
3416, 8mulcomd 9099 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( * `
 A )  / 
( abs `  A
) )  x.  A
)  =  ( A  x.  ( ( * `
 A )  / 
( abs `  A
) ) ) )
3526, 33, 343eqtr4d 2477 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =  ( ( ( * `  A
)  /  ( abs `  A ) )  x.  A ) )
36 fveq2 5720 . . . . . 6  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  ( abs `  x )  =  ( abs `  (
( * `  A
)  /  ( abs `  A ) ) ) )
3736eqeq1d 2443 . . . . 5  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
( abs `  x
)  =  1  <->  ( abs `  ( ( * `
 A )  / 
( abs `  A
) ) )  =  1 ) )
38 oveq1 6080 . . . . . 6  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
x  x.  A )  =  ( ( ( * `  A )  /  ( abs `  A
) )  x.  A
) )
3938eqeq2d 2446 . . . . 5  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
( abs `  A
)  =  ( x  x.  A )  <->  ( abs `  A )  =  ( ( ( * `  A )  /  ( abs `  A ) )  x.  A ) ) )
4037, 39anbi12d 692 . . . 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 3044 . . 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 1182 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  E. x  e.  CC  ( ( abs `  x
)  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) ) )
43 ax-icn 9039 . . . 4  |-  _i  e.  CC
44 absi 12081 . . . . 5  |-  ( abs `  _i )  =  1
4543mul01i 9246 . . . . . 6  |-  ( _i  x.  0 )  =  0
4645eqcomi 2439 . . . . 5  |-  0  =  ( _i  x.  0 )
4744, 46pm3.2i 442 . . . 4  |-  ( ( abs `  _i )  =  1  /\  0  =  ( _i  x.  0 ) )
48 fveq2 5720 . . . . . . 7  |-  ( x  =  _i  ->  ( abs `  x )  =  ( abs `  _i ) )
4948eqeq1d 2443 . . . . . 6  |-  ( x  =  _i  ->  (
( abs `  x
)  =  1  <->  ( abs `  _i )  =  1 ) )
50 oveq1 6080 . . . . . . 7  |-  ( x  =  _i  ->  (
x  x.  0 )  =  ( _i  x.  0 ) )
5150eqeq2d 2446 . . . . . 6  |-  ( x  =  _i  ->  (
0  =  ( x  x.  0 )  <->  0  =  ( _i  x.  0
) ) )
5249, 51anbi12d 692 . . . . 5  |-  ( x  =  _i  ->  (
( ( abs `  x
)  =  1  /\  0  =  ( x  x.  0 ) )  <-> 
( ( abs `  _i )  =  1  /\  0  =  ( _i  x.  0 ) ) ) )
5352rspcev 3044 . . . 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 654 . . 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 2673 1  |-  ( A  e.  CC  ->  E. x  e.  CC  ( ( abs `  x )  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   ` cfv 5446  (class class class)co 6073   CCcc 8978   RRcr 8979   0cc0 8980   1c1 8981   _ici 8982    x. cmul 8985    / cdiv 9667   2c2 10039   ^cexp 11372   *ccj 11891   abscabs 12029
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 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
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 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031
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