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Theorem abs1m 12094
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 5687 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
2 abs0 12045 . . . . . 6  |-  ( abs `  0 )  =  0
31, 2syl6eq 2452 . . . . 5  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
4 oveq2 6048 . . . . 5  |-  ( A  =  0  ->  (
x  x.  A )  =  ( x  x.  0 ) )
53, 4eqeq12d 2418 . . . 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 2687 . 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 11956 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( * `  A
)  e.  CC )
10 abscl 12038 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
1110adantr 452 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  RR )
1211recnd 9070 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  CC )
13 abs00 12049 . . . . . 6  |-  ( A  e.  CC  ->  (
( abs `  A
)  =  0  <->  A  =  0 ) )
1413necon3bid 2602 . . . . 5  |-  ( A  e.  CC  ->  (
( abs `  A
)  =/=  0  <->  A  =/=  0 ) )
1514biimpar 472 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =/=  0 )
169, 12, 15divcld 9746 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( * `  A )  /  ( abs `  A ) )  e.  CC )
17 absdiv 12055 . . . . 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 12039 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( * `  A ) )  =  ( abs `  A
) )
2019adantr 452 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
* `  A )
)  =  ( abs `  A ) )
21 absidm 12082 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( abs `  A
) )  =  ( abs `  A ) )
2221adantr 452 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  ( abs `  A ) )  =  ( abs `  A
) )
2320, 22oveq12d 6058 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  (
* `  A )
)  /  ( abs `  ( abs `  A
) ) )  =  ( ( abs `  A
)  /  ( abs `  A ) ) )
2412, 15dividd 9744 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
)  /  ( abs `  A ) )  =  1 )
2518, 23, 243eqtrd 2440 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
( * `  A
)  /  ( abs `  A ) ) )  =  1 )
268, 9, 12, 15divassd 9781 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  x.  ( * `  A
) )  /  ( abs `  A ) )  =  ( A  x.  ( ( * `  A )  /  ( abs `  A ) ) ) )
2712, 12, 15divcan3d 9751 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( abs `  A )  x.  ( abs `  A ) )  /  ( abs `  A
) )  =  ( abs `  A ) )
2812sqvald 11475 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
) ^ 2 )  =  ( ( abs `  A )  x.  ( abs `  A ) ) )
29 absvalsq 12040 . . . . . . . 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 2438 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
)  x.  ( abs `  A ) )  =  ( A  x.  (
* `  A )
) )
3231oveq1d 6055 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( abs `  A )  x.  ( abs `  A ) )  /  ( abs `  A
) )  =  ( ( A  x.  (
* `  A )
)  /  ( abs `  A ) ) )
3327, 32eqtr3d 2438 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =  ( ( A  x.  ( * `
 A ) )  /  ( abs `  A
) ) )
3416, 8mulcomd 9065 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( * `
 A )  / 
( abs `  A
) )  x.  A
)  =  ( A  x.  ( ( * `
 A )  / 
( abs `  A
) ) ) )
3526, 33, 343eqtr4d 2446 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =  ( ( ( * `  A
)  /  ( abs `  A ) )  x.  A ) )
36 fveq2 5687 . . . . . 6  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  ( abs `  x )  =  ( abs `  (
( * `  A
)  /  ( abs `  A ) ) ) )
3736eqeq1d 2412 . . . . 5  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
( abs `  x
)  =  1  <->  ( abs `  ( ( * `
 A )  / 
( abs `  A
) ) )  =  1 ) )
38 oveq1 6047 . . . . . 6  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
x  x.  A )  =  ( ( ( * `  A )  /  ( abs `  A
) )  x.  A
) )
3938eqeq2d 2415 . . . . 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 3012 . . 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 9005 . . . 4  |-  _i  e.  CC
44 absi 12046 . . . . 5  |-  ( abs `  _i )  =  1
4543mul01i 9212 . . . . . 6  |-  ( _i  x.  0 )  =  0
4645eqcomi 2408 . . . . 5  |-  0  =  ( _i  x.  0 )
4744, 46pm3.2i 442 . . . 4  |-  ( ( abs `  _i )  =  1  /\  0  =  ( _i  x.  0 ) )
48 fveq2 5687 . . . . . . 7  |-  ( x  =  _i  ->  ( abs `  x )  =  ( abs `  _i ) )
4948eqeq1d 2412 . . . . . 6  |-  ( x  =  _i  ->  (
( abs `  x
)  =  1  <->  ( abs `  _i )  =  1 ) )
50 oveq1 6047 . . . . . . 7  |-  ( x  =  _i  ->  (
x  x.  0 )  =  ( _i  x.  0 ) )
5150eqeq2d 2415 . . . . . 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 3012 . . . 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 2642 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 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   _ici 8948    x. cmul 8951    / cdiv 9633   2c2 10005   ^cexp 11337   *ccj 11856   abscabs 11994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996
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