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Theorem abs1m 11819
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 5525 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
2 abs0 11770 . . . . . 6  |-  ( abs `  0 )  =  0
31, 2syl6eq 2331 . . . . 5  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
4 oveq2 5866 . . . . 5  |-  ( A  =  0  ->  (
x  x.  A )  =  ( x  x.  0 ) )
53, 4eqeq12d 2297 . . . 4  |-  ( A  =  0  ->  (
( abs `  A
)  =  ( x  x.  A )  <->  0  =  ( x  x.  0
) ) )
65anbi2d 684 . . 3  |-  ( A  =  0  ->  (
( ( abs `  x
)  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) )  <-> 
( ( abs `  x
)  =  1  /\  0  =  ( x  x.  0 ) ) ) )
76rexbidv 2564 . 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 443 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  e.  CC )
98cjcld 11681 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( * `  A
)  e.  CC )
10 abscl 11763 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
1110adantr 451 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  RR )
1211recnd 8861 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  CC )
13 abs00 11774 . . . . . 6  |-  ( A  e.  CC  ->  (
( abs `  A
)  =  0  <->  A  =  0 ) )
1413necon3bid 2481 . . . . 5  |-  ( A  e.  CC  ->  (
( abs `  A
)  =/=  0  <->  A  =/=  0 ) )
1514biimpar 471 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =/=  0 )
169, 12, 15divcld 9536 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( * `  A )  /  ( abs `  A ) )  e.  CC )
17 absdiv 11780 . . . . 5  |-  ( ( ( * `  A
)  e.  CC  /\  ( abs `  A )  e.  CC  /\  ( abs `  A )  =/=  0 )  ->  ( abs `  ( ( * `
 A )  / 
( abs `  A
) ) )  =  ( ( abs `  (
* `  A )
)  /  ( abs `  ( abs `  A
) ) ) )
189, 12, 15, 17syl3anc 1182 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
( * `  A
)  /  ( abs `  A ) ) )  =  ( ( abs `  ( * `  A
) )  /  ( abs `  ( abs `  A
) ) ) )
19 abscj 11764 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( * `  A ) )  =  ( abs `  A
) )
2019adantr 451 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
* `  A )
)  =  ( abs `  A ) )
21 absidm 11807 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( abs `  A
) )  =  ( abs `  A ) )
2221adantr 451 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  ( abs `  A ) )  =  ( abs `  A
) )
2320, 22oveq12d 5876 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  (
* `  A )
)  /  ( abs `  ( abs `  A
) ) )  =  ( ( abs `  A
)  /  ( abs `  A ) ) )
2412, 15dividd 9534 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
)  /  ( abs `  A ) )  =  1 )
2518, 23, 243eqtrd 2319 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
( * `  A
)  /  ( abs `  A ) ) )  =  1 )
268, 9, 12, 15divassd 9571 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  x.  ( * `  A
) )  /  ( abs `  A ) )  =  ( A  x.  ( ( * `  A )  /  ( abs `  A ) ) ) )
2712, 12, 15divcan3d 9541 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( abs `  A )  x.  ( abs `  A ) )  /  ( abs `  A
) )  =  ( abs `  A ) )
2812sqvald 11242 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
) ^ 2 )  =  ( ( abs `  A )  x.  ( abs `  A ) ) )
29 absvalsq 11765 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
3029adantr 451 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
3128, 30eqtr3d 2317 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
)  x.  ( abs `  A ) )  =  ( A  x.  (
* `  A )
) )
3231oveq1d 5873 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( abs `  A )  x.  ( abs `  A ) )  /  ( abs `  A
) )  =  ( ( A  x.  (
* `  A )
)  /  ( abs `  A ) ) )
3327, 32eqtr3d 2317 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =  ( ( A  x.  ( * `
 A ) )  /  ( abs `  A
) ) )
3416, 8mulcomd 8856 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( * `
 A )  / 
( abs `  A
) )  x.  A
)  =  ( A  x.  ( ( * `
 A )  / 
( abs `  A
) ) ) )
3526, 33, 343eqtr4d 2325 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =  ( ( ( * `  A
)  /  ( abs `  A ) )  x.  A ) )
36 fveq2 5525 . . . . . 6  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  ( abs `  x )  =  ( abs `  (
( * `  A
)  /  ( abs `  A ) ) ) )
3736eqeq1d 2291 . . . . 5  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
( abs `  x
)  =  1  <->  ( abs `  ( ( * `
 A )  / 
( abs `  A
) ) )  =  1 ) )
38 oveq1 5865 . . . . . 6  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
x  x.  A )  =  ( ( ( * `  A )  /  ( abs `  A
) )  x.  A
) )
3938eqeq2d 2294 . . . . 5  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
( abs `  A
)  =  ( x  x.  A )  <->  ( abs `  A )  =  ( ( ( * `  A )  /  ( abs `  A ) )  x.  A ) ) )
4037, 39anbi12d 691 . . . 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 2884 . . 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 1180 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  E. x  e.  CC  ( ( abs `  x
)  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) ) )
43 ax-icn 8796 . . . 4  |-  _i  e.  CC
44 absi 11771 . . . . 5  |-  ( abs `  _i )  =  1
4543mul01i 9002 . . . . . 6  |-  ( _i  x.  0 )  =  0
4645eqcomi 2287 . . . . 5  |-  0  =  ( _i  x.  0 )
4744, 46pm3.2i 441 . . . 4  |-  ( ( abs `  _i )  =  1  /\  0  =  ( _i  x.  0 ) )
48 fveq2 5525 . . . . . . 7  |-  ( x  =  _i  ->  ( abs `  x )  =  ( abs `  _i ) )
4948eqeq1d 2291 . . . . . 6  |-  ( x  =  _i  ->  (
( abs `  x
)  =  1  <->  ( abs `  _i )  =  1 ) )
50 oveq1 5865 . . . . . . 7  |-  ( x  =  _i  ->  (
x  x.  0 )  =  ( _i  x.  0 ) )
5150eqeq2d 2294 . . . . . 6  |-  ( x  =  _i  ->  (
0  =  ( x  x.  0 )  <->  0  =  ( _i  x.  0
) ) )
5249, 51anbi12d 691 . . . . 5  |-  ( x  =  _i  ->  (
( ( abs `  x
)  =  1  /\  0  =  ( x  x.  0 ) )  <-> 
( ( abs `  _i )  =  1  /\  0  =  ( _i  x.  0 ) ) ) )
5352rspcev 2884 . . . 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 653 . . 3  |-  E. x  e.  CC  ( ( abs `  x )  =  1  /\  0  =  ( x  x.  0 ) )
5554a1i 10 . 2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( ( abs `  x )  =  1  /\  0  =  ( x  x.  0 ) ) )
567, 42, 55pm2.61ne 2521 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 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   _ici 8739    x. cmul 8742    / cdiv 9423   2c2 9795   ^cexp 11104   *ccj 11581   abscabs 11719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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