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Theorem abs1m 11749
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 5423 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
2 abs0 11700 . . . . . 6  |-  ( abs `  0 )  =  0
31, 2syl6eq 2304 . . . . 5  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
4 oveq2 5765 . . . . 5  |-  ( A  =  0  ->  (
x  x.  A )  =  ( x  x.  0 ) )
53, 4eqeq12d 2270 . . . 4  |-  ( A  =  0  ->  (
( abs `  A
)  =  ( x  x.  A )  <->  0  =  ( x  x.  0
) ) )
65anbi2d 687 . . 3  |-  ( A  =  0  ->  (
( ( abs `  x
)  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) )  <-> 
( ( abs `  x
)  =  1  /\  0  =  ( x  x.  0 ) ) ) )
76rexbidv 2535 . 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 445 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  e.  CC )
98cjcld 11611 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( * `  A
)  e.  CC )
10 abscl 11693 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
1110adantr 453 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  RR )
1211recnd 8794 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  CC )
13 abs00 11704 . . . . . 6  |-  ( A  e.  CC  ->  (
( abs `  A
)  =  0  <->  A  =  0 ) )
1413necon3bid 2454 . . . . 5  |-  ( A  e.  CC  ->  (
( abs `  A
)  =/=  0  <->  A  =/=  0 ) )
1514biimpar 473 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =/=  0 )
169, 12, 15divcld 9469 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( * `  A )  /  ( abs `  A ) )  e.  CC )
17 absdiv 11710 . . . . 5  |-  ( ( ( * `  A
)  e.  CC  /\  ( abs `  A )  e.  CC  /\  ( abs `  A )  =/=  0 )  ->  ( abs `  ( ( * `
 A )  / 
( abs `  A
) ) )  =  ( ( abs `  (
* `  A )
)  /  ( abs `  ( abs `  A
) ) ) )
189, 12, 15, 17syl3anc 1187 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
( * `  A
)  /  ( abs `  A ) ) )  =  ( ( abs `  ( * `  A
) )  /  ( abs `  ( abs `  A
) ) ) )
19 abscj 11694 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( * `  A ) )  =  ( abs `  A
) )
2019adantr 453 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
* `  A )
)  =  ( abs `  A ) )
21 absidm 11737 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( abs `  A
) )  =  ( abs `  A ) )
2221adantr 453 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  ( abs `  A ) )  =  ( abs `  A
) )
2320, 22oveq12d 5775 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  (
* `  A )
)  /  ( abs `  ( abs `  A
) ) )  =  ( ( abs `  A
)  /  ( abs `  A ) ) )
2412, 15dividd 9467 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
)  /  ( abs `  A ) )  =  1 )
2518, 23, 243eqtrd 2292 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  (
( * `  A
)  /  ( abs `  A ) ) )  =  1 )
268, 9, 12, 15divassd 9504 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  x.  ( * `  A
) )  /  ( abs `  A ) )  =  ( A  x.  ( ( * `  A )  /  ( abs `  A ) ) ) )
2712, 12, 15divcan3d 9474 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( abs `  A )  x.  ( abs `  A ) )  /  ( abs `  A
) )  =  ( abs `  A ) )
2812sqvald 11173 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
) ^ 2 )  =  ( ( abs `  A )  x.  ( abs `  A ) ) )
29 absvalsq 11695 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
3029adantr 453 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
3128, 30eqtr3d 2290 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( abs `  A
)  x.  ( abs `  A ) )  =  ( A  x.  (
* `  A )
) )
3231oveq1d 5772 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( abs `  A )  x.  ( abs `  A ) )  /  ( abs `  A
) )  =  ( ( A  x.  (
* `  A )
)  /  ( abs `  A ) ) )
3327, 32eqtr3d 2290 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =  ( ( A  x.  ( * `
 A ) )  /  ( abs `  A
) ) )
3416, 8mulcomd 8789 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( * `
 A )  / 
( abs `  A
) )  x.  A
)  =  ( A  x.  ( ( * `
 A )  / 
( abs `  A
) ) ) )
3526, 33, 343eqtr4d 2298 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  =  ( ( ( * `  A
)  /  ( abs `  A ) )  x.  A ) )
36 fveq2 5423 . . . . . 6  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  ( abs `  x )  =  ( abs `  (
( * `  A
)  /  ( abs `  A ) ) ) )
3736eqeq1d 2264 . . . . 5  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
( abs `  x
)  =  1  <->  ( abs `  ( ( * `
 A )  / 
( abs `  A
) ) )  =  1 ) )
38 oveq1 5764 . . . . . 6  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
x  x.  A )  =  ( ( ( * `  A )  /  ( abs `  A
) )  x.  A
) )
3938eqeq2d 2267 . . . . 5  |-  ( x  =  ( ( * `
 A )  / 
( abs `  A
) )  ->  (
( abs `  A
)  =  ( x  x.  A )  <->  ( abs `  A )  =  ( ( ( * `  A )  /  ( abs `  A ) )  x.  A ) ) )
4037, 39anbi12d 694 . . . 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
) ) ) )
4140rcla4ev 2835 . . 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 1185 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  E. x  e.  CC  ( ( abs `  x
)  =  1  /\  ( abs `  A
)  =  ( x  x.  A ) ) )
43 ax-icn 8729 . . . 4  |-  _i  e.  CC
44 absi 11701 . . . . 5  |-  ( abs `  _i )  =  1
4543mul01i 8935 . . . . . 6  |-  ( _i  x.  0 )  =  0
4645eqcomi 2260 . . . . 5  |-  0  =  ( _i  x.  0 )
4744, 46pm3.2i 443 . . . 4  |-  ( ( abs `  _i )  =  1  /\  0  =  ( _i  x.  0 ) )
48 fveq2 5423 . . . . . . 7  |-  ( x  =  _i  ->  ( abs `  x )  =  ( abs `  _i ) )
4948eqeq1d 2264 . . . . . 6  |-  ( x  =  _i  ->  (
( abs `  x
)  =  1  <->  ( abs `  _i )  =  1 ) )
50 oveq1 5764 . . . . . . 7  |-  ( x  =  _i  ->  (
x  x.  0 )  =  ( _i  x.  0 ) )
5150eqeq2d 2267 . . . . . 6  |-  ( x  =  _i  ->  (
0  =  ( x  x.  0 )  <->  0  =  ( _i  x.  0
) ) )
5249, 51anbi12d 694 . . . . 5  |-  ( x  =  _i  ->  (
( ( abs `  x
)  =  1  /\  0  =  ( x  x.  0 ) )  <-> 
( ( abs `  _i )  =  1  /\  0  =  ( _i  x.  0 ) ) ) )
5352rcla4ev 2835 . . . 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 656 . . 3  |-  E. x  e.  CC  ( ( abs `  x )  =  1  /\  0  =  ( x  x.  0 ) )
5554a1i 12 . 2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( ( abs `  x )  =  1  /\  0  =  ( x  x.  0 ) ) )
567, 42, 55pm2.61ne 2494 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 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   E.wrex 2517   ` cfv 4638  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670   1c1 8671   _ici 8672    x. cmul 8675    / cdiv 9356   2c2 9728   ^cexp 11035   *ccj 11511   abscabs 11649
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-sup 7127  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-seq 10978  df-exp 11036  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651
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