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Theorem sq01 12291
Description: If a complex number equals its square, it must be 0 or 1. (Contributed by NM, 6-Jun-2006.)
Assertion
Ref Expression
sq01  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  A  <->  ( A  =  0  \/  A  =  1 ) ) )

Proof of Theorem sq01
StepHypRef Expression
1 df-ne 2654 . . . . 5  |-  ( A  =/=  0  <->  -.  A  =  0 )
2 sqval 12230 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A ^ 2 )  =  ( A  x.  A
) )
3 mulid1 9610 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
43eqcomd 2465 . . . . . . . . . 10  |-  ( A  e.  CC  ->  A  =  ( A  x.  1 ) )
52, 4eqeq12d 2479 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  A  <->  ( A  x.  A )  =  ( A  x.  1 ) ) )
65adantr 465 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A ^
2 )  =  A  <-> 
( A  x.  A
)  =  ( A  x.  1 ) ) )
7 ax-1cn 9567 . . . . . . . . . 10  |-  1  e.  CC
8 mulcan 10207 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  1  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( A  x.  A )  =  ( A  x.  1 )  <-> 
A  =  1 ) )
97, 8mp3an2 1312 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) )  ->  ( ( A  x.  A )  =  ( A  x.  1 )  <->  A  = 
1 ) )
109anabss5 816 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  x.  A )  =  ( A  x.  1 )  <-> 
A  =  1 ) )
116, 10bitrd 253 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A ^
2 )  =  A  <-> 
A  =  1 ) )
1211biimpd 207 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A ^
2 )  =  A  ->  A  =  1 ) )
1312impancom 440 . . . . 5  |-  ( ( A  e.  CC  /\  ( A ^ 2 )  =  A )  -> 
( A  =/=  0  ->  A  =  1 ) )
141, 13syl5bir 218 . . . 4  |-  ( ( A  e.  CC  /\  ( A ^ 2 )  =  A )  -> 
( -.  A  =  0  ->  A  = 
1 ) )
1514orrd 378 . . 3  |-  ( ( A  e.  CC  /\  ( A ^ 2 )  =  A )  -> 
( A  =  0  \/  A  =  1 ) )
1615ex 434 . 2  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  A  -> 
( A  =  0  \/  A  =  1 ) ) )
17 sq0 12262 . . . 4  |-  ( 0 ^ 2 )  =  0
18 oveq1 6303 . . . 4  |-  ( A  =  0  ->  ( A ^ 2 )  =  ( 0 ^ 2 ) )
19 id 22 . . . 4  |-  ( A  =  0  ->  A  =  0 )
2017, 18, 193eqtr4a 2524 . . 3  |-  ( A  =  0  ->  ( A ^ 2 )  =  A )
21 sq1 12265 . . . 4  |-  ( 1 ^ 2 )  =  1
22 oveq1 6303 . . . 4  |-  ( A  =  1  ->  ( A ^ 2 )  =  ( 1 ^ 2 ) )
23 id 22 . . . 4  |-  ( A  =  1  ->  A  =  1 )
2421, 22, 233eqtr4a 2524 . . 3  |-  ( A  =  1  ->  ( A ^ 2 )  =  A )
2520, 24jaoi 379 . 2  |-  ( ( A  =  0  \/  A  =  1 )  ->  ( A ^
2 )  =  A )
2616, 25impbid1 203 1  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  A  <->  ( A  =  0  \/  A  =  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    x. cmul 9514   2c2 10606   ^cexp 12169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-seq 12111  df-exp 12170
This theorem is referenced by:  cphsubrglem  21750
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