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Theorem sq01 11503
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 2603 . . . . 5  |-  ( A  =/=  0  <->  -.  A  =  0 )
2 sqval 11443 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A ^ 2 )  =  ( A  x.  A
) )
3 mulid1 9090 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
43eqcomd 2443 . . . . . . . . . 10  |-  ( A  e.  CC  ->  A  =  ( A  x.  1 ) )
52, 4eqeq12d 2452 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  A  <->  ( A  x.  A )  =  ( A  x.  1 ) ) )
65adantr 453 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A ^
2 )  =  A  <-> 
( A  x.  A
)  =  ( A  x.  1 ) ) )
7 ax-1cn 9050 . . . . . . . . . 10  |-  1  e.  CC
8 mulcan 9661 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  1  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( A  x.  A )  =  ( A  x.  1 )  <-> 
A  =  1 ) )
97, 8mp3an2 1268 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) )  ->  ( ( A  x.  A )  =  ( A  x.  1 )  <->  A  = 
1 ) )
109anabss5 791 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  x.  A )  =  ( A  x.  1 )  <-> 
A  =  1 ) )
116, 10bitrd 246 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A ^
2 )  =  A  <-> 
A  =  1 ) )
1211biimpd 200 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A ^
2 )  =  A  ->  A  =  1 ) )
1312impancom 429 . . . . 5  |-  ( ( A  e.  CC  /\  ( A ^ 2 )  =  A )  -> 
( A  =/=  0  ->  A  =  1 ) )
141, 13syl5bir 211 . . . 4  |-  ( ( A  e.  CC  /\  ( A ^ 2 )  =  A )  -> 
( -.  A  =  0  ->  A  = 
1 ) )
1514orrd 369 . . 3  |-  ( ( A  e.  CC  /\  ( A ^ 2 )  =  A )  -> 
( A  =  0  \/  A  =  1 ) )
1615ex 425 . 2  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  A  -> 
( A  =  0  \/  A  =  1 ) ) )
17 sq0 11475 . . . 4  |-  ( 0 ^ 2 )  =  0
18 oveq1 6090 . . . 4  |-  ( A  =  0  ->  ( A ^ 2 )  =  ( 0 ^ 2 ) )
19 id 21 . . . 4  |-  ( A  =  0  ->  A  =  0 )
2017, 18, 193eqtr4a 2496 . . 3  |-  ( A  =  0  ->  ( A ^ 2 )  =  A )
21 sq1 11478 . . . 4  |-  ( 1 ^ 2 )  =  1
22 oveq1 6090 . . . 4  |-  ( A  =  1  ->  ( A ^ 2 )  =  ( 1 ^ 2 ) )
23 id 21 . . . 4  |-  ( A  =  1  ->  A  =  1 )
2421, 22, 233eqtr4a 2496 . . 3  |-  ( A  =  1  ->  ( A ^ 2 )  =  A )
2520, 24jaoi 370 . 2  |-  ( ( A  =  0  \/  A  =  1 )  ->  ( A ^
2 )  =  A )
2616, 25impbid1 196 1  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  A  <->  ( A  =  0  \/  A  =  1 ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601  (class class class)co 6083   CCcc 8990   0cc0 8992   1c1 8993    x. cmul 8997   2c2 10051   ^cexp 11384
This theorem is referenced by:  cphsubrglem  19142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-seq 11326  df-exp 11385
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