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Theorem sq01 12400
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 2616 . . . . 5  |-  ( A  =/=  0  <->  -.  A  =  0 )
2 sqval 12340 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A ^ 2 )  =  ( A  x.  A
) )
3 mulid1 9647 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
43eqcomd 2430 . . . . . . . . . 10  |-  ( A  e.  CC  ->  A  =  ( A  x.  1 ) )
52, 4eqeq12d 2444 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  A  <->  ( A  x.  A )  =  ( A  x.  1 ) ) )
65adantr 466 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A ^
2 )  =  A  <-> 
( A  x.  A
)  =  ( A  x.  1 ) ) )
7 ax-1cn 9604 . . . . . . . . . 10  |-  1  e.  CC
8 mulcan 10256 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  1  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( A  x.  A )  =  ( A  x.  1 )  <-> 
A  =  1 ) )
97, 8mp3an2 1348 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) )  ->  ( ( A  x.  A )  =  ( A  x.  1 )  <->  A  = 
1 ) )
109anabss5 823 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  x.  A )  =  ( A  x.  1 )  <-> 
A  =  1 ) )
116, 10bitrd 256 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A ^
2 )  =  A  <-> 
A  =  1 ) )
1211biimpd 210 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A ^
2 )  =  A  ->  A  =  1 ) )
1312impancom 441 . . . . 5  |-  ( ( A  e.  CC  /\  ( A ^ 2 )  =  A )  -> 
( A  =/=  0  ->  A  =  1 ) )
141, 13syl5bir 221 . . . 4  |-  ( ( A  e.  CC  /\  ( A ^ 2 )  =  A )  -> 
( -.  A  =  0  ->  A  = 
1 ) )
1514orrd 379 . . 3  |-  ( ( A  e.  CC  /\  ( A ^ 2 )  =  A )  -> 
( A  =  0  \/  A  =  1 ) )
1615ex 435 . 2  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  A  -> 
( A  =  0  \/  A  =  1 ) ) )
17 sq0 12372 . . . 4  |-  ( 0 ^ 2 )  =  0
18 oveq1 6312 . . . 4  |-  ( A  =  0  ->  ( A ^ 2 )  =  ( 0 ^ 2 ) )
19 id 22 . . . 4  |-  ( A  =  0  ->  A  =  0 )
2017, 18, 193eqtr4a 2489 . . 3  |-  ( A  =  0  ->  ( A ^ 2 )  =  A )
21 sq1 12375 . . . 4  |-  ( 1 ^ 2 )  =  1
22 oveq1 6312 . . . 4  |-  ( A  =  1  ->  ( A ^ 2 )  =  ( 1 ^ 2 ) )
23 id 22 . . . 4  |-  ( A  =  1  ->  A  =  1 )
2421, 22, 233eqtr4a 2489 . . 3  |-  ( A  =  1  ->  ( A ^ 2 )  =  A )
2520, 24jaoi 380 . 2  |-  ( ( A  =  0  \/  A  =  1 )  ->  ( A ^
2 )  =  A )
2616, 25impbid1 206 1  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  A  <->  ( A  =  0  \/  A  =  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614  (class class class)co 6305   CCcc 9544   0cc0 9546   1c1 9547    x. cmul 9551   2c2 10666   ^cexp 12278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-seq 12220  df-exp 12279
This theorem is referenced by:  cphsubrglem  22153
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