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Theorem sqeqori 12262
Description: The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of [Gleason] p. 311 and its converse. (Contributed by NM, 15-Jan-2006.)
Hypotheses
Ref Expression
binom2.1  |-  A  e.  CC
binom2.2  |-  B  e.  CC
Assertion
Ref Expression
sqeqori  |-  ( ( A ^ 2 )  =  ( B ^
2 )  <->  ( A  =  B  \/  A  =  -u B ) )

Proof of Theorem sqeqori
StepHypRef Expression
1 binom2.1 . . . . 5  |-  A  e.  CC
2 binom2.2 . . . . 5  |-  B  e.  CC
31, 2subsqi 12261 . . . 4  |-  ( ( A ^ 2 )  -  ( B ^
2 ) )  =  ( ( A  +  B )  x.  ( A  -  B )
)
43eqeq1i 2461 . . 3  |-  ( ( ( A ^ 2 )  -  ( B ^ 2 ) )  =  0  <->  ( ( A  +  B )  x.  ( A  -  B
) )  =  0 )
51sqcli 12230 . . . 4  |-  ( A ^ 2 )  e.  CC
62sqcli 12230 . . . 4  |-  ( B ^ 2 )  e.  CC
75, 6subeq0i 9890 . . 3  |-  ( ( ( A ^ 2 )  -  ( B ^ 2 ) )  =  0  <->  ( A ^ 2 )  =  ( B ^ 2 ) )
81, 2addcli 9589 . . . 4  |-  ( A  +  B )  e.  CC
91, 2subcli 9886 . . . 4  |-  ( A  -  B )  e.  CC
108, 9mul0ori 10193 . . 3  |-  ( ( ( A  +  B
)  x.  ( A  -  B ) )  =  0  <->  ( ( A  +  B )  =  0  \/  ( A  -  B )  =  0 ) )
114, 7, 103bitr3i 275 . 2  |-  ( ( A ^ 2 )  =  ( B ^
2 )  <->  ( ( A  +  B )  =  0  \/  ( A  -  B )  =  0 ) )
12 orcom 385 . 2  |-  ( ( ( A  +  B
)  =  0  \/  ( A  -  B
)  =  0 )  <-> 
( ( A  -  B )  =  0  \/  ( A  +  B )  =  0 ) )
131, 2subeq0i 9890 . . 3  |-  ( ( A  -  B )  =  0  <->  A  =  B )
141, 2subnegi 9889 . . . . 5  |-  ( A  -  -u B )  =  ( A  +  B
)
1514eqeq1i 2461 . . . 4  |-  ( ( A  -  -u B
)  =  0  <->  ( A  +  B )  =  0 )
162negcli 9878 . . . . 5  |-  -u B  e.  CC
171, 16subeq0i 9890 . . . 4  |-  ( ( A  -  -u B
)  =  0  <->  A  =  -u B )
1815, 17bitr3i 251 . . 3  |-  ( ( A  +  B )  =  0  <->  A  =  -u B )
1913, 18orbi12i 519 . 2  |-  ( ( ( A  -  B
)  =  0  \/  ( A  +  B
)  =  0 )  <-> 
( A  =  B  \/  A  =  -u B ) )
2011, 12, 193bitri 271 1  |-  ( ( A ^ 2 )  =  ( B ^
2 )  <->  ( A  =  B  \/  A  =  -u B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 366    = wceq 1398    e. wcel 1823  (class class class)co 6270   CCcc 9479   0cc0 9481    + caddc 9484    x. cmul 9486    - cmin 9796   -ucneg 9797   2c2 10581   ^cexp 12148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-seq 12090  df-exp 12149
This theorem is referenced by:  subsq0i  12263  sqeqor  12264  sinhalfpilem  23022
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