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Theorem sqeqori 11217
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 11216 . . . 4  |-  ( ( A ^ 2 )  -  ( B ^
2 ) )  =  ( ( A  +  B )  x.  ( A  -  B )
)
43eqeq1i 2292 . . 3  |-  ( ( ( A ^ 2 )  -  ( B ^ 2 ) )  =  0  <->  ( ( A  +  B )  x.  ( A  -  B
) )  =  0 )
51sqcli 11186 . . . 4  |-  ( A ^ 2 )  e.  CC
62sqcli 11186 . . . 4  |-  ( B ^ 2 )  e.  CC
75, 6subeq0i 9128 . . 3  |-  ( ( ( A ^ 2 )  -  ( B ^ 2 ) )  =  0  <->  ( A ^ 2 )  =  ( B ^ 2 ) )
81, 2addcli 8843 . . . 4  |-  ( A  +  B )  e.  CC
91, 2subcli 9124 . . . 4  |-  ( A  -  B )  e.  CC
108, 9mul0ori 9418 . . 3  |-  ( ( ( A  +  B
)  x.  ( A  -  B ) )  =  0  <->  ( ( A  +  B )  =  0  \/  ( A  -  B )  =  0 ) )
114, 7, 103bitr3i 266 . 2  |-  ( ( A ^ 2 )  =  ( B ^
2 )  <->  ( ( A  +  B )  =  0  \/  ( A  -  B )  =  0 ) )
12 orcom 376 . 2  |-  ( ( ( A  +  B
)  =  0  \/  ( A  -  B
)  =  0 )  <-> 
( ( A  -  B )  =  0  \/  ( A  +  B )  =  0 ) )
131, 2subeq0i 9128 . . 3  |-  ( ( A  -  B )  =  0  <->  A  =  B )
141, 2subnegi 9127 . . . . 5  |-  ( A  -  -u B )  =  ( A  +  B
)
1514eqeq1i 2292 . . . 4  |-  ( ( A  -  -u B
)  =  0  <->  ( A  +  B )  =  0 )
162negcli 9116 . . . . 5  |-  -u B  e.  CC
171, 16subeq0i 9128 . . . 4  |-  ( ( A  -  -u B
)  =  0  <->  A  =  -u B )
1815, 17bitr3i 242 . . 3  |-  ( ( A  +  B )  =  0  <->  A  =  -u B )
1913, 18orbi12i 507 . 2  |-  ( ( ( A  -  B
)  =  0  \/  ( A  +  B
)  =  0 )  <-> 
( A  =  B  \/  A  =  -u B ) )
2011, 12, 193bitri 262 1  |-  ( ( A ^ 2 )  =  ( B ^
2 )  <->  ( A  =  B  \/  A  =  -u B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    = wceq 1625    e. wcel 1686  (class class class)co 5860   CCcc 8737   0cc0 8739    + caddc 8742    x. cmul 8744    - cmin 9039   -ucneg 9040   2c2 9797   ^cexp 11106
This theorem is referenced by:  subsq0i  11218  sqeqor  11219  sinhalfpilem  19836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-n0 9968  df-z 10027  df-uz 10233  df-seq 11049  df-exp 11107
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