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Theorem sqeqori 11211
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 11210 . . . 4  |-  ( ( A ^ 2 )  -  ( B ^
2 ) )  =  ( ( A  +  B )  x.  ( A  -  B )
)
43eqeq1i 2291 . . 3  |-  ( ( ( A ^ 2 )  -  ( B ^ 2 ) )  =  0  <->  ( ( A  +  B )  x.  ( A  -  B
) )  =  0 )
51sqcli 11180 . . . 4  |-  ( A ^ 2 )  e.  CC
62sqcli 11180 . . . 4  |-  ( B ^ 2 )  e.  CC
75, 6subeq0i 9122 . . 3  |-  ( ( ( A ^ 2 )  -  ( B ^ 2 ) )  =  0  <->  ( A ^ 2 )  =  ( B ^ 2 ) )
81, 2addcli 8837 . . . 4  |-  ( A  +  B )  e.  CC
91, 2subcli 9118 . . . 4  |-  ( A  -  B )  e.  CC
108, 9mul0ori 9412 . . 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 9122 . . 3  |-  ( ( A  -  B )  =  0  <->  A  =  B )
141, 2subnegi 9121 . . . . 5  |-  ( A  -  -u B )  =  ( A  +  B
)
1514eqeq1i 2291 . . . 4  |-  ( ( A  -  -u B
)  =  0  <->  ( A  +  B )  =  0 )
162negcli 9110 . . . . 5  |-  -u B  e.  CC
171, 16subeq0i 9122 . . . 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 1623    e. wcel 1685  (class class class)co 5820   CCcc 8731   0cc0 8733    + caddc 8736    x. cmul 8738    - cmin 9033   -ucneg 9034   2c2 9791   ^cexp 11100
This theorem is referenced by:  subsq0i  11212  sqeqor  11213  sinhalfpilem  19830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-2 9800  df-n0 9962  df-z 10021  df-uz 10227  df-seq 11043  df-exp 11101
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