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Theorem sqeqd 11647
Description: A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.)
Hypotheses
Ref Expression
sqeqd.1  |-  ( ph  ->  A  e.  CC )
sqeqd.2  |-  ( ph  ->  B  e.  CC )
sqeqd.3  |-  ( ph  ->  ( A ^ 2 )  =  ( B ^ 2 ) )
sqeqd.4  |-  ( ph  ->  0  <_  ( Re `  A ) )
sqeqd.5  |-  ( ph  ->  0  <_  ( Re `  B ) )
sqeqd.6  |-  ( (
ph  /\  ( Re `  A )  =  0  /\  ( Re `  B )  =  0 )  ->  A  =  B )
Assertion
Ref Expression
sqeqd  |-  ( ph  ->  A  =  B )

Proof of Theorem sqeqd
StepHypRef Expression
1 sqeqd.3 . . . . 5  |-  ( ph  ->  ( A ^ 2 )  =  ( B ^ 2 ) )
2 sqeqd.1 . . . . . 6  |-  ( ph  ->  A  e.  CC )
3 sqeqd.2 . . . . . 6  |-  ( ph  ->  B  e.  CC )
4 sqeqor 11213 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( A  =  B  \/  A  =  -u B ) ) )
52, 3, 4syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( A  =  B  \/  A  =  -u B ) ) )
61, 5mpbid 201 . . . 4  |-  ( ph  ->  ( A  =  B  \/  A  =  -u B ) )
76ord 366 . . 3  |-  ( ph  ->  ( -.  A  =  B  ->  A  =  -u B ) )
8 simpl 443 . . . . 5  |-  ( (
ph  /\  A  =  -u B )  ->  ph )
9 fveq2 5486 . . . . . . 7  |-  ( A  =  -u B  ->  (
Re `  A )  =  ( Re `  -u B ) )
10 reneg 11606 . . . . . . . 8  |-  ( B  e.  CC  ->  (
Re `  -u B )  =  -u ( Re `  B ) )
113, 10syl 15 . . . . . . 7  |-  ( ph  ->  ( Re `  -u B
)  =  -u (
Re `  B )
)
129, 11sylan9eqr 2338 . . . . . 6  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  A )  =  -u ( Re `  B ) )
13 sqeqd.4 . . . . . . . . . . . 12  |-  ( ph  ->  0  <_  ( Re `  A ) )
1413adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  -u B )  ->  0  <_  ( Re `  A
) )
1514, 12breqtrd 4048 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  -u B )  ->  0  <_ 
-u ( Re `  B ) )
163adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =  -u B )  ->  B  e.  CC )
17 recl 11591 . . . . . . . . . . . 12  |-  ( B  e.  CC  ->  (
Re `  B )  e.  RR )
1816, 17syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  B )  e.  RR )
1918le0neg1d 9340 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  -u B )  ->  (
( Re `  B
)  <_  0  <->  0  <_  -u ( Re `  B ) ) )
2015, 19mpbird 223 . . . . . . . . 9  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  B )  <_  0 )
21 sqeqd.5 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( Re `  B ) )
2221adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  A  =  -u B )  ->  0  <_  ( Re `  B
) )
23 0re 8834 . . . . . . . . . 10  |-  0  e.  RR
24 letri3 8903 . . . . . . . . . 10  |-  ( ( ( Re `  B
)  e.  RR  /\  0  e.  RR )  ->  ( ( Re `  B )  =  0  <-> 
( ( Re `  B )  <_  0  /\  0  <_  ( Re
`  B ) ) ) )
2518, 23, 24sylancl 643 . . . . . . . . 9  |-  ( (
ph  /\  A  =  -u B )  ->  (
( Re `  B
)  =  0  <->  (
( Re `  B
)  <_  0  /\  0  <_  ( Re `  B ) ) ) )
2620, 22, 25mpbir2and 888 . . . . . . . 8  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  B )  =  0 )
2726negeqd 9042 . . . . . . 7  |-  ( (
ph  /\  A  =  -u B )  ->  -u (
Re `  B )  =  -u 0 )
28 neg0 9089 . . . . . . 7  |-  -u 0  =  0
2927, 28syl6eq 2332 . . . . . 6  |-  ( (
ph  /\  A  =  -u B )  ->  -u (
Re `  B )  =  0 )
3012, 29eqtrd 2316 . . . . 5  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  A )  =  0 )
31 sqeqd.6 . . . . 5  |-  ( (
ph  /\  ( Re `  A )  =  0  /\  ( Re `  B )  =  0 )  ->  A  =  B )
328, 30, 26, 31syl3anc 1182 . . . 4  |-  ( (
ph  /\  A  =  -u B )  ->  A  =  B )
3332ex 423 . . 3  |-  ( ph  ->  ( A  =  -u B  ->  A  =  B ) )
347, 33syld 40 . 2  |-  ( ph  ->  ( -.  A  =  B  ->  A  =  B ) )
3534pm2.18d 103 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   CCcc 8731   RRcr 8732   0cc0 8733    <_ cle 8864   -ucneg 9034   2c2 9791   ^cexp 11100   Recre 11578
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-rmo 2552  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-div 9420  df-nn 9743  df-2 9800  df-n0 9962  df-z 10021  df-uz 10227  df-seq 11043  df-exp 11101  df-cj 11580  df-re 11581  df-im 11582
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