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Theorem sqeqd 11972
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 11496 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( A  =  B  \/  A  =  -u B ) ) )
52, 3, 4syl2anc 644 . . . . 5  |-  ( ph  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( A  =  B  \/  A  =  -u B ) ) )
61, 5mpbid 203 . . . 4  |-  ( ph  ->  ( A  =  B  \/  A  =  -u B ) )
76ord 368 . . 3  |-  ( ph  ->  ( -.  A  =  B  ->  A  =  -u B ) )
8 simpl 445 . . . . 5  |-  ( (
ph  /\  A  =  -u B )  ->  ph )
9 fveq2 5729 . . . . . . 7  |-  ( A  =  -u B  ->  (
Re `  A )  =  ( Re `  -u B ) )
10 reneg 11931 . . . . . . . 8  |-  ( B  e.  CC  ->  (
Re `  -u B )  =  -u ( Re `  B ) )
113, 10syl 16 . . . . . . 7  |-  ( ph  ->  ( Re `  -u B
)  =  -u (
Re `  B )
)
129, 11sylan9eqr 2491 . . . . . 6  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  A )  =  -u ( Re `  B ) )
13 sqeqd.4 . . . . . . . . . . . 12  |-  ( ph  ->  0  <_  ( Re `  A ) )
1413adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  -u B )  ->  0  <_  ( Re `  A
) )
1514, 12breqtrd 4237 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  -u B )  ->  0  <_ 
-u ( Re `  B ) )
163adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =  -u B )  ->  B  e.  CC )
17 recl 11916 . . . . . . . . . . . 12  |-  ( B  e.  CC  ->  (
Re `  B )  e.  RR )
1816, 17syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  B )  e.  RR )
1918le0neg1d 9599 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  -u B )  ->  (
( Re `  B
)  <_  0  <->  0  <_  -u ( Re `  B ) ) )
2015, 19mpbird 225 . . . . . . . . 9  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  B )  <_  0 )
21 sqeqd.5 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( Re `  B ) )
2221adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  A  =  -u B )  ->  0  <_  ( Re `  B
) )
23 0re 9092 . . . . . . . . . 10  |-  0  e.  RR
24 letri3 9161 . . . . . . . . . 10  |-  ( ( ( Re `  B
)  e.  RR  /\  0  e.  RR )  ->  ( ( Re `  B )  =  0  <-> 
( ( Re `  B )  <_  0  /\  0  <_  ( Re
`  B ) ) ) )
2518, 23, 24sylancl 645 . . . . . . . . 9  |-  ( (
ph  /\  A  =  -u B )  ->  (
( Re `  B
)  =  0  <->  (
( Re `  B
)  <_  0  /\  0  <_  ( Re `  B ) ) ) )
2620, 22, 25mpbir2and 890 . . . . . . . 8  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  B )  =  0 )
2726negeqd 9301 . . . . . . 7  |-  ( (
ph  /\  A  =  -u B )  ->  -u (
Re `  B )  =  -u 0 )
28 neg0 9348 . . . . . . 7  |-  -u 0  =  0
2927, 28syl6eq 2485 . . . . . 6  |-  ( (
ph  /\  A  =  -u B )  ->  -u (
Re `  B )  =  0 )
3012, 29eqtrd 2469 . . . . 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 1185 . . . 4  |-  ( (
ph  /\  A  =  -u B )  ->  A  =  B )
3332ex 425 . . 3  |-  ( ph  ->  ( A  =  -u B  ->  A  =  B ) )
347, 33syld 43 . 2  |-  ( ph  ->  ( -.  A  =  B  ->  A  =  B ) )
3534pm2.18d 106 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   CCcc 8989   RRcr 8990   0cc0 8991    <_ cle 9122   -ucneg 9293   2c2 10050   ^cexp 11383   Recre 11903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-n0 10223  df-z 10284  df-uz 10490  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907
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