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Theorem sqeqd 12951
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 12239 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( A  =  B  \/  A  =  -u B ) ) )
52, 3, 4syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( A  =  B  \/  A  =  -u B ) ) )
61, 5mpbid 210 . . . 4  |-  ( ph  ->  ( A  =  B  \/  A  =  -u B ) )
76ord 377 . . 3  |-  ( ph  ->  ( -.  A  =  B  ->  A  =  -u B ) )
8 simpl 457 . . . . 5  |-  ( (
ph  /\  A  =  -u B )  ->  ph )
9 fveq2 5859 . . . . . . 7  |-  ( A  =  -u B  ->  (
Re `  A )  =  ( Re `  -u B ) )
10 reneg 12910 . . . . . . . 8  |-  ( B  e.  CC  ->  (
Re `  -u B )  =  -u ( Re `  B ) )
113, 10syl 16 . . . . . . 7  |-  ( ph  ->  ( Re `  -u B
)  =  -u (
Re `  B )
)
129, 11sylan9eqr 2525 . . . . . 6  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  A )  =  -u ( Re `  B ) )
13 sqeqd.4 . . . . . . . . . . . 12  |-  ( ph  ->  0  <_  ( Re `  A ) )
1413adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  -u B )  ->  0  <_  ( Re `  A
) )
1514, 12breqtrd 4466 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  -u B )  ->  0  <_ 
-u ( Re `  B ) )
163adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =  -u B )  ->  B  e.  CC )
17 recl 12895 . . . . . . . . . . . 12  |-  ( B  e.  CC  ->  (
Re `  B )  e.  RR )
1816, 17syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  B )  e.  RR )
1918le0neg1d 10115 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  -u B )  ->  (
( Re `  B
)  <_  0  <->  0  <_  -u ( Re `  B ) ) )
2015, 19mpbird 232 . . . . . . . . 9  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  B )  <_  0 )
21 sqeqd.5 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( Re `  B ) )
2221adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  A  =  -u B )  ->  0  <_  ( Re `  B
) )
23 0re 9587 . . . . . . . . . 10  |-  0  e.  RR
24 letri3 9661 . . . . . . . . . 10  |-  ( ( ( Re `  B
)  e.  RR  /\  0  e.  RR )  ->  ( ( Re `  B )  =  0  <-> 
( ( Re `  B )  <_  0  /\  0  <_  ( Re
`  B ) ) ) )
2518, 23, 24sylancl 662 . . . . . . . . 9  |-  ( (
ph  /\  A  =  -u B )  ->  (
( Re `  B
)  =  0  <->  (
( Re `  B
)  <_  0  /\  0  <_  ( Re `  B ) ) ) )
2620, 22, 25mpbir2and 915 . . . . . . . 8  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  B )  =  0 )
2726negeqd 9805 . . . . . . 7  |-  ( (
ph  /\  A  =  -u B )  ->  -u (
Re `  B )  =  -u 0 )
28 neg0 9856 . . . . . . 7  |-  -u 0  =  0
2927, 28syl6eq 2519 . . . . . 6  |-  ( (
ph  /\  A  =  -u B )  ->  -u (
Re `  B )  =  0 )
3012, 29eqtrd 2503 . . . . 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 1223 . . . 4  |-  ( (
ph  /\  A  =  -u B )  ->  A  =  B )
3332ex 434 . . 3  |-  ( ph  ->  ( A  =  -u B  ->  A  =  B ) )
347, 33syld 44 . 2  |-  ( ph  ->  ( -.  A  =  B  ->  A  =  B ) )
3534pm2.18d 111 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483    <_ cle 9620   -ucneg 9797   2c2 10576   ^cexp 12124   Recre 12882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-seq 12066  df-exp 12125  df-cj 12884  df-re 12885  df-im 12886
This theorem is referenced by: (None)
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