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Theorem sqabs 12114
Description: The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)
Assertion
Ref Expression
sqabs  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( abs `  A
)  =  ( abs `  B ) ) )

Proof of Theorem sqabs
StepHypRef Expression
1 resqcl 11451 . . . . 5  |-  ( A  e.  RR  ->  ( A ^ 2 )  e.  RR )
2 sqge0 11460 . . . . 5  |-  ( A  e.  RR  ->  0  <_  ( A ^ 2 ) )
3 absid 12103 . . . . 5  |-  ( ( ( A ^ 2 )  e.  RR  /\  0  <_  ( A ^
2 ) )  -> 
( abs `  ( A ^ 2 ) )  =  ( A ^
2 ) )
41, 2, 3syl2anc 644 . . . 4  |-  ( A  e.  RR  ->  ( abs `  ( A ^
2 ) )  =  ( A ^ 2 ) )
5 recn 9082 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
6 2nn0 10240 . . . . 5  |-  2  e.  NN0
7 absexp 12111 . . . . 5  |-  ( ( A  e.  CC  /\  2  e.  NN0 )  -> 
( abs `  ( A ^ 2 ) )  =  ( ( abs `  A ) ^ 2 ) )
85, 6, 7sylancl 645 . . . 4  |-  ( A  e.  RR  ->  ( abs `  ( A ^
2 ) )  =  ( ( abs `  A
) ^ 2 ) )
94, 8eqtr3d 2472 . . 3  |-  ( A  e.  RR  ->  ( A ^ 2 )  =  ( ( abs `  A
) ^ 2 ) )
10 resqcl 11451 . . . . 5  |-  ( B  e.  RR  ->  ( B ^ 2 )  e.  RR )
11 sqge0 11460 . . . . 5  |-  ( B  e.  RR  ->  0  <_  ( B ^ 2 ) )
12 absid 12103 . . . . 5  |-  ( ( ( B ^ 2 )  e.  RR  /\  0  <_  ( B ^
2 ) )  -> 
( abs `  ( B ^ 2 ) )  =  ( B ^
2 ) )
1310, 11, 12syl2anc 644 . . . 4  |-  ( B  e.  RR  ->  ( abs `  ( B ^
2 ) )  =  ( B ^ 2 ) )
14 recn 9082 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
15 absexp 12111 . . . . 5  |-  ( ( B  e.  CC  /\  2  e.  NN0 )  -> 
( abs `  ( B ^ 2 ) )  =  ( ( abs `  B ) ^ 2 ) )
1614, 6, 15sylancl 645 . . . 4  |-  ( B  e.  RR  ->  ( abs `  ( B ^
2 ) )  =  ( ( abs `  B
) ^ 2 ) )
1713, 16eqtr3d 2472 . . 3  |-  ( B  e.  RR  ->  ( B ^ 2 )  =  ( ( abs `  B
) ^ 2 ) )
189, 17eqeqan12d 2453 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( ( abs `  A
) ^ 2 )  =  ( ( abs `  B ) ^ 2 ) ) )
19 abscl 12085 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
20 absge0 12094 . . . . 5  |-  ( A  e.  CC  ->  0  <_  ( abs `  A
) )
2119, 20jca 520 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
22 abscl 12085 . . . . 5  |-  ( B  e.  CC  ->  ( abs `  B )  e.  RR )
23 absge0 12094 . . . . 5  |-  ( B  e.  CC  ->  0  <_  ( abs `  B
) )
2422, 23jca 520 . . . 4  |-  ( B  e.  CC  ->  (
( abs `  B
)  e.  RR  /\  0  <_  ( abs `  B
) ) )
25 sq11 11456 . . . 4  |-  ( ( ( ( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) )  /\  (
( abs `  B
)  e.  RR  /\  0  <_  ( abs `  B
) ) )  -> 
( ( ( abs `  A ) ^ 2 )  =  ( ( abs `  B ) ^ 2 )  <->  ( abs `  A )  =  ( abs `  B ) ) )
2621, 24, 25syl2an 465 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( abs `  A ) ^ 2 )  =  ( ( abs `  B ) ^ 2 )  <->  ( abs `  A )  =  ( abs `  B ) ) )
275, 14, 26syl2an 465 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( abs `  A ) ^ 2 )  =  ( ( abs `  B ) ^ 2 )  <->  ( abs `  A )  =  ( abs `  B ) ) )
2818, 27bitrd 246 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( abs `  A
)  =  ( abs `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992    <_ cle 9123   2c2 10051   NN0cn0 10223   ^cexp 11384   abscabs 12041
This theorem is referenced by:  coskpi  20430  dvreacos  26293
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 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043
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