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Theorem sqabs 13165
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 12161 . . . . 5  |-  ( A  e.  RR  ->  ( A ^ 2 )  e.  RR )
2 sqge0 12170 . . . . 5  |-  ( A  e.  RR  ->  0  <_  ( A ^ 2 ) )
3 absid 13154 . . . . 5  |-  ( ( ( A ^ 2 )  e.  RR  /\  0  <_  ( A ^
2 ) )  -> 
( abs `  ( A ^ 2 ) )  =  ( A ^
2 ) )
41, 2, 3syl2anc 659 . . . 4  |-  ( A  e.  RR  ->  ( abs `  ( A ^
2 ) )  =  ( A ^ 2 ) )
5 recn 9515 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
6 2nn0 10751 . . . . 5  |-  2  e.  NN0
7 absexp 13162 . . . . 5  |-  ( ( A  e.  CC  /\  2  e.  NN0 )  -> 
( abs `  ( A ^ 2 ) )  =  ( ( abs `  A ) ^ 2 ) )
85, 6, 7sylancl 660 . . . 4  |-  ( A  e.  RR  ->  ( abs `  ( A ^
2 ) )  =  ( ( abs `  A
) ^ 2 ) )
94, 8eqtr3d 2439 . . 3  |-  ( A  e.  RR  ->  ( A ^ 2 )  =  ( ( abs `  A
) ^ 2 ) )
10 resqcl 12161 . . . . 5  |-  ( B  e.  RR  ->  ( B ^ 2 )  e.  RR )
11 sqge0 12170 . . . . 5  |-  ( B  e.  RR  ->  0  <_  ( B ^ 2 ) )
12 absid 13154 . . . . 5  |-  ( ( ( B ^ 2 )  e.  RR  /\  0  <_  ( B ^
2 ) )  -> 
( abs `  ( B ^ 2 ) )  =  ( B ^
2 ) )
1310, 11, 12syl2anc 659 . . . 4  |-  ( B  e.  RR  ->  ( abs `  ( B ^
2 ) )  =  ( B ^ 2 ) )
14 recn 9515 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
15 absexp 13162 . . . . 5  |-  ( ( B  e.  CC  /\  2  e.  NN0 )  -> 
( abs `  ( B ^ 2 ) )  =  ( ( abs `  B ) ^ 2 ) )
1614, 6, 15sylancl 660 . . . 4  |-  ( B  e.  RR  ->  ( abs `  ( B ^
2 ) )  =  ( ( abs `  B
) ^ 2 ) )
1713, 16eqtr3d 2439 . . 3  |-  ( B  e.  RR  ->  ( B ^ 2 )  =  ( ( abs `  B
) ^ 2 ) )
189, 17eqeqan12d 2419 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( ( abs `  A
) ^ 2 )  =  ( ( abs `  B ) ^ 2 ) ) )
19 abscl 13136 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
20 absge0 13145 . . . . 5  |-  ( A  e.  CC  ->  0  <_  ( abs `  A
) )
2119, 20jca 530 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
22 abscl 13136 . . . . 5  |-  ( B  e.  CC  ->  ( abs `  B )  e.  RR )
23 absge0 13145 . . . . 5  |-  ( B  e.  CC  ->  0  <_  ( abs `  B
) )
2422, 23jca 530 . . . 4  |-  ( B  e.  CC  ->  (
( abs `  B
)  e.  RR  /\  0  <_  ( abs `  B
) ) )
25 sq11 12166 . . . 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 475 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( abs `  A ) ^ 2 )  =  ( ( abs `  B ) ^ 2 )  <->  ( abs `  A )  =  ( abs `  B ) ) )
275, 14, 26syl2an 475 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( abs `  A ) ^ 2 )  =  ( ( abs `  B ) ^ 2 )  <->  ( abs `  A )  =  ( abs `  B ) ) )
2818, 27bitrd 253 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( abs `  A
)  =  ( abs `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836   class class class wbr 4384   ` cfv 5513  (class class class)co 6218   CCcc 9423   RRcr 9424   0cc0 9425    <_ cle 9562   2c2 10524   NN0cn0 10734   ^cexp 12092   abscabs 13092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502  ax-pre-sup 9503
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-2nd 6722  df-recs 6982  df-rdg 7016  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-sup 7838  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-div 10146  df-nn 10475  df-2 10533  df-3 10534  df-n0 10735  df-z 10804  df-uz 11024  df-rp 11162  df-seq 12034  df-exp 12093  df-cj 12957  df-re 12958  df-im 12959  df-sqrt 13093  df-abs 13094
This theorem is referenced by:  coskpi  23021
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