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Theorem sqreu 12853
Description: Existence and uniqueness for the square root function in general. (Contributed by Mario Carneiro, 9-Jul-2013.)
Assertion
Ref Expression
sqreu  |-  ( A  e.  CC  ->  E! x  e.  CC  (
( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )
Distinct variable group:    x, A

Proof of Theorem sqreu
StepHypRef Expression
1 abscl 12772 . . . . . . . 8  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
21recnd 9417 . . . . . . 7  |-  ( A  e.  CC  ->  ( abs `  A )  e.  CC )
3 subneg 9663 . . . . . . 7  |-  ( ( ( abs `  A
)  e.  CC  /\  A  e.  CC )  ->  ( ( abs `  A
)  -  -u A
)  =  ( ( abs `  A )  +  A ) )
42, 3mpancom 669 . . . . . 6  |-  ( A  e.  CC  ->  (
( abs `  A
)  -  -u A
)  =  ( ( abs `  A )  +  A ) )
54eqeq1d 2451 . . . . 5  |-  ( A  e.  CC  ->  (
( ( abs `  A
)  -  -u A
)  =  0  <->  (
( abs `  A
)  +  A )  =  0 ) )
6 negcl 9615 . . . . . 6  |-  ( A  e.  CC  ->  -u A  e.  CC )
72, 6subeq0ad 9734 . . . . 5  |-  ( A  e.  CC  ->  (
( ( abs `  A
)  -  -u A
)  =  0  <->  ( abs `  A )  = 
-u A ) )
85, 7bitr3d 255 . . . 4  |-  ( A  e.  CC  ->  (
( ( abs `  A
)  +  A )  =  0  <->  ( abs `  A )  =  -u A ) )
9 ax-icn 9346 . . . . . . 7  |-  _i  e.  CC
10 absge0 12781 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  0  <_  ( abs `  A
) )
111, 10jca 532 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
12 eleq1 2503 . . . . . . . . . . . 12  |-  ( ( abs `  A )  =  -u A  ->  (
( abs `  A
)  e.  RR  <->  -u A  e.  RR ) )
13 breq2 4301 . . . . . . . . . . . 12  |-  ( ( abs `  A )  =  -u A  ->  (
0  <_  ( abs `  A )  <->  0  <_  -u A ) )
1412, 13anbi12d 710 . . . . . . . . . . 11  |-  ( ( abs `  A )  =  -u A  ->  (
( ( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) )  <->  ( -u A  e.  RR  /\  0  <_  -u A ) ) )
1511, 14syl5ib 219 . . . . . . . . . 10  |-  ( ( abs `  A )  =  -u A  ->  ( A  e.  CC  ->  (
-u A  e.  RR  /\  0  <_  -u A ) ) )
1615impcom 430 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  -> 
( -u A  e.  RR  /\  0  <_  -u A ) )
17 resqrcl 12748 . . . . . . . . 9  |-  ( (
-u A  e.  RR  /\  0  <_  -u A )  ->  ( sqr `  -u A
)  e.  RR )
1816, 17syl 16 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  -> 
( sqr `  -u A
)  e.  RR )
1918recnd 9417 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  -> 
( sqr `  -u A
)  e.  CC )
20 mulcl 9371 . . . . . . 7  |-  ( ( _i  e.  CC  /\  ( sqr `  -u A
)  e.  CC )  ->  ( _i  x.  ( sqr `  -u A
) )  e.  CC )
219, 19, 20sylancr 663 . . . . . 6  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  -> 
( _i  x.  ( sqr `  -u A ) )  e.  CC )
22 sqrneglem 12761 . . . . . . . 8  |-  ( (
-u A  e.  RR  /\  0  <_  -u A )  ->  ( ( ( _i  x.  ( sqr `  -u A ) ) ^ 2 )  = 
-u -u A  /\  0  <_  ( Re `  (
_i  x.  ( sqr `  -u A ) ) )  /\  ( _i  x.  ( _i  x.  ( sqr `  -u A ) ) )  e/  RR+ )
)
2316, 22syl 16 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  -> 
( ( ( _i  x.  ( sqr `  -u A
) ) ^ 2 )  =  -u -u A  /\  0  <_  ( Re
`  ( _i  x.  ( sqr `  -u A
) ) )  /\  ( _i  x.  (
_i  x.  ( sqr `  -u A ) ) )  e/  RR+ ) )
24 negneg 9664 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u -u A  =  A )
2524adantr 465 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  ->  -u -u A  =  A
)
2625eqeq2d 2454 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  -> 
( ( ( _i  x.  ( sqr `  -u A
) ) ^ 2 )  =  -u -u A  <->  ( ( _i  x.  ( sqr `  -u A ) ) ^ 2 )  =  A ) )
27263anbi1d 1293 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  -> 
( ( ( ( _i  x.  ( sqr `  -u A ) ) ^ 2 )  = 
-u -u A  /\  0  <_  ( Re `  (
_i  x.  ( sqr `  -u A ) ) )  /\  ( _i  x.  ( _i  x.  ( sqr `  -u A ) ) )  e/  RR+ )  <->  ( ( ( _i  x.  ( sqr `  -u A
) ) ^ 2 )  =  A  /\  0  <_  ( Re `  ( _i  x.  ( sqr `  -u A ) ) )  /\  ( _i  x.  ( _i  x.  ( sqr `  -u A
) ) )  e/  RR+ ) ) )
2823, 27mpbid 210 . . . . . 6  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  -> 
( ( ( _i  x.  ( sqr `  -u A
) ) ^ 2 )  =  A  /\  0  <_  ( Re `  ( _i  x.  ( sqr `  -u A ) ) )  /\  ( _i  x.  ( _i  x.  ( sqr `  -u A
) ) )  e/  RR+ ) )
29 oveq1 6103 . . . . . . . . 9  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
x ^ 2 )  =  ( ( _i  x.  ( sqr `  -u A
) ) ^ 2 ) )
3029eqeq1d 2451 . . . . . . . 8  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
( x ^ 2 )  =  A  <->  ( (
_i  x.  ( sqr `  -u A ) ) ^
2 )  =  A ) )
31 fveq2 5696 . . . . . . . . 9  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
Re `  x )  =  ( Re `  ( _i  x.  ( sqr `  -u A ) ) ) )
3231breq2d 4309 . . . . . . . 8  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
0  <_  ( Re `  x )  <->  0  <_  ( Re `  ( _i  x.  ( sqr `  -u A
) ) ) ) )
33 oveq2 6104 . . . . . . . . 9  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
_i  x.  x )  =  ( _i  x.  ( _i  x.  ( sqr `  -u A ) ) ) )
34 neleq1 2714 . . . . . . . . 9  |-  ( ( _i  x.  x )  =  ( _i  x.  ( _i  x.  ( sqr `  -u A ) ) )  ->  ( (
_i  x.  x )  e/  RR+  <->  ( _i  x.  ( _i  x.  ( sqr `  -u A ) ) )  e/  RR+ )
)
3533, 34syl 16 . . . . . . . 8  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
( _i  x.  x
)  e/  RR+  <->  ( _i  x.  ( _i  x.  ( sqr `  -u A ) ) )  e/  RR+ )
)
3630, 32, 353anbi123d 1289 . . . . . . 7  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ )  <->  ( (
( _i  x.  ( sqr `  -u A ) ) ^ 2 )  =  A  /\  0  <_ 
( Re `  (
_i  x.  ( sqr `  -u A ) ) )  /\  ( _i  x.  ( _i  x.  ( sqr `  -u A ) ) )  e/  RR+ )
) )
3736rspcev 3078 . . . . . 6  |-  ( ( ( _i  x.  ( sqr `  -u A ) )  e.  CC  /\  (
( ( _i  x.  ( sqr `  -u A
) ) ^ 2 )  =  A  /\  0  <_  ( Re `  ( _i  x.  ( sqr `  -u A ) ) )  /\  ( _i  x.  ( _i  x.  ( sqr `  -u A
) ) )  e/  RR+ ) )  ->  E. x  e.  CC  ( ( x ^ 2 )  =  A  /\  0  <_ 
( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )
)
3821, 28, 37syl2anc 661 . . . . 5  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  ->  E. x  e.  CC  ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) )
3938ex 434 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
)  =  -u A  ->  E. x  e.  CC  ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) ) )
408, 39sylbid 215 . . 3  |-  ( A  e.  CC  ->  (
( ( abs `  A
)  +  A )  =  0  ->  E. x  e.  CC  ( ( x ^ 2 )  =  A  /\  0  <_ 
( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )
) )
41 resqrcl 12748 . . . . . . . . 9  |-  ( ( ( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) )  ->  ( sqr `  ( abs `  A
) )  e.  RR )
421, 10, 41syl2anc 661 . . . . . . . 8  |-  ( A  e.  CC  ->  ( sqr `  ( abs `  A
) )  e.  RR )
4342recnd 9417 . . . . . . 7  |-  ( A  e.  CC  ->  ( sqr `  ( abs `  A
) )  e.  CC )
4443adantr 465 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  -> 
( sqr `  ( abs `  A ) )  e.  CC )
45 addcl 9369 . . . . . . . . 9  |-  ( ( ( abs `  A
)  e.  CC  /\  A  e.  CC )  ->  ( ( abs `  A
)  +  A )  e.  CC )
462, 45mpancom 669 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( abs `  A
)  +  A )  e.  CC )
4746adantr 465 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  -> 
( ( abs `  A
)  +  A )  e.  CC )
48 abscl 12772 . . . . . . . . . 10  |-  ( ( ( abs `  A
)  +  A )  e.  CC  ->  ( abs `  ( ( abs `  A )  +  A
) )  e.  RR )
4946, 48syl 16 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( abs `  ( ( abs `  A )  +  A
) )  e.  RR )
5049recnd 9417 . . . . . . . 8  |-  ( A  e.  CC  ->  ( abs `  ( ( abs `  A )  +  A
) )  e.  CC )
5150adantr 465 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  -> 
( abs `  (
( abs `  A
)  +  A ) )  e.  CC )
5246abs00ad 12784 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( abs `  (
( abs `  A
)  +  A ) )  =  0  <->  (
( abs `  A
)  +  A )  =  0 ) )
5352necon3bid 2648 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( abs `  (
( abs `  A
)  +  A ) )  =/=  0  <->  (
( abs `  A
)  +  A )  =/=  0 ) )
5453biimpar 485 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  -> 
( abs `  (
( abs `  A
)  +  A ) )  =/=  0 )
5547, 51, 54divcld 10112 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  -> 
( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) )  e.  CC )
5644, 55mulcld 9411 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  -> 
( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) )  e.  CC )
57 eqid 2443 . . . . . 6  |-  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) ) )  =  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) )
5857sqreulem 12852 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  -> 
( ( ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) ) ) ^
2 )  =  A  /\  0  <_  (
Re `  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) ) )  /\  ( _i  x.  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) )  e/  RR+ ) )
59 oveq1 6103 . . . . . . . 8  |-  ( x  =  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) )  -> 
( x ^ 2 )  =  ( ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) ^
2 ) )
6059eqeq1d 2451 . . . . . . 7  |-  ( x  =  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) )  -> 
( ( x ^
2 )  =  A  <-> 
( ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) ) ^
2 )  =  A ) )
61 fveq2 5696 . . . . . . . 8  |-  ( x  =  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) )  -> 
( Re `  x
)  =  ( Re
`  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) ) ) )
6261breq2d 4309 . . . . . . 7  |-  ( x  =  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) )  -> 
( 0  <_  (
Re `  x )  <->  0  <_  ( Re `  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) ) ) )
63 oveq2 6104 . . . . . . . 8  |-  ( x  =  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) )  -> 
( _i  x.  x
)  =  ( _i  x.  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) ) ) )
64 neleq1 2714 . . . . . . . 8  |-  ( ( _i  x.  x )  =  ( _i  x.  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) )  ->  ( ( _i  x.  x )  e/  RR+  <->  ( _i  x.  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) ) ) )  e/  RR+ ) )
6563, 64syl 16 . . . . . . 7  |-  ( x  =  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) )  -> 
( ( _i  x.  x )  e/  RR+  <->  ( _i  x.  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) )  e/  RR+ ) )
6660, 62, 653anbi123d 1289 . . . . . 6  |-  ( x  =  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) )  -> 
( ( ( x ^ 2 )  =  A  /\  0  <_ 
( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )  <->  ( ( ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) ) ^
2 )  =  A  /\  0  <_  (
Re `  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) ) )  /\  ( _i  x.  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) )  e/  RR+ ) ) )
6766rspcev 3078 . . . . 5  |-  ( ( ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) )  e.  CC  /\  ( ( ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) ^
2 )  =  A  /\  0  <_  (
Re `  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) ) )  /\  ( _i  x.  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) )  e/  RR+ ) )  ->  E. x  e.  CC  ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) )
6856, 58, 67syl2anc 661 . . . 4  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  ->  E. x  e.  CC  ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) )
6968ex 434 . . 3  |-  ( A  e.  CC  ->  (
( ( abs `  A
)  +  A )  =/=  0  ->  E. x  e.  CC  ( ( x ^ 2 )  =  A  /\  0  <_ 
( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )
) )
7040, 69pm2.61dne 2693 . 2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( ( x ^ 2 )  =  A  /\  0  <_ 
( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )
)
71 sqrmo 12746 . 2  |-  ( A  e.  CC  ->  E* x  e.  CC  (
( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )
72 reu5 2941 . 2  |-  ( E! x  e.  CC  (
( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ )  <->  ( E. x  e.  CC  (
( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ )  /\  E* x  e.  CC  (
( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) ) )
7370, 71, 72sylanbrc 664 1  |-  ( A  e.  CC  ->  E! x  e.  CC  (
( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611    e/ wnel 2612   E.wrex 2721   E!wreu 2722   E*wrmo 2723   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287   _ici 9289    + caddc 9290    x. cmul 9292    <_ cle 9424    - cmin 9600   -ucneg 9601    / cdiv 9998   2c2 10376   RR+crp 10996   ^cexp 11870   Recre 12591   sqrcsqr 12727   abscabs 12728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730
This theorem is referenced by:  sqrcl  12854  sqrthlem  12855  eqsqrd  12860
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