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Theorem sqreu 13149
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 13068 . . . . . . . 8  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
21recnd 9618 . . . . . . 7  |-  ( A  e.  CC  ->  ( abs `  A )  e.  CC )
3 subneg 9864 . . . . . . 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 2469 . . . . 5  |-  ( A  e.  CC  ->  (
( ( abs `  A
)  -  -u A
)  =  0  <->  (
( abs `  A
)  +  A )  =  0 ) )
6 negcl 9816 . . . . . 6  |-  ( A  e.  CC  ->  -u A  e.  CC )
72, 6subeq0ad 9936 . . . . 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 9547 . . . . . . 7  |-  _i  e.  CC
10 absge0 13077 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  0  <_  ( abs `  A
) )
111, 10jca 532 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
12 eleq1 2539 . . . . . . . . . . . 12  |-  ( ( abs `  A )  =  -u A  ->  (
( abs `  A
)  e.  RR  <->  -u A  e.  RR ) )
13 breq2 4451 . . . . . . . . . . . 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 resqrtcl 13044 . . . . . . . . 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 9618 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  -> 
( sqr `  -u A
)  e.  CC )
20 mulcl 9572 . . . . . . 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 sqrtneglem 13057 . . . . . . . 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 9865 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u -u A  =  A )
2524adantr 465 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  ->  -u -u A  =  A
)
2625eqeq2d 2481 . . . . . . . 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 1303 . . . . . . 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 6289 . . . . . . . . 9  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
x ^ 2 )  =  ( ( _i  x.  ( sqr `  -u A
) ) ^ 2 ) )
3029eqeq1d 2469 . . . . . . . 8  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
( x ^ 2 )  =  A  <->  ( (
_i  x.  ( sqr `  -u A ) ) ^
2 )  =  A ) )
31 fveq2 5864 . . . . . . . . 9  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
Re `  x )  =  ( Re `  ( _i  x.  ( sqr `  -u A ) ) ) )
3231breq2d 4459 . . . . . . . 8  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
0  <_  ( Re `  x )  <->  0  <_  ( Re `  ( _i  x.  ( sqr `  -u A
) ) ) ) )
33 oveq2 6290 . . . . . . . . 9  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
_i  x.  x )  =  ( _i  x.  ( _i  x.  ( sqr `  -u A ) ) ) )
34 neleq1 2805 . . . . . . . . 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 1299 . . . . . . 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 3214 . . . . . 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 resqrtcl 13044 . . . . . . . . 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 9618 . . . . . . 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 9570 . . . . . . . . 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 13068 . . . . . . . . . 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 9618 . . . . . . . 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 13080 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( abs `  (
( abs `  A
)  +  A ) )  =  0  <->  (
( abs `  A
)  +  A )  =  0 ) )
5352necon3bid 2725 . . . . . . . 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 10316 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  -> 
( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) )  e.  CC )
5644, 55mulcld 9612 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  -> 
( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) )  e.  CC )
57 eqid 2467 . . . . . 6  |-  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) ) )  =  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) )
5857sqreulem 13148 . . . . 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 6289 . . . . . . . 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 2469 . . . . . . 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 5864 . . . . . . . 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 4459 . . . . . . 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 6290 . . . . . . . 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 2805 . . . . . . . 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 1299 . . . . . 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 3214 . . . . 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 2784 . 2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( ( x ^ 2 )  =  A  /\  0  <_ 
( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )
)
71 sqrmo 13042 . 2  |-  ( A  e.  CC  ->  E* x  e.  CC  (
( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )
72 reu5 3077 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662    e/ wnel 2663   E.wrex 2815   E!wreu 2816   E*wrmo 2817   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   _ici 9490    + caddc 9491    x. cmul 9493    <_ cle 9625    - cmin 9801   -ucneg 9802    / cdiv 10202   2c2 10581   RR+crp 11216   ^cexp 12129   Recre 12887   sqrcsqrt 13023   abscabs 13024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-seq 12071  df-exp 12130  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026
This theorem is referenced by:  sqrtcl  13150  sqrtthlem  13151  eqsqrtd  13156
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