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Theorem sqreu 13204
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 13122 . . . . . . . 8  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
21recnd 9639 . . . . . . 7  |-  ( A  e.  CC  ->  ( abs `  A )  e.  CC )
3 subneg 9887 . . . . . . 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 2459 . . . . 5  |-  ( A  e.  CC  ->  (
( ( abs `  A
)  -  -u A
)  =  0  <->  (
( abs `  A
)  +  A )  =  0 ) )
6 negcl 9839 . . . . . 6  |-  ( A  e.  CC  ->  -u A  e.  CC )
72, 6subeq0ad 9960 . . . . 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 9568 . . . . . . 7  |-  _i  e.  CC
10 absge0 13131 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  0  <_  ( abs `  A
) )
111, 10jca 532 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
12 eleq1 2529 . . . . . . . . . . . 12  |-  ( ( abs `  A )  =  -u A  ->  (
( abs `  A
)  e.  RR  <->  -u A  e.  RR ) )
13 breq2 4460 . . . . . . . . . . . 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 13098 . . . . . . . . 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 9639 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  -> 
( sqr `  -u A
)  e.  CC )
20 mulcl 9593 . . . . . . 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 13111 . . . . . . . 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 9888 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u -u A  =  A )
2524adantr 465 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  -u A )  ->  -u -u A  =  A
)
2625eqeq2d 2471 . . . . . . . 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 6303 . . . . . . . . 9  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
x ^ 2 )  =  ( ( _i  x.  ( sqr `  -u A
) ) ^ 2 ) )
3029eqeq1d 2459 . . . . . . . 8  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
( x ^ 2 )  =  A  <->  ( (
_i  x.  ( sqr `  -u A ) ) ^
2 )  =  A ) )
31 fveq2 5872 . . . . . . . . 9  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
Re `  x )  =  ( Re `  ( _i  x.  ( sqr `  -u A ) ) ) )
3231breq2d 4468 . . . . . . . 8  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
0  <_  ( Re `  x )  <->  0  <_  ( Re `  ( _i  x.  ( sqr `  -u A
) ) ) ) )
33 oveq2 6304 . . . . . . . . 9  |-  ( x  =  ( _i  x.  ( sqr `  -u A
) )  ->  (
_i  x.  x )  =  ( _i  x.  ( _i  x.  ( sqr `  -u A ) ) ) )
34 neleq1 2795 . . . . . . . . 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 3210 . . . . . 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 13098 . . . . . . . . 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 9639 . . . . . . 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 9591 . . . . . . . . 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 13122 . . . . . . . . . 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 9639 . . . . . . . 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 13134 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( abs `  (
( abs `  A
)  +  A ) )  =  0  <->  (
( abs `  A
)  +  A )  =  0 ) )
5352necon3bid 2715 . . . . . . . 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 10341 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  -> 
( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) )  e.  CC )
5644, 55mulcld 9633 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  -> 
( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) )  e.  CC )
57 eqid 2457 . . . . . 6  |-  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) ) )  =  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) )
5857sqreulem 13203 . . . . 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 6303 . . . . . . . 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 2459 . . . . . . 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 5872 . . . . . . . 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 4468 . . . . . . 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 6304 . . . . . . . 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 2795 . . . . . . . 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 3210 . . . . 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 2774 . 2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( ( x ^ 2 )  =  A  /\  0  <_ 
( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )
)
71 sqrmo 13096 . 2  |-  ( A  e.  CC  ->  E* x  e.  CC  (
( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )
72 reu5 3073 . 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 1395    e. wcel 1819    =/= wne 2652    e/ wnel 2653   E.wrex 2808   E!wreu 2809   E*wrmo 2810   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   _ici 9511    + caddc 9512    x. cmul 9514    <_ cle 9646    - cmin 9824   -ucneg 9825    / cdiv 10227   2c2 10606   RR+crp 11245   ^cexp 12168   Recre 12941   sqrcsqrt 13077   abscabs 13078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080
This theorem is referenced by:  sqrtcl  13205  sqrtthlem  13206  eqsqrtd  13211
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