Theorem sqreu 14100

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

Step	Hyp	Ref	Expression
1		abscl 14018	. . . . . . . 8 |- ( A e. CC -> ( abs ` A ) e. RR )
2	1	recnd 10068	. . . . . . 7 |- ( A e. CC -> ( abs ` A ) e. CC )
3		subneg 10330	. . . . . . 7 |- ( ( ( abs ` A ) e. CC /\ A e. CC ) -> ( ( abs ` A ) - -u A ) = ( ( abs ` A ) + A ) )
4	2, 3	mpancom 703	. . . . . 6 |- ( A e. CC -> ( ( abs ` A ) - -u A ) = ( ( abs ` A ) + A ) )
5	4	eqeq1d 2624	. . . . 5 |- ( A e. CC -> ( ( ( abs ` A ) - -u A ) = 0 <-> ( ( abs ` A ) + A ) = 0 ) )
6		negcl 10281	. . . . . 6 |- ( A e. CC -> -u A e. CC )
7	2, 6	subeq0ad 10402	. . . . 5 |- ( A e. CC -> ( ( ( abs ` A ) - -u A ) = 0 <-> ( abs ` A ) = -u A ) )
8	5, 7	bitr3d 270	. . . 4 |- ( A e. CC -> ( ( ( abs ` A ) + A ) = 0 <-> ( abs ` A ) = -u A ) )
9		ax-icn 9995	. . . . . . 7 |- _i e. CC
10		absge0 14027	. . . . . . . . . . . 12 |- ( A e. CC -> 0 <_ ( abs ` A ) )
11	1, 10	jca 554	. . . . . . . . . . 11 |- ( A e. CC -> ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) )
12		eleq1 2689	. . . . . . . . . . . 12 |- ( ( abs ` A ) = -u A -> ( ( abs ` A ) e. RR <-> -u A e. RR ) )
13		breq2 4657	. . . . . . . . . . . 12 |- ( ( abs ` A ) = -u A -> ( 0 <_ ( abs ` A ) <-> 0 <_ -u A ) )
14	12, 13	anbi12d 747	. . . . . . . . . . 11 |- ( ( abs ` A ) = -u A -> ( ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) <-> ( -u A e. RR /\ 0 <_ -u A ) ) )
15	11, 14	syl5ib 234	. . . . . . . . . 10 |- ( ( abs ` A ) = -u A -> ( A e. CC -> ( -u A e. RR /\ 0 <_ -u A ) ) )
16	15	impcom 446	. . . . . . . . 9 |- ( ( A e. CC /\ ( abs ` A ) = -u A ) -> ( -u A e. RR /\ 0 <_ -u A ) )
17		resqrtcl 13994	. . . . . . . . 9 |- ( ( -u A e. RR /\ 0 <_ -u A ) -> ( sqr ` -u A ) e. RR )
18	16, 17	syl 17	. . . . . . . 8 |- ( ( A e. CC /\ ( abs ` A ) = -u A ) -> ( sqr ` -u A ) e. RR )
19	18	recnd 10068	. . . . . . 7 |- ( ( A e. CC /\ ( abs ` A ) = -u A ) -> ( sqr ` -u A ) e. CC )
20		mulcl 10020	. . . . . . 7 |- ( ( _i e. CC /\ ( sqr ` -u A ) e. CC ) -> ( _i x. ( sqr ` -u A ) ) e. CC )
21	9, 19, 20	sylancr 695	. . . . . 6 |- ( ( A e. CC /\ ( abs ` A ) = -u A ) -> ( _i x. ( sqr ` -u A ) ) e. CC )
22		sqrtneglem 14007	. . . . . . . 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+ ) )
23	16, 22	syl 17	. . . . . . 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 10331	. . . . . . . . . 10 |- ( A e. CC -> -u -u A = A )
25	24	adantr 481	. . . . . . . . 9 |- ( ( A e. CC /\ ( abs ` A ) = -u A ) -> -u -u A = A )
26	25	eqeq2d 2632	. . . . . . . 8 |- ( ( A e. CC /\ ( abs ` A ) = -u A ) -> ( ( ( _i x. ( sqr ` -u A ) ) ^ 2 ) = -u -u A <-> ( ( _i x. ( sqr ` -u A ) ) ^ 2 ) = A ) )
27	26	3anbi1d 1403	. . . . . . 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+ ) ) )
28	23, 27	mpbid 222	. . . . . 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 6657	. . . . . . . . 9 |- ( x = ( _i x. ( sqr ` -u A ) ) -> ( x ^ 2 ) = ( ( _i x. ( sqr ` -u A ) ) ^ 2 ) )
30	29	eqeq1d 2624	. . . . . . . 8 |- ( x = ( _i x. ( sqr ` -u A ) ) -> ( ( x ^ 2 ) = A <-> ( ( _i x. ( sqr ` -u A ) ) ^ 2 ) = A ) )
31		fveq2 6191	. . . . . . . . 9 |- ( x = ( _i x. ( sqr ` -u A ) ) -> ( Re ` x ) = ( Re ` ( _i x. ( sqr ` -u A ) ) ) )
32	31	breq2d 4665	. . . . . . . 8 |- ( x = ( _i x. ( sqr ` -u A ) ) -> ( 0 <_ ( Re ` x ) <-> 0 <_ ( Re ` ( _i x. ( sqr ` -u A ) ) ) ) )
33		oveq2 6658	. . . . . . . . 9 |- ( x = ( _i x. ( sqr ` -u A ) ) -> ( _i x. x ) = ( _i x. ( _i x. ( sqr ` -u A ) ) ) )
34		neleq1 2902	. . . . . . . . 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+ ) )
35	33, 34	syl 17	. . . . . . . 8 |- ( x = ( _i x. ( sqr ` -u A ) ) -> ( ( _i x. x ) e/ RR+ <-> ( _i x. ( _i x. ( sqr ` -u A ) ) ) e/ RR+ ) )
36	30, 32, 35	3anbi123d 1399	. . . . . . 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+ ) ) )
37	36	rspcev 3309	. . . . . 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+ ) )
38	21, 28, 37	syl2anc 693	. . . . 5 |- ( ( A e. CC /\ ( abs ` A ) = -u A ) -> E. x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )
39	38	ex 450	. . . 4 |- ( A e. CC -> ( ( abs ` A ) = -u A -> E. x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) )
40	8, 39	sylbid 230	. . 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 13994	. . . . . . . . 9 |- ( ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) -> ( sqr ` ( abs ` A ) ) e. RR )
42	1, 10, 41	syl2anc 693	. . . . . . . 8 |- ( A e. CC -> ( sqr ` ( abs ` A ) ) e. RR )
43	42	recnd 10068	. . . . . . 7 |- ( A e. CC -> ( sqr ` ( abs ` A ) ) e. CC )
44	43	adantr 481	. . . . . 6 |- ( ( A e. CC /\ ( ( abs ` A ) + A ) =/= 0 ) -> ( sqr ` ( abs ` A ) ) e. CC )
45		addcl 10018	. . . . . . . . 9 |- ( ( ( abs ` A ) e. CC /\ A e. CC ) -> ( ( abs ` A ) + A ) e. CC )
46	2, 45	mpancom 703	. . . . . . . 8 |- ( A e. CC -> ( ( abs ` A ) + A ) e. CC )
47	46	adantr 481	. . . . . . 7 |- ( ( A e. CC /\ ( ( abs ` A ) + A ) =/= 0 ) -> ( ( abs ` A ) + A ) e. CC )
48		abscl 14018	. . . . . . . . . 10 |- ( ( ( abs ` A ) + A ) e. CC -> ( abs ` ( ( abs ` A ) + A ) ) e. RR )
49	46, 48	syl 17	. . . . . . . . 9 |- ( A e. CC -> ( abs ` ( ( abs ` A ) + A ) ) e. RR )
50	49	recnd 10068	. . . . . . . 8 |- ( A e. CC -> ( abs ` ( ( abs ` A ) + A ) ) e. CC )
51	50	adantr 481	. . . . . . 7 |- ( ( A e. CC /\ ( ( abs ` A ) + A ) =/= 0 ) -> ( abs ` ( ( abs ` A ) + A ) ) e. CC )
52	46	abs00ad 14030	. . . . . . . . 9 |- ( A e. CC -> ( ( abs ` ( ( abs ` A ) + A ) ) = 0 <-> ( ( abs ` A ) + A ) = 0 ) )
53	52	necon3bid 2838	. . . . . . . 8 |- ( A e. CC -> ( ( abs ` ( ( abs ` A ) + A ) ) =/= 0 <-> ( ( abs ` A ) + A ) =/= 0 ) )
54	53	biimpar 502	. . . . . . 7 |- ( ( A e. CC /\ ( ( abs ` A ) + A ) =/= 0 ) -> ( abs ` ( ( abs ` A ) + A ) ) =/= 0 )
55	47, 51, 54	divcld 10801	. . . . . 6 |- ( ( A e. CC /\ ( ( abs ` A ) + A ) =/= 0 ) -> ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) e. CC )
56	44, 55	mulcld 10060	. . . . 5 |- ( ( A e. CC /\ ( ( abs ` A ) + A ) =/= 0 ) -> ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) e. CC )
57		eqid 2622	. . . . . 6 |- ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) = ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) )
58	57	sqreulem 14099	. . . . 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 6657	. . . . . . . 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 ) )
60	59	eqeq1d 2624	. . . . . . 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 6191	. . . . . . . 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 ) ) ) ) ) )
62	61	breq2d 4665	. . . . . . 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 6658	. . . . . . . 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 2902	. . . . . . . 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+ ) )
65	63, 64	syl 17	. . . . . . 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+ ) )
66	60, 62, 65	3anbi123d 1399	. . . . . 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+ ) ) )
67	66	rspcev 3309	. . . . 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+ ) )
68	56, 58, 67	syl2anc 693	. . . 4 |- ( ( A e. CC /\ ( ( abs ` A ) + A ) =/= 0 ) -> E. x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )
69	68	ex 450	. . 3 |- ( A e. CC -> ( ( ( abs ` A ) + A ) =/= 0 -> E. x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) )
70	40, 69	pm2.61dne 2880	. 2 |- ( A e. CC -> E. x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )
71		sqrmo 13992	. 2 |- ( A e. CC -> E* x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )
72		reu5 3159	. 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+ ) ) )
73	70, 71, 72	sylanbrc 698	1 |- ( A e. CC -> E! x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   E.wrex 2913   E!wreu 2914   E*wrmo 2915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   _ici 9938   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   RR+crp 11832   ^cexp 12860   Recre 13837   sqrcsqrt 13973   abscabs 13974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976

This theorem is referenced by:  sqrtcl  14101  sqrtthlem  14102  eqsqrtd  14107