Theorem sqrlem4 6676

Description: Lemma for square root theorem.

Hypotheses

Ref	Expression
sqrlem1.1	|- A e. RR
sqrlem1.2	|- 0 < A
sqrlem4.3	|- S = {x e. RR | (0 <_ x /\ (x x. x) <_ A)}

Assertion

Ref	Expression
sqrlem4	|- (D e. S <-> (D e. RR /\ (0 <_ D /\ (D x. D) <_ A)))

Distinct variable groups:   x,A   x,S   x,D

Proof of Theorem sqrlem4

Step	Hyp	Ref	Expression
1		breq2 2623	. . 3 |- (x = D -> (0 <_ x <-> 0 <_ D))
2		opreq12 3970	. . . . 5 |- ((x = D /\ x = D) -> (x x. x) = (D x. D))
3	2	anidms 434	. . . 4 |- (x = D -> (x x. x) = (D x. D))
4	3	breq1d 2629	. . 3 |- (x = D -> ((x x. x) <_ A <-> (D x. D) <_ A))
5	1, 4	anbi12d 628	. 2 |- (x = D -> ((0 <_ x /\ (x x. x) <_ A) <-> (0 <_ D /\ (D x. D) <_ A)))
6		sqrlem4.3	. 2 |- S = {x e. RR | (0 <_ x /\ (x x. x) <_ A)}
7	5, 6	elrab2 1907	1 |- (D e. S <-> (D e. RR /\ (0 <_ D /\ (D x. D) <_ A)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {crab 1648   class class class wbr 2619  (class class class)co 3963  RRcr 5233  0cc0 5234   x. cmul 5239   <_ cle 5295   < clt 5486

This theorem is referenced by:  sqrlem5 6677  sqrlem6 6678  sqrlem12 6684  sqrlem13 6685

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965

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