Theorem sqrlem4 6606

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 2613	. . 3 |- (x = D -> (0 <_ x <-> 0 <_ D))
2		opreq12 3955	. . . . 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 2619	. . 3 |- (x = D -> ((x x. x) <_ A <-> (D x. D) <_ A))
5	1, 4	anbi12d 626	. 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 1898	1 |- (D e. S <-> (D e. RR /\ (0 <_ D /\ (D x. D) <_ A)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  {crab 1640   class class class wbr 2609  (class class class)co 3948  RRcr 5205  0cc0 5206   x. cmul 5211   <_ cle 5267   < clt 5458

This theorem is referenced by:  sqrlem5 6607  sqrlem6 6608  sqrlem12 6614  sqrlem13 6615

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fv 3188  df-opr 3950

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