### Nuprl Lemma : rabs-bounds

`∀[x:ℝ]. ((-(|x|) ≤ x) ∧ (x ≤ |x|))`

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` rabs: `|x|` rminus: `-(x)` real: `ℝ` uall: `∀[x:A]. B[x]` and: `P ∧ Q`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` and: `P ∧ Q` rleq: `x ≤ y` rnonneg: `rnonneg(x)` all: `∀x:A. B[x]` le: `A ≤ B` not: `¬A` implies: `P `` Q` false: `False` subtype_rel: `A ⊆r B` real: `ℝ` prop: `ℙ` top: `Top` cand: `A c∧ B` uimplies: `b supposing a` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` uiff: `uiff(P;Q)`
Lemmas referenced :  rleq-rmax rminus_wf less_than'_wf rsub_wf rabs_wf real_wf nat_plus_wf rleq-implies-rleq real_term_polynomial itermSubtract_wf itermVar_wf itermMinus_wf int-to-real_wf real_term_value_const_lemma real_term_value_sub_lemma real_term_value_var_lemma real_term_value_minus_lemma req-iff-rsub-is-0 rabs-as-rmax
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis sqequalRule productElimination independent_pairEquality lambdaEquality dependent_functionElimination voidElimination applyEquality setElimination rename minusEquality natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality voidEquality because_Cache independent_isectElimination computeAll int_eqEquality intEquality independent_pairFormation

Latex:
\mforall{}[x:\mBbbR{}].  ((-(|x|)  \mleq{}  x)  \mwedge{}  (x  \mleq{}  |x|))

Date html generated: 2017_10_03-AM-08_37_26
Last ObjectModification: 2017_07_28-AM-07_30_08

Theory : reals

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