### Nuprl Lemma : r-triangle-inequality

`∀[x,y:ℝ].  (|x + y| ≤ (|x| + |y|))`

This theorem is one of freek's list of 100 theorems

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` rabs: `|x|` radd: `a + b` real: `ℝ` uall: `∀[x:A]. B[x]`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` rleq: `x ≤ y` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` rnonneg: `rnonneg(x)` all: `∀x:A. B[x]` le: `A ≤ B` not: `¬A` false: `False` subtype_rel: `A ⊆r B` real: `ℝ` prop: `ℙ` rsub: `x - y` rabs: `|x|` nat: `ℕ` reg-seq-add: `reg-seq-add(x;y)` rminus: `-(x)` rnonneg2: `rnonneg2(x)` exists: `∃x:A. B[x]` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` so_lambda: `λ2x.t[x]` int_upper: `{i...}` guard: `{T}` uimplies: `b supposing a` so_apply: `x[s]` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` top: `Top` satisfiable_int_formula: `satisfiable_int_formula(fmla)` rev_uimplies: `rev_uimplies(P;Q)` ge: `i ≥ j `
Lemmas referenced :  int-triangle-inequality minus_functionality_wrt_le multiply_functionality_wrt_le le_weakening le_functionality int_term_value_mul_lemma int_formula_prop_and_lemma itermMultiply_wf intformand_wf int_formula_prop_wf int_term_value_minus_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma itermMinus_wf itermVar_wf itermAdd_wf itermConstant_wf intformle_wf intformnot_wf satisfiable-full-omega-tt decidable__le nat_plus_properties int_upper_properties nat_plus_subtype_nat le-add-cancel zero-add add-commutes add_functionality_wrt_le not-lt-2 false_wf decidable__lt less_than_transitivity1 le_wf all_wf int_upper_wf less_than_wf bdd-diff_weakening reg-seq-add_functionality_wrt_bdd-diff reg-seq-add_wf rabs_functionality_wrt_bdd-diff nat_wf absval_wf rminus_functionality_wrt_bdd-diff radd-bdd-diff rminus_wf rnonneg2_functionality nat_plus_wf real_wf less_than'_wf rabs_wf radd_wf rsub_wf rnonneg-iff
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis productElimination independent_functionElimination sqequalRule lambdaEquality dependent_functionElimination independent_pairEquality because_Cache applyEquality setElimination rename minusEquality natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality voidElimination addEquality lambdaFormation dependent_pairFormation dependent_set_memberEquality independent_pairFormation imageMemberEquality baseClosed multiplyEquality independent_isectElimination unionElimination voidEquality intEquality int_eqEquality computeAll

Latex:
\mforall{}[x,y:\mBbbR{}].    (|x  +  y|  \mleq{}  (|x|  +  |y|))

Date html generated: 2016_05_18-AM-07_14_23
Last ObjectModification: 2016_01_17-AM-01_55_03

Theory : reals

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