### Nuprl Lemma : rpower-nonzero

`∀x:ℝ. (x ≠ r0 `` (∀n:ℕ. x^n ≠ r0))`

Proof

Definitions occuring in Statement :  rneq: `x ≠ y` rnexp: `x^k1` int-to-real: `r(n)` real: `ℝ` nat: `ℕ` all: `∀x:A. B[x]` implies: `P `` Q` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` top: `Top` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` prop: `ℙ` uall: `∀[x:A]. B[x]` nat: `ℕ` decidable: `Dec(P)` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T `
Lemmas referenced :  rnexp_zero_lemma rless-int rless_wf int-to-real_wf rneq_wf rnexp_wf decidable__le subtract_wf satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf le_wf set_wf less_than_wf primrec-wf2 nat_wf real_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int rmul_wf intformeq_wf int_formula_prop_eq_lemma rmul-neq-zero rneq_functionality rnexp-req req_weakening
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin sqequalRule introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination isect_memberEquality voidElimination voidEquality hypothesis inrFormation because_Cache productElimination independent_functionElimination independent_pairFormation natural_numberEquality imageMemberEquality hypothesisEquality baseClosed isectElimination rename setElimination dependent_set_memberEquality unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality computeAll equalityElimination equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity

Latex:
\mforall{}x:\mBbbR{}.  (x  \mneq{}  r0  {}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  x\^{}n  \mneq{}  r0))

Date html generated: 2017_10_03-AM-08_33_38
Last ObjectModification: 2017_07_28-AM-07_28_27

Theory : reals

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