### Nuprl Lemma : zero-div-rem

`∀[x:ℤ-o]. ((0 ÷ x ~ 0) ∧ (0 rem x ~ 0))`

Proof

Definitions occuring in Statement :  int_nzero: `ℤ-o` uall: `∀[x:A]. B[x]` and: `P ∧ Q` remainder: `n rem m` divide: `n ÷ m` natural_number: `\$n` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` and: `P ∧ Q` cand: `A c∧ B` uimplies: `b supposing a` sq_type: `SQType(T)` all: `∀x:A. B[x]` implies: `P `` Q` guard: `{T}` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` exists: `∃x:A. B[x]` int_nzero: `ℤ-o` prop: `ℙ` nat: `ℕ` uiff: `uiff(P;Q)` true: `True` squash: `↓T` not: `¬A` false: `False` absval: `|i|` less_than': `less_than'(a;b)` le: `A ≤ B` so_apply: `x[s]` so_lambda: `λ2x.t[x]` nequal: `a ≠ b ∈ T `
Lemmas referenced :  subtype_base_sq int_subtype_base rem-zero iff_weakening_equal int_nzero_wf add-zero zero-mul le_wf less_than_wf absval_wf nat_wf equal-wf-base-T div_unique3 absval_pos true_wf squash_wf equal_wf false_wf set_subtype_base absval-positive
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin instantiate extract_by_obid sqequalHypSubstitution isectElimination cumulativity intEquality independent_isectElimination hypothesis dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination independent_pairFormation sqequalIntensionalEquality hypothesisEquality applyEquality sqequalRule baseClosed productElimination independent_pairEquality axiomSqEquality natural_numberEquality dependent_pairFormation because_Cache multiplyEquality setElimination rename lambdaFormation productEquality lambdaEquality addEquality functionEquality imageMemberEquality universeEquality imageElimination dependent_set_memberEquality voidElimination

Latex:
\mforall{}[x:\mBbbZ{}\msupminus{}\msupzero{}].  ((0  \mdiv{}  x  \msim{}  0)  \mwedge{}  (0  rem  x  \msim{}  0))

Date html generated: 2019_06_20-AM-11_24_56
Last ObjectModification: 2018_08_21-PM-10_43_08

Theory : arithmetic

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