Nuprl Lemma : p-1-mul

`∀[p:{2...}]. ∀[a:p-adics(p)].  (a * 1(p) = a ∈ p-adics(p))`

Proof

Definitions occuring in Statement :  p-int: `k(p)` p-mul: `x * y` p-adics: `p-adics(p)` int_upper: `{i...}` uall: `∀[x:A]. B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` crng: `CRng` rng: `Rng` p-adic-ring: `ℤ(p)` ring_p: `IsRing(T;plus;zero;neg;times;one)` rng_car: `|r|` pi1: `fst(t)` rng_plus: `+r` pi2: `snd(t)` rng_zero: `0` rng_minus: `-r` rng_times: `*` rng_one: `1` monoid_p: `IsMonoid(T;op;id)` group_p: `IsGroup(T;op;id;inv)` bilinear: `BiLinear(T;pl;tm)` ident: `Ident(T;op;id)` assoc: `Assoc(T;op)` inverse: `Inverse(T;op;id;inv)` infix_ap: `x f y` comm: `Comm(T;op)` and: `P ∧ Q` all: `∀x:A. B[x]` nat_plus: `ℕ+` int_upper: `{i...}` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` implies: `P `` Q` false: `False` prop: `ℙ` uiff: `uiff(P;Q)` uimplies: `b supposing a` subtype_rel: `A ⊆r B` top: `Top` less_than': `less_than'(a;b)` true: `True` guard: `{T}` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` p-adics: `p-adics(p)` so_lambda: `λ2x.t[x]` subtract: `n - m` int_seg: `{i..j-}` nat: `ℕ` lelt: `i ≤ j < k` so_apply: `x[s]`
Lemmas referenced :  p-adic-ring_wf crng_properties rng_properties p-adic-property decidable__lt false_wf not-lt-2 add_functionality_wrt_le add-commutes zero-add le-add-cancel less_than_wf p-mul_wf p-int_wf nat_plus_properties int_upper_properties decidable__le full-omega-unsat intformnot_wf intformle_wf itermVar_wf itermAdd_wf itermConstant_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_wf le_wf nat_plus_wf p-adics_wf all_wf eqmod_wf exp_wf2 nat_plus_subtype_nat less-iff-le condition-implies-le minus-add minus-one-mul minus-one-mul-top add-associates add-zero int_seg_wf int_seg_properties intformand_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_upper_wf
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality equalityTransitivity equalitySymmetry applyLambdaEquality setElimination rename sqequalRule productElimination lambdaFormation dependent_functionElimination dependent_set_memberEquality natural_numberEquality unionElimination independent_pairFormation voidElimination independent_functionElimination independent_isectElimination applyEquality lambdaEquality isect_memberEquality voidEquality intEquality because_Cache addEquality approximateComputation dependent_pairFormation int_eqEquality minusEquality

Latex:
\mforall{}[p:\{2...\}].  \mforall{}[a:p-adics(p)].    (a  *  1(p)  =  a)

Date html generated: 2018_05_21-PM-03_21_10
Last ObjectModification: 2018_05_19-AM-08_18_26

Theory : rings_1

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