### Nuprl Lemma : pa-mul_functionality

`∀[p,q:{2...}]. ∀[x1,y1,x2,y2:basic-padic(p)].`
`  (pa-mul(p;x1;y1) = pa-mul(q;x2;y2) ∈ padic(p)) supposing ((p = q ∈ ℤ) and bpa-equiv(p;x1;x2) and bpa-equiv(p;y1;y2))`

Proof

Definitions occuring in Statement :  pa-mul: `pa-mul(p;x;y)` padic: `padic(p)` bpa-equiv: `bpa-equiv(p;x;y)` basic-padic: `basic-padic(p)` int_upper: `{i...}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` sq_type: `SQType(T)` all: `∀x:A. B[x]` implies: `P `` Q` guard: `{T}` uiff: `uiff(P;Q)` and: `P ∧ Q` rev_uimplies: `rev_uimplies(P;Q)` pa-mul: `pa-mul(p;x;y)` nat_plus: `ℕ+` int_upper: `{i...}` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` false: `False` prop: `ℙ` subtype_rel: `A ⊆r B` less_than': `less_than'(a;b)` true: `True` basic-padic: `basic-padic(p)` bpa-equiv: `bpa-equiv(p;x;y)` bpa-mul: `bpa-mul(p;x;y)` top: `Top` nat: `ℕ` squash: `↓T` ge: `i ≥ j ` 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-}` lelt: `i ≤ j < k` so_apply: `x[s]`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut thin instantiate introduction extract_by_obid sqequalHypSubstitution isectElimination cumulativity intEquality independent_isectElimination hypothesis dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination hypothesisEquality productElimination dependent_set_memberEquality setElimination rename because_Cache natural_numberEquality unionElimination independent_pairFormation lambdaFormation voidElimination sqequalRule applyEquality lambdaEquality isect_memberEquality voidEquality imageElimination universeEquality imageMemberEquality baseClosed hyp_replacement addEquality approximateComputation dependent_pairFormation int_eqEquality minusEquality applyLambdaEquality

Latex:
(pa-mul(p;x1;y1)  =  pa-mul(q;x2;y2))  supposing
((p  =  q)  and
bpa-equiv(p;x1;x2)  and
bpa-equiv(p;y1;y2))

Date html generated: 2018_05_21-PM-03_27_00
Last ObjectModification: 2018_05_19-AM-08_24_23

Theory : rings_1

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