`∀[p:ℕ+]. ∀[x,y:p-adics(p)].  uiff(x = y ∈ p-adics(p);x = y ∈ (ℕ+ ⟶ ℤ))`

Proof

Definitions occuring in Statement :  p-adics: `p-adics(p)` nat_plus: `ℕ+` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` function: `x:A ⟶ B[x]` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` p-adics: `p-adics(p)` squash: `↓T` so_lambda: `λ2x.t[x]` nat: `ℕ` nat_plus: `ℕ+` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` prop: `ℙ` so_apply: `x[s]` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` respects-equality: `respects-equality(S;T)`
Lemmas referenced :  p-adics-subtype subtype_rel_dep_function nat_plus_wf int_seg_wf exp_wf2 nat_plus_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf istype-le equal_functionality_wrt_subtype_rel2 decidable__lt istype-less_than int_seg_properties exp_wf_nat_plus respects-equality-function subtype-base-respects-equality set_subtype_base lelt_wf int_subtype_base eqmod_wf itermAdd_wf int_term_value_add_lemma p-adics_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis independent_pairFormation setElimination rename applyLambdaEquality sqequalRule imageMemberEquality baseClosed imageElimination lambdaEquality_alt natural_numberEquality dependent_set_memberEquality_alt dependent_functionElimination unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination universeIsType inhabitedIsType because_Cache intEquality lambdaFormation_alt functionEquality equalityIstype functionExtensionality_alt applyEquality equalityTransitivity equalitySymmetry productElimination productIsType functionIsType addEquality

Latex:
\mforall{}[p:\mBbbN{}\msupplus{}].  \mforall{}[x,y:p-adics(p)].    uiff(x  =  y;x  =  y)

Date html generated: 2019_10_15-AM-10_34_19
Last ObjectModification: 2018_12_08-AM-11_57_07

Theory : rings_1

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