Nuprl Lemma : qadd_wf

[r,s:ℚ].  (r s ∈ ℚ)


Proof




Definitions occuring in Statement :  qadd: s rationals: uall: [x:A]. B[x] member: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T rationals: quotient: x,y:A//B[x; y] and: P ∧ Q all: x:A. B[x] implies:  Q so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] uimplies: supposing a b-union: A ⋃ B tunion: x:A.B[x] bool: 𝔹 unit: Unit ifthenelse: if then else fi  pi2: snd(t) qadd: s so_lambda: λ2x.t[x] so_apply: x[s] int_nzero: -o callbyvalueall: callbyvalueall has-value: (a)↓ has-valueall: has-valueall(a) btrue: tt qeq: qeq(r;s) bfalse: ff uiff: uiff(P;Q) rev_uimplies: rev_uimplies(P;Q) sq_type: SQType(T) guard: {T} subtype_rel: A ⊆B nequal: a ≠ b ∈  not: ¬A satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False prop: decidable: Dec(P) or: P ∨ Q
Lemmas referenced :  rationals_wf quotient-member-eq b-union_wf int_nzero_wf equal-wf-T-base bool_wf qeq_wf qeq-equiv valueall-type-has-valueall bunion-valueall-type int-valueall-type product-valueall-type istype-int set-valueall-type nequal_wf evalall-reduce isint-int eqtt_to_assert eq_int_wf assert_of_eq_int subtype_base_sq int_subtype_base mul-distributes-right mul-commutes add-commutes btrue_wf subtype_rel_b-union-left subtype_rel_b-union-right trivial-equal bfalse_wf int_entire_a int_nzero_properties full-omega-unsat intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_not_lemma int_formula_prop_wf ifthenelse_wf mul-swap mul-associates mul_preserves_eq decidable__equal_int itermAdd_wf itermMultiply_wf int_term_value_add_lemma int_term_value_mul_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut sqequalHypSubstitution pointwiseFunctionalityForEquality extract_by_obid hypothesis sqequalRule pertypeElimination promote_hyp thin productElimination equalityTransitivity equalitySymmetry inhabitedIsType lambdaFormation_alt rename isectElimination intEquality productEquality lambdaEquality_alt hypothesisEquality baseClosed independent_isectElimination dependent_functionElimination imageElimination unionElimination equalityElimination independent_functionElimination because_Cache natural_numberEquality callbyvalueReduce isintReduceTrue addEquality independent_pairEquality multiplyEquality setElimination Error :memTop,  instantiate cumulativity equalityIstype productIsType sqequalBase universeIsType axiomEquality isect_memberEquality_alt isectIsTypeImplies applyEquality imageMemberEquality dependent_pairEquality_alt dependent_set_memberEquality_alt approximateComputation dependent_pairFormation_alt int_eqEquality independent_pairFormation voidElimination universeEquality

Latex:
\mforall{}[r,s:\mBbbQ{}].    (r  +  s  \mmember{}  \mBbbQ{})



Date html generated: 2020_05_20-AM-09_12_51
Last ObjectModification: 2019_12_31-PM-08_19_57

Theory : rationals


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