Nuprl Lemma : equipollent-multiply
∀a,b:ℕ. ℕa × ℕb ~ ℕa * b
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
product: x:A × B[x],
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
nat: ℕ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
prop: ℙ,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
guard: {T},
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
so_lambda: λ2x.t[x],
so_apply: x[s],
surject: Surj(A;B;f),
nat_plus: ℕ+,
div_nrel: Div(a;n;q),
sq_type: SQType(T),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtract: n - m,
true: True,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
add-member-int_seg1,
istype-le,
subtract_wf,
istype-less_than,
int_seg_wf,
biject_wf,
istype-nat,
int_term_value_mul_lemma,
itermMultiply_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
mul_preserves_le,
istype-void,
int_seg_subtype_nat,
mul_bounds_1a,
decidable__lt,
set_subtype_base,
le_wf,
int_subtype_base,
product_subtype_base,
lelt_wf,
div_unique,
istype-false,
int_term_value_add_lemma,
itermAdd_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
less_than_wf,
decidable__equal_int,
subtype_base_sq,
remainder_wfa,
nequal_wf,
rem_bounds_1,
minus-zero,
minus-add,
add-commutes,
condition-implies-le,
le-add-cancel,
zero-add,
add-zero,
add-associates,
add_functionality_wrt_le,
not-equal-2,
not-lt-2,
div_rem_sum,
div_bounds_1,
false_wf,
multiply-is-int-iff,
add-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
multiplyEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
closedConclusion,
natural_numberEquality,
because_Cache,
independent_isectElimination,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
Error :productIsType,
imageElimination,
Error :universeIsType,
productEquality,
Error :inhabitedIsType,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
independent_functionElimination,
approximateComputation,
unionElimination,
dependent_functionElimination,
applyEquality,
Error :equalityIstype,
baseApply,
baseClosed,
intEquality,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
applyLambdaEquality,
addEquality,
independent_pairEquality,
cumulativity,
instantiate,
minusEquality,
divideEquality,
promote_hyp,
pointwiseFunctionality
Latex:
\mforall{}a,b:\mBbbN{}. \mBbbN{}a \mtimes{} \mBbbN{}b \msim{} \mBbbN{}a * b
Date html generated:
2019_06_20-PM-02_17_08
Last ObjectModification:
2019_06_19-PM-06_34_45
Theory : equipollence!!cardinality!
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