Nuprl Lemma : sb-equipollent
ℕ2 List ~ {p:ℕ+ × ℕ+| let m,n = p in gcd(m;n) = 1 ∈ ℤ}
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
list: T List,
gcd: gcd(a;b),
int_seg: {i..j-},
nat_plus: ℕ+,
set: {x:A| B[x]} ,
spread: spread def,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
nat_plus: ℕ+,
surject: Surj(A;B;f),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
sbdecode_wf_gcd,
list_wf,
int_seg_wf,
equal_wf,
nat_plus_wf,
equal-wf-T-base,
gcd_wf,
set_wf,
biject_wf,
sbcode_wf,
sbcode-decode,
sbdecode-code,
subtype_base_sq,
int_subtype_base,
div-one,
spread_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_pairFormation,
lambdaEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
sqequalRule,
setEquality,
productEquality,
spreadEquality,
productElimination,
independent_pairEquality,
intEquality,
dependent_functionElimination,
setElimination,
rename,
baseClosed,
functionExtensionality,
applyEquality,
applyLambdaEquality,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_set_memberEquality,
because_Cache,
universeEquality
Latex:
\mBbbN{}2 List \msim{} \{p:\mBbbN{}\msupplus{} \mtimes{} \mBbbN{}\msupplus{}| let m,n = p in gcd(m;n) = 1\}
Date html generated:
2018_05_21-PM-11_40_33
Last ObjectModification:
2017_07_26-PM-06_42_51
Theory : rationals
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