Nuprl Lemma : qle_complement_qorder
∀[a,b:ℚ]. uiff(¬(a ≤ b);b < a)
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qless: r < s,
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
qadd_grp: <ℚ+>,
grp_car: |g|,
pi1: fst(t),
qless: r < s,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
qle: r ≤ s,
grp_lt: a < b,
set_lt: a <p b,
guard: {T},
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
implies: P ⇒ Q,
not: ¬A,
grp_leq: a ≤ b,
infix_ap: x f y,
false: False
Lemmas referenced :
grp_leq_complement,
qadd_grp_wf2,
ocgrp_subtype_ocmon,
assert_witness,
set_blt_wf,
oset_of_ocmon_wf0,
mon_subtype_grp_sig,
dmon_subtype_mon,
abdmonoid_dmon,
ocmon_subtype_abdmonoid,
subtype_rel_transitivity,
ocgrp_wf,
ocmon_wf,
abdmonoid_wf,
dmon_wf,
mon_wf,
grp_sig_wf,
istype-assert,
grp_le_wf,
istype-void,
rationals_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
applyEquality,
sqequalRule,
isect_memberFormation_alt,
independent_pairFormation,
hypothesisEquality,
productElimination,
independent_isectElimination,
instantiate,
independent_functionElimination,
functionIsType,
because_Cache,
lambdaFormation_alt,
voidElimination,
lambdaEquality_alt,
dependent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType
Latex:
\mforall{}[a,b:\mBbbQ{}]. uiff(\mneg{}(a \mleq{} b);b < a)
Date html generated:
2020_05_20-AM-09_15_01
Last ObjectModification:
2020_01_27-PM-04_01_34
Theory : rationals
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