Nuprl Lemma : ocmon_trans
∀[g:OCMon]. ∀[a,b,c:|g|]. (↑(a ≤b c)) supposing ((↑(b ≤b c)) and (↑(a ≤b b)))
Proof
Definitions occuring in Statement :
ocmon: OCMon,
grp_le: ≤b,
grp_car: |g|,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
ulinorder: UniformLinorder(T;x,y.R[x; y]),
uorder: UniformOrder(T;x,y.R[x; y]),
eqfun_p: IsEqFun(T;eq),
monot: monot(T;x,y.R[x; y];f),
cancel: Cancel(T;S;op),
connex: Connex(T;x,y.R[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
implies: P ⇒ Q,
infix_ap: x f y,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
prop: ℙ,
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
ocmon_properties,
assert_witness,
grp_le_wf,
assert_wf,
grp_car_wf,
ocmon_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
isect_memberEquality,
isectElimination,
lambdaEquality,
applyEquality,
setElimination,
rename,
independent_functionElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[g:OCMon]. \mforall{}[a,b,c:|g|]. (\muparrow{}(a \mleq{}\msubb{} c)) supposing ((\muparrow{}(b \mleq{}\msubb{} c)) and (\muparrow{}(a \mleq{}\msubb{} b)))
Date html generated:
2016_05_15-PM-00_11_16
Last ObjectModification:
2015_12_26-PM-11_44_26
Theory : groups_1
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