Nuprl Lemma : ocmon_anti_sym
∀[g:OCMon]. ∀[x,y:|g|]. (x = y ∈ |g|) supposing ((↑(y ≤b x)) and (↑(x ≤b y)))
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,
equal: s = t ∈ T
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]),
uimplies: b supposing a,
prop: ℙ,
infix_ap: x f y,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon
Lemmas referenced :
ocmon_properties,
assert_wf,
grp_le_wf,
grp_car_wf,
ocmon_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
isect_memberEquality,
isectElimination,
axiomEquality,
applyEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[g:OCMon]. \mforall{}[x,y:|g|]. (x = y) supposing ((\muparrow{}(y \mleq{}\msubb{} x)) and (\muparrow{}(x \mleq{}\msubb{} y)))
Date html generated:
2016_05_15-PM-00_11_19
Last ObjectModification:
2015_12_26-PM-11_43_33
Theory : groups_1
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