Nuprl Lemma : triple_txpose_perm
∀n:ℕ. ∀i,j,k:ℕn.
((¬(i = j ∈ ℤ))
⇒ (¬(j = k ∈ ℤ))
⇒ (txpose_perm(i;j) = txpose_perm(i;k) O txpose_perm(j;k) O txpose_perm(i;k) ∈ Sym(n)))
Proof
Definitions occuring in Statement :
txpose_perm: txpose_perm,
sym_grp: Sym(n),
comp_perm: comp_perm,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
txpose_perm: txpose_perm,
comp_perm: comp_perm,
mk_perm: mk_perm(f;b),
perm_f: p.f,
pi1: fst(t),
perm_b: p.b,
pi2: snd(t),
sym_grp: Sym(n),
perm: Perm(T),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
not_wf,
equal-wf-base,
set_subtype_base,
lelt_wf,
istype-int,
int_subtype_base,
int_seg_wf,
nat_wf,
txpose_perm_wf,
perm_properties,
inv_funs_wf,
perm_f_wf,
perm_b_wf,
mk_perm_wf,
squash_wf,
true_wf,
istype-universe,
triple_swap,
equal_wf,
swap_wf,
comp_assoc,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
lambdaEquality_alt,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
inhabitedIsType,
dependent_functionElimination,
applyLambdaEquality,
dependent_set_memberEquality_alt,
because_Cache,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionIsType,
universeEquality,
imageMemberEquality,
independent_functionElimination,
functionEquality,
instantiate,
productElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}i,j,k:\mBbbN{}n.
((\mneg{}(i = j))
{}\mRightarrow{} (\mneg{}(j = k))
{}\mRightarrow{} (txpose\_perm(i;j) = txpose\_perm(i;k) O txpose\_perm(j;k) O txpose\_perm(i;k)))
Date html generated:
2019_10_16-PM-00_59_35
Last ObjectModification:
2018_10_08-AM-09_20_33
Theory : perms_1
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