Nuprl Lemma : rng_mssum_functionality_wrt_equal
∀s:DSet. ∀r:Rng. ∀f,f':|s| ⟶ |r|. ∀a,a':MSet{s}.
((a = a' ∈ MSet{s}) ⇒ (∀x:|s|. ((↑(x ∈b a)) ⇒ (f[x] = f'[x] ∈ |r|))) ⇒ ((Σx ∈ a. f[x]) = (Σx ∈ a'. f'[x]) ∈ |r|))
Proof
Definitions occuring in Statement :
rng_mssum: rng_mssum,
mset_mem: mset_mem,
mset: MSet{s},
assert: ↑b,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
equal: s = t ∈ T,
rng: Rng,
rng_car: |r|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
rng_mssum: rng_mssum,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
abgrp: AbGrp,
grp: Group{i},
mon: Mon,
iabmonoid: IAbMonoid,
imon: IMonoid,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
pi1: fst(t),
rng: Rng,
dset: DSet
Lemmas referenced :
mset_for_functionality,
add_grp_of_rng_wf_b,
subtype_rel_sets,
grp_sig_wf,
monoid_p_wf,
grp_car_wf,
grp_op_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
comm_wf,
set_wf,
rng_car_wf,
set_car_wf,
assert_wf,
mset_mem_wf,
add_grp_of_rng_wf,
all_wf,
equal_wf,
mset_wf,
rng_wf,
dset_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
applyEquality,
instantiate,
setEquality,
cumulativity,
setElimination,
rename,
lambdaEquality,
independent_isectElimination,
independent_functionElimination,
functionEquality
Latex:
\mforall{}s:DSet. \mforall{}r:Rng. \mforall{}f,f':|s| {}\mrightarrow{} |r|. \mforall{}a,a':MSet\{s\}.
((a = a') {}\mRightarrow{} (\mforall{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} a)) {}\mRightarrow{} (f[x] = f'[x]))) {}\mRightarrow{} ((\mSigma{}x \mmember{} a. f[x]) = (\mSigma{}x \mmember{} a'. f'[x])))
Date html generated:
2016_05_16-AM-08_11_49
Last ObjectModification:
2015_12_28-PM-06_06_27
Theory : list_3
Home
Index