Nuprl Lemma : rng_plus_inv
∀[r:Rng]. ∀[a:|r|]. (((a +r (-r a)) = 0 ∈ |r|) ∧ (((-r a) +r a) = 0 ∈ |r|))
Proof
Definitions occuring in Statement :
rng: Rng,
rng_minus: -r,
rng_zero: 0,
rng_plus: +r,
rng_car: |r|,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
pi1: fst(t),
grp_op: *,
pi2: snd(t),
grp_inv: ~,
grp_id: e,
and: P ∧ Q,
rng: Rng
Lemmas referenced :
grp_inverse,
add_grp_of_rng_wf_a,
grp_subtype_igrp,
rng_car_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
isect_memberEquality,
productElimination,
independent_pairEquality,
axiomEquality,
setElimination,
rename
Latex:
\mforall{}[r:Rng]. \mforall{}[a:|r|]. (((a +r (-r a)) = 0) \mwedge{} (((-r a) +r a) = 0))
Date html generated:
2016_05_15-PM-00_21_19
Last ObjectModification:
2015_12_27-AM-00_02_20
Theory : rings_1
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