Nuprl Lemma : rminus-as-rmul
∀[r:ℝ]. (-(r) = (r(-1) * r))
Proof
Definitions occuring in Statement :
req: x = y,
rmul: a * b,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
real: ℝ,
bdd-diff: bdd-diff(f;g),
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
all: ∀x:A. B[x],
int-to-real: r(n),
reg-seq-mul: reg-seq-mul(x;y),
rminus: -(x),
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
sq_type: SQType(T),
guard: {T},
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
absval: |i|,
subtract: n - m
Lemmas referenced :
zero-mul,
add-mul-special,
add-commutes,
minus-one-mul,
minus-minus,
nat_wf,
nequal_wf,
equal_wf,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformless_wf,
intformand_wf,
div-cancel,
int_formula_prop_wf,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermMinus_wf,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
nat_plus_properties,
int_subtype_base,
subtype_base_sq,
rmul-bdd-diff-reg-seq-mul,
bdd-diff_weakening,
bdd-diff_functionality,
subtract_wf,
absval_wf,
all_wf,
nat_plus_wf,
le_wf,
false_wf,
reg-seq-mul_wf,
real_wf,
req_witness,
int-to-real_wf,
rmul_wf,
rminus_wf,
req-iff-bdd-diff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
minusEquality,
natural_numberEquality,
productElimination,
independent_isectElimination,
independent_functionElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
because_Cache,
dependent_pairFormation,
dependent_set_memberEquality,
independent_pairFormation,
lambdaFormation,
dependent_functionElimination,
instantiate,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
multiplyEquality
Latex:
\mforall{}[r:\mBbbR{}]. (-(r) = (r(-1) * r))
Date html generated:
2016_05_18-AM-06_52_32
Last ObjectModification:
2016_01_17-AM-01_46_57
Theory : reals
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