Nuprl Lemma : quadratic-1-2-equal-implies2
∀[a,b,c:ℝ].
({((b * b) = (r(4) * a * c)) ∧ (quadratic1(a;b;c) = (-(b)/r(2) * a))}) supposing
((quadratic1(a;b;c) = quadratic2(a;b;c)) and
(r0 ≤ ((b * b) - r(4) * a * c)) and
a ≠ r0)
Proof
Definitions occuring in Statement :
quadratic2: quadratic2(a;b;c),
quadratic1: quadratic1(a;b;c),
rdiv: (x/y),
rneq: x ≠ y,
rleq: x ≤ y,
rsub: x - y,
req: x = y,
rmul: a * b,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
rneq: x ≠ y,
or: P ∨ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
false: False,
not: ¬A,
rat_term_to_real: rat_term_to_real(f;t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermDivide: num "/" denom,
rtermMultiply: left "*" right,
rtermConstant: "const",
pi2: snd(t),
stable: Stable{P},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
req_int_terms: t1 ≡ t2
Latex:
\mforall{}[a,b,c:\mBbbR{}].
(\{((b * b) = (r(4) * a * c)) \mwedge{} (quadratic1(a;b;c) = (-(b)/r(2) * a))\}) supposing
((quadratic1(a;b;c) = quadratic2(a;b;c)) and
(r0 \mleq{} ((b * b) - r(4) * a * c)) and
a \mneq{} r0)
Date html generated:
2020_05_20-PM-00_34_58
Last ObjectModification:
2020_01_06-PM-00_25_26
Theory : reals
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