Nuprl Lemma : rexp-radd
∀[x,y:ℝ]. (e^x + y = (e^x * e^y))
Proof
Definitions occuring in Statement :
rexp: e^x,
req: x = y,
rmul: a * b,
radd: a + b,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
prop: ℙ,
so_apply: x[s],
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rge: x ≥ y,
guard: {T},
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rfun: I ⟶ℝ,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermVar: rtermVar(var),
pi1: fst(t),
rtermMultiply: left "*" right,
rtermAdd: left "+" right,
rtermConstant: "const",
pi2: snd(t),
exists: ∃x:A. B[x],
cand: A c∧ B,
subtype_rel: A ⊆r B
Lemmas referenced :
rexp-of-nonneg,
rexp-unique,
rdiv_wf,
rexp_wf,
radd_wf,
rless_wf,
int-to-real_wf,
req_functionality,
rdiv_functionality,
rexp_functionality,
radd_functionality,
req_weakening,
req_wf,
rleq_wf,
real_wf,
rless-int,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rinv_wf2,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
rmul_preserves_req,
rmul_wf,
itermMultiply_wf,
rmul_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
req_transitivity,
rmul-rinv,
real_term_value_mul_lemma,
derivative-rdiv-const,
riiint_wf,
i-member_wf,
derivative-rexp,
derivative-function-radd-const,
derivative_functionality,
assert-rat-term-eq2,
rtermDivide_wf,
rtermMultiply_wf,
rtermVar_wf,
rtermAdd_wf,
rtermConstant_wf,
req_inversion,
req_witness,
rmax_wf,
rminus_wf,
rleq-rmax,
radd-preserves-rleq,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd-ac,
radd-rminus-both,
radd-zero-both,
rmul-assoc,
squash_wf,
true_wf,
radd_comm_eq,
iff_weakening_equal,
radd-assoc
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
lambdaEquality_alt,
isectElimination,
because_Cache,
independent_isectElimination,
inrFormation_alt,
universeIsType,
natural_numberEquality,
productElimination,
inhabitedIsType,
independent_pairFormation,
imageMemberEquality,
baseClosed,
closedConclusion,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
setIsType,
equalityIstype,
isect_memberFormation,
isect_memberEquality,
dependent_pairFormation,
productEquality,
inrFormation,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality
Latex:
\mforall{}[x,y:\mBbbR{}]. (e\^{}x + y = (e\^{}x * e\^{}y))
Date html generated:
2019_10_30-AM-11_40_10
Last ObjectModification:
2019_04_03-AM-00_21_52
Theory : reals_2
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