Theorem List for Intuitionistic Logic Explorer - 7801-7900 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | lensymd 7801 |
'Less than or equal to' implies 'not less than'. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
|
Theorem | mulgt0d 7802 |
The product of two positive numbers is positive. (Contributed by
Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴)
& ⊢ (𝜑 → 0 < 𝐵) ⇒ ⊢ (𝜑 → 0 < (𝐴 · 𝐵)) |
|
Theorem | letrd 7803 |
Transitive law deduction for 'less than or equal to'. (Contributed by
NM, 20-May-2005.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵)
& ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
|
Theorem | lelttrd 7804 |
Transitive law deduction for 'less than or equal to', 'less than'.
(Contributed by NM, 8-Jan-2006.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵)
& ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) |
|
Theorem | lttrd 7805 |
Transitive law deduction for 'less than'. (Contributed by NM,
9-Jan-2006.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) |
|
Theorem | 0lt1 7806 |
0 is less than 1. Theorem I.21 of [Apostol] p.
20. Part of definition
11.2.7(vi) of [HoTT], p. (varies).
(Contributed by NM, 17-Jan-1997.)
|
⊢ 0 < 1 |
|
Theorem | ltntri 7807 |
Negative trichotomy property for real numbers. It is well known that we
cannot prove real number trichotomy, 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴. Does
that mean there is a pair of real numbers where none of those hold (that
is, where we can refute each of those three relationships)? Actually, no,
as shown here. This is another example of distinguishing between being
unable to prove something, or being able to refute it. (Contributed by
Jim Kingdon, 13-Aug-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴)) |
|
3.2.5 Initial properties of the complex
numbers
|
|
Theorem | mul12 7808 |
Commutative/associative law for multiplication. (Contributed by NM,
30-Apr-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) |
|
Theorem | mul32 7809 |
Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)) |
|
Theorem | mul31 7810 |
Commutative/associative law. (Contributed by Scott Fenton,
3-Jan-2013.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴)) |
|
Theorem | mul4 7811 |
Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))) |
|
Theorem | muladd11 7812 |
A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = ((1 + 𝐴) + (𝐵 + (𝐴 · 𝐵)))) |
|
Theorem | 1p1times 7813 |
Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝐴 ∈ ℂ → ((1 + 1) ·
𝐴) = (𝐴 + 𝐴)) |
|
Theorem | peano2cn 7814 |
A theorem for complex numbers analogous the second Peano postulate
peano2 4467. (Contributed by NM, 17-Aug-2005.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ) |
|
Theorem | peano2re 7815 |
A theorem for reals analogous the second Peano postulate peano2 4467.
(Contributed by NM, 5-Jul-2005.)
|
⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) |
|
Theorem | addcom 7816 |
Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
|
Theorem | addid1 7817 |
0 is an additive identity. (Contributed by Jim
Kingdon,
16-Jan-2020.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) |
|
Theorem | addid2 7818 |
0 is a left identity for addition. (Contributed by
Scott Fenton,
3-Jan-2013.)
|
⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) |
|
Theorem | readdcan 7819 |
Cancellation law for addition over the reals. (Contributed by Scott
Fenton, 3-Jan-2013.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) ↔ 𝐴 = 𝐵)) |
|
Theorem | 00id 7820 |
0 is its own additive identity. (Contributed by Scott
Fenton,
3-Jan-2013.)
|
⊢ (0 + 0) = 0 |
|
Theorem | addid1i 7821 |
0 is an additive identity. (Contributed by NM,
23-Nov-1994.)
(Revised by Scott Fenton, 3-Jan-2013.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 + 0) = 𝐴 |
|
Theorem | addid2i 7822 |
0 is a left identity for addition. (Contributed by NM,
3-Jan-2013.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (0 + 𝐴) = 𝐴 |
|
Theorem | addcomi 7823 |
Addition commutes. Based on ideas by Eric Schmidt. (Contributed by
Scott Fenton, 3-Jan-2013.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴) |
|
Theorem | addcomli 7824 |
Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 + 𝐵) = 𝐶 ⇒ ⊢ (𝐵 + 𝐴) = 𝐶 |
|
Theorem | mul12i 7825 |
Commutative/associative law that swaps the first two factors in a triple
product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew
Salmon, 19-Nov-2011.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)) |
|
Theorem | mul32i 7826 |
Commutative/associative law that swaps the last two factors in a triple
product. (Contributed by NM, 11-May-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵) |
|
Theorem | mul4i 7827 |
Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈
ℂ ⇒ ⊢ ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷)) |
|
Theorem | addid1d 7828 |
0 is an additive identity. (Contributed by Mario
Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 + 0) = 𝐴) |
|
Theorem | addid2d 7829 |
0 is a left identity for addition. (Contributed by
Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (0 + 𝐴) = 𝐴) |
|
Theorem | addcomd 7830 |
Addition commutes. Based on ideas by Eric Schmidt. (Contributed by
Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
|
Theorem | mul12d 7831 |
Commutative/associative law that swaps the first two factors in a triple
product. (Contributed by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) |
|
Theorem | mul32d 7832 |
Commutative/associative law that swaps the last two factors in a triple
product. (Contributed by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)) |
|
Theorem | mul31d 7833 |
Commutative/associative law. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴)) |
|
Theorem | mul4d 7834 |
Rearrangement of 4 factors. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))) |
|
Theorem | muladd11r 7835 |
A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 1) · (𝐵 + 1)) = (((𝐴 · 𝐵) + (𝐴 + 𝐵)) + 1)) |
|
Theorem | comraddd 7836 |
Commute RHS addition, in deduction form. (Contributed by David A.
Wheeler, 11-Oct-2018.)
|
⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐵)) |
|
3.3 Real and complex numbers - basic
operations
|
|
3.3.1 Addition
|
|
Theorem | add12 7837 |
Commutative/associative law that swaps the first two terms in a triple
sum. (Contributed by NM, 11-May-2004.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))) |
|
Theorem | add32 7838 |
Commutative/associative law that swaps the last two terms in a triple sum.
(Contributed by NM, 13-Nov-1999.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵)) |
|
Theorem | add32r 7839 |
Commutative/associative law that swaps the last two terms in a triple sum,
rearranging the parentheses. (Contributed by Paul Chapman,
18-May-2007.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 + 𝐶)) = ((𝐴 + 𝐶) + 𝐵)) |
|
Theorem | add4 7840 |
Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.)
(Proof shortened by Andrew Salmon, 22-Oct-2011.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))) |
|
Theorem | add42 7841 |
Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))) |
|
Theorem | add12i 7842 |
Commutative/associative law that swaps the first two terms in a triple
sum. (Contributed by NM, 21-Jan-1997.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)) |
|
Theorem | add32i 7843 |
Commutative/associative law that swaps the last two terms in a triple
sum. (Contributed by NM, 21-Jan-1997.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵) |
|
Theorem | add4i 7844 |
Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈
ℂ ⇒ ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)) |
|
Theorem | add42i 7845 |
Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈
ℂ ⇒ ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)) |
|
Theorem | add12d 7846 |
Commutative/associative law that swaps the first two terms in a triple
sum. (Contributed by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))) |
|
Theorem | add32d 7847 |
Commutative/associative law that swaps the last two terms in a triple
sum. (Contributed by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵)) |
|
Theorem | add4d 7848 |
Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))) |
|
Theorem | add42d 7849 |
Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))) |
|
3.3.2 Subtraction
|
|
Syntax | cmin 7850 |
Extend class notation to include subtraction.
|
class − |
|
Syntax | cneg 7851 |
Extend class notation to include unary minus. The symbol - is not a
class by itself but part of a compound class definition. We do this
rather than making it a formal function since it is so commonly used.
Note: We use different symbols for unary minus (-) and subtraction
cmin 7850 (−) to prevent
syntax ambiguity. For example, looking at the
syntax definition co 5726, if we used the same symbol
then "( − 𝐴 − 𝐵) " could mean either
"− 𝐴 " minus "𝐵",
or
it could represent the (meaningless) operation of
classes "− " and "− 𝐵
" connected with "operation" "𝐴".
On the other hand, "(-𝐴 − 𝐵) " is unambiguous.
|
class -𝐴 |
|
Definition | df-sub 7852* |
Define subtraction. Theorem subval 7871 shows its value (and describes how
this definition works), theorem subaddi 7966 relates it to addition, and
theorems subcli 7955 and resubcli 7942 prove its closure laws. (Contributed
by NM, 26-Nov-1994.)
|
⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
|
Definition | df-neg 7853 |
Define the negative of a number (unary minus). We use different symbols
for unary minus (-) and subtraction (−) to prevent syntax
ambiguity. See cneg 7851 for a discussion of this. (Contributed by
NM,
10-Feb-1995.)
|
⊢ -𝐴 = (0 − 𝐴) |
|
Theorem | cnegexlem1 7854 |
Addition cancellation of a real number from two complex numbers. Lemma
for cnegex 7857. (Contributed by Eric Schmidt, 22-May-2007.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)) |
|
Theorem | cnegexlem2 7855 |
Existence of a real number which produces a real number when multiplied
by i. (Hint: zero is such a number, although we
don't need to
prove that yet). Lemma for cnegex 7857. (Contributed by Eric Schmidt,
22-May-2007.)
|
⊢ ∃𝑦 ∈ ℝ (i · 𝑦) ∈
ℝ |
|
Theorem | cnegexlem3 7856* |
Existence of real number difference. Lemma for cnegex 7857. (Contributed
by Eric Schmidt, 22-May-2007.)
|
⊢ ((𝑏 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ∃𝑐 ∈ ℝ (𝑏 + 𝑐) = 𝑦) |
|
Theorem | cnegex 7857* |
Existence of the negative of a complex number. (Contributed by Eric
Schmidt, 21-May-2007.)
|
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) |
|
Theorem | cnegex2 7858* |
Existence of a left inverse for addition. (Contributed by Scott Fenton,
3-Jan-2013.)
|
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) |
|
Theorem | addcan 7859 |
Cancellation law for addition. Theorem I.1 of [Apostol] p. 18.
(Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro,
27-May-2016.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)) |
|
Theorem | addcan2 7860 |
Cancellation law for addition. (Contributed by NM, 30-Jul-2004.)
(Revised by Scott Fenton, 3-Jan-2013.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)) |
|
Theorem | addcani 7861 |
Cancellation law for addition. Theorem I.1 of [Apostol] p. 18.
(Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton,
3-Jan-2013.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶) |
|
Theorem | addcan2i 7862 |
Cancellation law for addition. Theorem I.1 of [Apostol] p. 18.
(Contributed by NM, 14-May-2003.) (Revised by Scott Fenton,
3-Jan-2013.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵) |
|
Theorem | addcand 7863 |
Cancellation law for addition. Theorem I.1 of [Apostol] p. 18.
(Contributed by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)) |
|
Theorem | addcan2d 7864 |
Cancellation law for addition. Theorem I.1 of [Apostol] p. 18.
(Contributed by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)) |
|
Theorem | addcanad 7865 |
Cancelling a term on the left-hand side of a sum in an equality.
Consequence of addcand 7863. (Contributed by David Moews,
28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐴 + 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) |
|
Theorem | addcan2ad 7866 |
Cancelling a term on the right-hand side of a sum in an equality.
Consequence of addcan2d 7864. (Contributed by David Moews,
28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | addneintrd 7867 |
Introducing a term on the left-hand side of a sum in a negated
equality. Contrapositive of addcanad 7865. Consequence of addcand 7863.
(Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ≠ (𝐴 + 𝐶)) |
|
Theorem | addneintr2d 7868 |
Introducing a term on the right-hand side of a sum in a negated
equality. Contrapositive of addcan2ad 7866. Consequence of
addcan2d 7864. (Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 + 𝐶) ≠ (𝐵 + 𝐶)) |
|
Theorem | 0cnALT 7869 |
Alternate proof of 0cn 7676. (Contributed by NM, 19-Feb-2005.) (Revised
by
Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ 0 ∈ ℂ |
|
Theorem | negeu 7870* |
Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18.
(Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro,
27-May-2016.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵) |
|
Theorem | subval 7871* |
Value of subtraction, which is the (unique) element 𝑥 such that
𝐵 +
𝑥 = 𝐴. (Contributed by NM, 4-Aug-2007.)
(Revised by Mario
Carneiro, 2-Nov-2013.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)) |
|
Theorem | negeq 7872 |
Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
|
⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) |
|
Theorem | negeqi 7873 |
Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ -𝐴 = -𝐵 |
|
Theorem | negeqd 7874 |
Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → -𝐴 = -𝐵) |
|
Theorem | nfnegd 7875 |
Deduction version of nfneg 7876. (Contributed by NM, 29-Feb-2008.)
(Revised by Mario Carneiro, 15-Oct-2016.)
|
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥-𝐴) |
|
Theorem | nfneg 7876 |
Bound-variable hypothesis builder for the negative of a complex number.
(Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro,
15-Oct-2016.)
|
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥-𝐴 |
|
Theorem | csbnegg 7877 |
Move class substitution in and out of the negative of a number.
(Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon,
22-Oct-2011.)
|
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌-𝐵 = -⦋𝐴 / 𝑥⦌𝐵) |
|
Theorem | subcl 7878 |
Closure law for subtraction. (Contributed by NM, 10-May-1999.)
(Revised by Mario Carneiro, 21-Dec-2013.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) |
|
Theorem | negcl 7879 |
Closure law for negative. (Contributed by NM, 6-Aug-2003.)
|
⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) |
|
Theorem | negicn 7880 |
-i is a complex number (common case). (Contributed by
David A.
Wheeler, 7-Dec-2018.)
|
⊢ -i ∈ ℂ |
|
Theorem | subf 7881 |
Subtraction is an operation on the complex numbers. (Contributed by NM,
4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
|
⊢ − :(ℂ ×
ℂ)⟶ℂ |
|
Theorem | subadd 7882 |
Relationship between subtraction and addition. (Contributed by NM,
20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴)) |
|
Theorem | subadd2 7883 |
Relationship between subtraction and addition. (Contributed by Scott
Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴)) |
|
Theorem | subsub23 7884 |
Swap subtrahend and result of subtraction. (Contributed by NM,
14-Dec-2007.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐴 − 𝐶) = 𝐵)) |
|
Theorem | pncan 7885 |
Cancellation law for subtraction. (Contributed by NM, 10-May-2004.)
(Revised by Mario Carneiro, 27-May-2016.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
|
Theorem | pncan2 7886 |
Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵) |
|
Theorem | pncan3 7887 |
Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
|
Theorem | npcan 7888 |
Cancellation law for subtraction. (Contributed by NM, 10-May-2004.)
(Revised by Mario Carneiro, 27-May-2016.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
|
Theorem | addsubass 7889 |
Associative-type law for addition and subtraction. (Contributed by NM,
6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵 − 𝐶))) |
|
Theorem | addsub 7890 |
Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.)
(Proof shortened by Andrew Salmon, 22-Oct-2011.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵)) |
|
Theorem | subadd23 7891 |
Commutative/associative law for addition and subtraction. (Contributed by
NM, 1-Feb-2007.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐶) = (𝐴 + (𝐶 − 𝐵))) |
|
Theorem | addsub12 7892 |
Commutative/associative law for addition and subtraction. (Contributed by
NM, 8-Feb-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 − 𝐶)) = (𝐵 + (𝐴 − 𝐶))) |
|
Theorem | 2addsub 7893 |
Law for subtraction and addition. (Contributed by NM, 20-Nov-2005.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴 + 𝐵) + 𝐶) − 𝐷) = (((𝐴 + 𝐶) − 𝐷) + 𝐵)) |
|
Theorem | addsubeq4 7894 |
Relation between sums and differences. (Contributed by Jeff Madsen,
17-Jun-2010.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐶 − 𝐴) = (𝐵 − 𝐷))) |
|
Theorem | pncan3oi 7895 |
Subtraction and addition of equals. Almost but not exactly the same as
pncan3i 7956 and pncan 7885, this order happens often when
applying
"operations to both sides" so create a theorem specifically
for it. A
deduction version of this is available as pncand 7991. (Contributed by
David A. Wheeler, 11-Oct-2018.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ ((𝐴 + 𝐵) − 𝐵) = 𝐴 |
|
Theorem | mvrraddi 7896 |
Move RHS right addition to LHS. (Contributed by David A. Wheeler,
11-Oct-2018.)
|
⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐴 = (𝐵 + 𝐶) ⇒ ⊢ (𝐴 − 𝐶) = 𝐵 |
|
Theorem | mvlladdi 7897 |
Move LHS left addition to RHS. (Contributed by David A. Wheeler,
11-Oct-2018.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 + 𝐵) = 𝐶 ⇒ ⊢ 𝐵 = (𝐶 − 𝐴) |
|
Theorem | subid 7898 |
Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.)
(Revised by Mario Carneiro, 27-May-2016.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 − 𝐴) = 0) |
|
Theorem | subid1 7899 |
Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised
by Mario Carneiro, 27-May-2016.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴) |
|
Theorem | npncan 7900 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐵 − 𝐶)) = (𝐴 − 𝐶)) |