Theorem List for Intuitionistic Logic Explorer - 7301-7400 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | lelttrd 7301 |
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 7302 |
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 7303 |
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 |
|
3.2.5 Initial properties of the complex
numbers
|
|
Theorem | mul12 7304 |
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 7305 |
Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)) |
|
Theorem | mul31 7306 |
Commutative/associative law. (Contributed by Scott Fenton,
3-Jan-2013.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴)) |
|
Theorem | mul4 7307 |
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 7308 |
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 7309 |
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 7310 |
A theorem for complex numbers analogous the second Peano postulate
peano2 4344. (Contributed by NM, 17-Aug-2005.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ) |
|
Theorem | peano2re 7311 |
A theorem for reals analogous the second Peano postulate peano2 4344.
(Contributed by NM, 5-Jul-2005.)
|
⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) |
|
Theorem | addcom 7312 |
Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
|
Theorem | addid1 7313 |
0 is an additive identity. (Contributed by Jim
Kingdon,
16-Jan-2020.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) |
|
Theorem | addid2 7314 |
0 is a left identity for addition. (Contributed by
Scott Fenton,
3-Jan-2013.)
|
⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) |
|
Theorem | readdcan 7315 |
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 7316 |
0 is its own additive identity. (Contributed by Scott
Fenton,
3-Jan-2013.)
|
⊢ (0 + 0) = 0 |
|
Theorem | addid1i 7317 |
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 7318 |
0 is a left identity for addition. (Contributed by NM,
3-Jan-2013.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (0 + 𝐴) = 𝐴 |
|
Theorem | addcomi 7319 |
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 7320 |
Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 + 𝐵) = 𝐶 ⇒ ⊢ (𝐵 + 𝐴) = 𝐶 |
|
Theorem | mul12i 7321 |
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 7322 |
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 7323 |
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 7324 |
0 is an additive identity. (Contributed by Mario
Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 + 0) = 𝐴) |
|
Theorem | addid2d 7325 |
0 is a left identity for addition. (Contributed by
Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (0 + 𝐴) = 𝐴) |
|
Theorem | addcomd 7326 |
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 7327 |
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 7328 |
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 7329 |
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 7330 |
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 7331 |
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 7332 |
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 7333 |
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 7334 |
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 7335 |
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 7336 |
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 7337 |
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 7338 |
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 7339 |
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 7340 |
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 7341 |
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 7342 |
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 7343 |
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 7344 |
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 7345 |
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 7346 |
Extend class notation to include subtraction.
|
class − |
|
Syntax | cneg 7347 |
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 7346 (−) to prevent
syntax ambiguity. For example, looking at the
syntax definition co 5543, 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 7348* |
Define subtraction. Theorem subval 7367 shows its value (and describes how
this definition works), theorem subaddi 7462 relates it to addition, and
theorems subcli 7451 and resubcli 7438 prove its closure laws. (Contributed
by NM, 26-Nov-1994.)
|
⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
|
Definition | df-neg 7349 |
Define the negative of a number (unary minus). We use different symbols
for unary minus (-) and subtraction (−) to prevent syntax
ambiguity. See cneg 7347 for a discussion of this. (Contributed by
NM,
10-Feb-1995.)
|
⊢ -𝐴 = (0 − 𝐴) |
|
Theorem | cnegexlem1 7350 |
Addition cancellation of a real number from two complex numbers. Lemma
for cnegex 7353. (Contributed by Eric Schmidt, 22-May-2007.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)) |
|
Theorem | cnegexlem2 7351 |
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 7353. (Contributed by Eric Schmidt,
22-May-2007.)
|
⊢ ∃𝑦 ∈ ℝ (i · 𝑦) ∈
ℝ |
|
Theorem | cnegexlem3 7352* |
Existence of real number difference. Lemma for cnegex 7353. (Contributed
by Eric Schmidt, 22-May-2007.)
|
⊢ ((𝑏 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ∃𝑐 ∈ ℝ (𝑏 + 𝑐) = 𝑦) |
|
Theorem | cnegex 7353* |
Existence of the negative of a complex number. (Contributed by Eric
Schmidt, 21-May-2007.)
|
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) |
|
Theorem | cnegex2 7354* |
Existence of a left inverse for addition. (Contributed by Scott Fenton,
3-Jan-2013.)
|
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) |
|
Theorem | addcan 7355 |
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 7356 |
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 7357 |
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 7358 |
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 7359 |
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 7360 |
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 7361 |
Cancelling a term on the left-hand side of a sum in an equality.
Consequence of addcand 7359. (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 7362 |
Cancelling a term on the right-hand side of a sum in an equality.
Consequence of addcan2d 7360. (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 7363 |
Introducing a term on the left-hand side of a sum in a negated
equality. Contrapositive of addcanad 7361. Consequence of addcand 7359.
(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 7364 |
Introducing a term on the right-hand side of a sum in a negated
equality. Contrapositive of addcan2ad 7362. Consequence of
addcan2d 7360. (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 7365 |
Alternate proof of 0cn 7173. (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 7366* |
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 7367* |
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 7368 |
Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
|
⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) |
|
Theorem | negeqi 7369 |
Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ -𝐴 = -𝐵 |
|
Theorem | negeqd 7370 |
Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → -𝐴 = -𝐵) |
|
Theorem | nfnegd 7371 |
Deduction version of nfneg 7372. (Contributed by NM, 29-Feb-2008.)
(Revised by Mario Carneiro, 15-Oct-2016.)
|
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥-𝐴) |
|
Theorem | nfneg 7372 |
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 7373 |
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 7374 |
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 7375 |
Closure law for negative. (Contributed by NM, 6-Aug-2003.)
|
⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) |
|
Theorem | negicn 7376 |
-i is a complex number (common case). (Contributed by
David A.
Wheeler, 7-Dec-2018.)
|
⊢ -i ∈ ℂ |
|
Theorem | subf 7377 |
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 7378 |
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 7379 |
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 7380 |
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 7381 |
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 7382 |
Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵) |
|
Theorem | pncan3 7383 |
Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
|
Theorem | npcan 7384 |
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 7385 |
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 7386 |
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 7387 |
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 7388 |
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 7389 |
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 7390 |
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 7391 |
Subtraction and addition of equals. Almost but not exactly the same as
pncan3i 7452 and pncan 7381, 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 7487. (Contributed by
David A. Wheeler, 11-Oct-2018.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ ((𝐴 + 𝐵) − 𝐵) = 𝐴 |
|
Theorem | mvrraddi 7392 |
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 7393 |
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 7394 |
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 7395 |
Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised
by Mario Carneiro, 27-May-2016.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴) |
|
Theorem | npncan 7396 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐵 − 𝐶)) = (𝐴 − 𝐶)) |
|
Theorem | nppcan 7397 |
Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 − 𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶)) |
|
Theorem | nnpcan 7398 |
Cancellation law for subtraction: ((a-b)-c)+b = a-c holds for complex
numbers a,b,c. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 − 𝐵) − 𝐶) + 𝐵) = (𝐴 − 𝐶)) |
|
Theorem | nppcan3 7399 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
14-Sep-2015.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐶 + 𝐵)) = (𝐴 + 𝐶)) |
|
Theorem | subcan2 7400 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵)) |