Theorem List for Intuitionistic Logic Explorer - 7901-8000 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | negcli 7901 |
Closure law for negative. (Contributed by NM, 26-Nov-1994.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ -𝐴 ∈ ℂ |
|
Theorem | negidi 7902 |
Addition of a number and its negative. (Contributed by NM,
26-Nov-1994.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 + -𝐴) = 0 |
|
Theorem | negnegi 7903 |
A number is equal to the negative of its negative. Theorem I.4 of
[Apostol] p. 18. (Contributed by NM,
8-Feb-1995.) (Proof shortened by
Andrew Salmon, 22-Oct-2011.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ --𝐴 = 𝐴 |
|
Theorem | subidi 7904 |
Subtraction of a number from itself. (Contributed by NM,
26-Nov-1994.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 − 𝐴) = 0 |
|
Theorem | subid1i 7905 |
Identity law for subtraction. (Contributed by NM, 29-May-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 − 0) = 𝐴 |
|
Theorem | negne0bi 7906 |
A number is nonzero iff its negative is nonzero. (Contributed by NM,
10-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 ≠ 0 ↔ -𝐴 ≠ 0) |
|
Theorem | negrebi 7907 |
The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ) |
|
Theorem | negne0i 7908 |
The negative of a nonzero number is nonzero. (Contributed by NM,
30-Jul-2004.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠
0 ⇒ ⊢ -𝐴 ≠ 0 |
|
Theorem | subcli 7909 |
Closure law for subtraction. (Contributed by NM, 26-Nov-1994.)
(Revised by Mario Carneiro, 21-Dec-2013.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐴 − 𝐵) ∈ ℂ |
|
Theorem | pncan3i 7910 |
Subtraction and addition of equals. (Contributed by NM,
26-Nov-1994.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐴 + (𝐵 − 𝐴)) = 𝐵 |
|
Theorem | negsubi 7911 |
Relationship between subtraction and negative. Theorem I.3 of [Apostol]
p. 18. (Contributed by NM, 26-Nov-1994.) (Proof shortened by Andrew
Salmon, 22-Oct-2011.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐴 + -𝐵) = (𝐴 − 𝐵) |
|
Theorem | subnegi 7912 |
Relationship between subtraction and negative. (Contributed by NM,
1-Dec-2005.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐴 − -𝐵) = (𝐴 + 𝐵) |
|
Theorem | subeq0i 7913 |
If the difference between two numbers is zero, they are equal.
(Contributed by NM, 8-May-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵) |
|
Theorem | neg11i 7914 |
Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵) |
|
Theorem | negcon1i 7915 |
Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴) |
|
Theorem | negcon2i 7916 |
Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐴 = -𝐵 ↔ 𝐵 = -𝐴) |
|
Theorem | negdii 7917 |
Distribution of negative over addition. (Contributed by NM,
28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ -(𝐴 + 𝐵) = (-𝐴 + -𝐵) |
|
Theorem | negsubdii 7918 |
Distribution of negative over subtraction. (Contributed by NM,
6-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ -(𝐴 − 𝐵) = (-𝐴 + 𝐵) |
|
Theorem | negsubdi2i 7919 |
Distribution of negative over subtraction. (Contributed by NM,
1-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ -(𝐴 − 𝐵) = (𝐵 − 𝐴) |
|
Theorem | subaddi 7920 |
Relationship between subtraction and addition. (Contributed by NM,
26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 − 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴) |
|
Theorem | subadd2i 7921 |
Relationship between subtraction and addition. (Contributed by NM,
15-Dec-2006.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 − 𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴) |
|
Theorem | subaddrii 7922 |
Relationship between subtraction and addition. (Contributed by NM,
16-Dec-2006.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ (𝐵 + 𝐶) = 𝐴 ⇒ ⊢ (𝐴 − 𝐵) = 𝐶 |
|
Theorem | subsub23i 7923 |
Swap subtrahend and result of subtraction. (Contributed by NM,
7-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 − 𝐵) = 𝐶 ↔ (𝐴 − 𝐶) = 𝐵) |
|
Theorem | addsubassi 7924 |
Associative-type law for subtraction and addition. (Contributed by NM,
16-Sep-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵 − 𝐶)) |
|
Theorem | addsubi 7925 |
Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵) |
|
Theorem | subcani 7926 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 − 𝐵) = (𝐴 − 𝐶) ↔ 𝐵 = 𝐶) |
|
Theorem | subcan2i 7927 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵) |
|
Theorem | pnncani 7928 |
Cancellation law for mixed addition and subtraction. (Contributed by
NM, 14-Jan-2006.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 + 𝐵) − (𝐴 − 𝐶)) = (𝐵 + 𝐶) |
|
Theorem | addsub4i 7929 |
Rearrangement of 4 terms in a mixed addition and subtraction.
(Contributed by NM, 17-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈
ℂ ⇒ ⊢ ((𝐴 + 𝐵) − (𝐶 + 𝐷)) = ((𝐴 − 𝐶) + (𝐵 − 𝐷)) |
|
Theorem | 0reALT 7930 |
Alternate proof of 0re 7638. (Contributed by NM, 19-Feb-2005.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ 0 ∈ ℝ |
|
Theorem | negcld 7931 |
Closure law for negative. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → -𝐴 ∈ ℂ) |
|
Theorem | subidd 7932 |
Subtraction of a number from itself. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
|
Theorem | subid1d 7933 |
Identity law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − 0) = 𝐴) |
|
Theorem | negidd 7934 |
Addition of a number and its negative. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 + -𝐴) = 0) |
|
Theorem | negnegd 7935 |
A number is equal to the negative of its negative. Theorem I.4 of
[Apostol] p. 18. (Contributed by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → --𝐴 = 𝐴) |
|
Theorem | negeq0d 7936 |
A number is zero iff its negative is zero. (Contributed by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 = 0 ↔ -𝐴 = 0)) |
|
Theorem | negne0bd 7937 |
A number is nonzero iff its negative is nonzero. (Contributed by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 ≠ 0 ↔ -𝐴 ≠ 0)) |
|
Theorem | negcon1d 7938 |
Contraposition law for unary minus. Deduction form of negcon1 7885.
(Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)) |
|
Theorem | negcon1ad 7939 |
Contraposition law for unary minus. One-way deduction form of
negcon1 7885. (Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → -𝐴 = 𝐵) ⇒ ⊢ (𝜑 → -𝐵 = 𝐴) |
|
Theorem | neg11ad 7940 |
The negatives of two complex numbers are equal iff they are equal.
Deduction form of neg11 7884. Generalization of neg11d 7956.
(Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵)) |
|
Theorem | negned 7941 |
If two complex numbers are unequal, so are their negatives.
Contrapositive of neg11d 7956. (Contributed by David Moews,
28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → -𝐴 ≠ -𝐵) |
|
Theorem | negne0d 7942 |
The negative of a nonzero number is nonzero. (Contributed by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → -𝐴 ≠ 0) |
|
Theorem | negrebd 7943 |
The negative of a real is real. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → -𝐴 ∈ ℝ)
⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) |
|
Theorem | subcld 7944 |
Closure law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
|
Theorem | pncand 7945 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
|
Theorem | pncan2d 7946 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐴) = 𝐵) |
|
Theorem | pncan3d 7947 |
Subtraction and addition of equals. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
|
Theorem | npcand 7948 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
|
Theorem | nncand 7949 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) = 𝐵) |
|
Theorem | negsubd 7950 |
Relationship between subtraction and negative. Theorem I.3 of [Apostol]
p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 + -𝐵) = (𝐴 − 𝐵)) |
|
Theorem | subnegd 7951 |
Relationship between subtraction and negative. (Contributed by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵)) |
|
Theorem | subeq0d 7952 |
If the difference between two numbers is zero, they are equal.
(Contributed by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 − 𝐵) = 0) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | subne0d 7953 |
Two unequal numbers have nonzero difference. See also subap0d 8271 which
is the same thing for apartness rather than negated equality.
(Contributed by Mario Carneiro, 1-Jan-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0) |
|
Theorem | subeq0ad 7954 |
The difference of two complex numbers is zero iff they are equal.
Deduction form of subeq0 7859. Generalization of subeq0d 7952.
(Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
|
Theorem | subne0ad 7955 |
If the difference of two complex numbers is nonzero, they are unequal.
Converse of subne0d 7953. Contrapositive of subeq0bd 8008. (Contributed
by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
|
Theorem | neg11d 7956 |
If the difference between two numbers is zero, they are equal.
(Contributed by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → -𝐴 = -𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | negdid 7957 |
Distribution of negative over addition. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → -(𝐴 + 𝐵) = (-𝐴 + -𝐵)) |
|
Theorem | negdi2d 7958 |
Distribution of negative over addition. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → -(𝐴 + 𝐵) = (-𝐴 − 𝐵)) |
|
Theorem | negsubdid 7959 |
Distribution of negative over subtraction. (Contributed by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → -(𝐴 − 𝐵) = (-𝐴 + 𝐵)) |
|
Theorem | negsubdi2d 7960 |
Distribution of negative over subtraction. (Contributed by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) |
|
Theorem | neg2subd 7961 |
Relationship between subtraction and negative. (Contributed by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (-𝐴 − -𝐵) = (𝐵 − 𝐴)) |
|
Theorem | subaddd 7962 |
Relationship between subtraction and addition. (Contributed by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴)) |
|
Theorem | subadd2d 7963 |
Relationship between subtraction and addition. (Contributed by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴)) |
|
Theorem | addsubassd 7964 |
Associative-type law for subtraction and addition. (Contributed by
Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵 − 𝐶))) |
|
Theorem | addsubd 7965 |
Law for subtraction and addition. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵)) |
|
Theorem | subadd23d 7966 |
Commutative/associative law for addition and subtraction. (Contributed
by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐶) = (𝐴 + (𝐶 − 𝐵))) |
|
Theorem | addsub12d 7967 |
Commutative/associative law for addition and subtraction. (Contributed
by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐶)) = (𝐵 + (𝐴 − 𝐶))) |
|
Theorem | npncand 7968 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) + (𝐵 − 𝐶)) = (𝐴 − 𝐶)) |
|
Theorem | nppcand 7969 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → (((𝐴 − 𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶)) |
|
Theorem | nppcan2d 7970 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − (𝐵 + 𝐶)) + 𝐶) = (𝐴 − 𝐵)) |
|
Theorem | nppcan3d 7971 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) + (𝐶 + 𝐵)) = (𝐴 + 𝐶)) |
|
Theorem | subsubd 7972 |
Law for double subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 − 𝐵) + 𝐶)) |
|
Theorem | subsub2d 7973 |
Law for double subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − (𝐵 − 𝐶)) = (𝐴 + (𝐶 − 𝐵))) |
|
Theorem | subsub3d 7974 |
Law for double subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 + 𝐶) − 𝐵)) |
|
Theorem | subsub4d 7975 |
Law for double subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) − 𝐶) = (𝐴 − (𝐵 + 𝐶))) |
|
Theorem | sub32d 7976 |
Swap the second and third terms in a double subtraction. (Contributed
by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) − 𝐶) = ((𝐴 − 𝐶) − 𝐵)) |
|
Theorem | nnncand 7977 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − (𝐵 − 𝐶)) − 𝐶) = (𝐴 − 𝐵)) |
|
Theorem | nnncan1d 7978 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) − (𝐴 − 𝐶)) = (𝐶 − 𝐵)) |
|
Theorem | nnncan2d 7979 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐶) − (𝐵 − 𝐶)) = (𝐴 − 𝐵)) |
|
Theorem | npncan3d 7980 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) + (𝐶 − 𝐴)) = (𝐶 − 𝐵)) |
|
Theorem | pnpcand 7981 |
Cancellation law for mixed addition and subtraction. (Contributed by
Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵 − 𝐶)) |
|
Theorem | pnpcan2d 7982 |
Cancellation law for mixed addition and subtraction. (Contributed by
Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐶) − (𝐵 + 𝐶)) = (𝐴 − 𝐵)) |
|
Theorem | pnncand 7983 |
Cancellation law for mixed addition and subtraction. (Contributed by
Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐴 − 𝐶)) = (𝐵 + 𝐶)) |
|
Theorem | ppncand 7984 |
Cancellation law for mixed addition and subtraction. (Contributed by
Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 − 𝐵)) = (𝐴 + 𝐶)) |
|
Theorem | subcand 7985 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) |
|
Theorem | subcan2d 7986 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
22-Sep-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐵 − 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | subcanad 7987 |
Cancellation law for subtraction. Deduction form of subcan 7888.
Generalization of subcand 7985. (Contributed by David Moews,
28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐴 − 𝐶) ↔ 𝐵 = 𝐶)) |
|
Theorem | subneintrd 7988 |
Introducing subtraction on both sides of a statement of inequality.
Contrapositive of subcand 7985. (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 | subcan2ad 7989 |
Cancellation law for subtraction. Deduction form of subcan2 7858.
Generalization of subcan2d 7986. (Contributed by David Moews,
28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵)) |
|
Theorem | subneintr2d 7990 |
Introducing subtraction on both sides of a statement of inequality.
Contrapositive of subcan2d 7986. (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 | addsub4d 7991 |
Rearrangement of 4 terms in a mixed addition and subtraction.
(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 | subadd4d 7992 |
Rearrangement of 4 terms in a mixed addition and subtraction.
(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 + 𝐷) − (𝐵 + 𝐶))) |
|
Theorem | sub4d 7993 |
Rearrangement of 4 terms in a subtraction. (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 | 2addsubd 7994 |
Law for subtraction and addition. (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)) |
|
Theorem | addsubeq4d 7995 |
Relation between sums and differences. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐶 − 𝐴) = (𝐵 − 𝐷))) |
|
Theorem | mvlraddd 7996 |
Move LHS right addition to RHS. (Contributed by David A. Wheeler,
15-Oct-2018.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = (𝐶 − 𝐵)) |
|
Theorem | mvrraddd 7997 |
Move RHS right addition to LHS. (Contributed by David A. Wheeler,
15-Oct-2018.)
|
⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) = 𝐵) |
|
Theorem | subaddeqd 7998 |
Transfer two terms of a subtraction to an addition in an equality.
(Contributed by Thierry Arnoux, 2-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐷) = (𝐶 − 𝐵)) |
|
Theorem | addlsub 7999 |
Left-subtraction: Subtraction of the left summand from the result of an
addition. (Contributed by BJ, 6-Jun-2019.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) = 𝐶 ↔ 𝐴 = (𝐶 − 𝐵))) |
|
Theorem | addrsub 8000 |
Right-subtraction: Subtraction of the right summand from the result of
an addition. (Contributed by BJ, 6-Jun-2019.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) = 𝐶 ↔ 𝐵 = (𝐶 − 𝐴))) |