P. 118 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 11701-11800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	le2addi 11701	Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 𝐷 ∈ ℝ    ⇒   ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))
Theorem	gt0ne0d 11702	Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 0 < 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0)
Theorem	lt0ne0d 11703	Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 < 0)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0)
Theorem	leidd 11704	'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐴)
Theorem	msqgt0d 11705	A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → 0 < (𝐴 · 𝐴))
Theorem	msqge0d 11706	A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐴))
Theorem	lt0neg1d 11707	Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 < 0 ↔ 0 < -𝐴))
Theorem	lt0neg2d 11708	Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (0 < 𝐴 ↔ -𝐴 < 0))
Theorem	le0neg1d 11709	Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
Theorem	le0neg2d 11710	Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0))
Theorem	addgegt0d 11711	Addition of nonnegative and 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	addgtge0d 11712	Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 0 < (𝐴 + 𝐵))
Theorem	addgt0d 11713	Addition of 2 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	addge0d 11714	Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵))
Theorem	mulge0d 11715	The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵))
Theorem	ltnegd 11716	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴))
Theorem	lenegd 11717	Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴))
Theorem	ltnegcon1d 11718	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → -𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → -𝐵 < 𝐴)
Theorem	ltnegcon2d 11719	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < -𝐵)    ⇒   ⊢ (𝜑 → 𝐵 < -𝐴)
Theorem	lenegcon1d 11720	Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → -𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → -𝐵 ≤ 𝐴)
Theorem	lenegcon2d 11721	Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ -𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ≤ -𝐴)
Theorem	ltaddposd 11722	Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴)))
Theorem	ltaddpos2d 11723	Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (0 < 𝐴 ↔ 𝐵 < (𝐴 + 𝐵)))
Theorem	ltsubposd 11724	Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (0 < 𝐴 ↔ (𝐵 − 𝐴) < 𝐵))
Theorem	posdifd 11725	Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
Theorem	addge01d 11726	A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵)))
Theorem	addge02d 11727	A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 + 𝐴)))
Theorem	subge0d 11728	Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴))
Theorem	suble0d 11729	Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 0 ↔ 𝐴 ≤ 𝐵))
Theorem	subge02d 11730	Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (0 ≤ 𝐵 ↔ (𝐴 − 𝐵) ≤ 𝐴))
Theorem	ltadd1d 11731	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶)))
Theorem	leadd1d 11732	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)))
Theorem	leadd2d 11733	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)))
Theorem	ltsubaddd 11734	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 + 𝐵)))
Theorem	lesubaddd 11735	'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵)))
Theorem	ltsubadd2d 11736	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐵 + 𝐶)))
Theorem	lesubadd2d 11737	'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐵 + 𝐶)))
Theorem	ltaddsubd 11738	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 − 𝐵)))
Theorem	ltaddsub2d 11739	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐵 < (𝐶 − 𝐴)))
Theorem	leaddsub2d 11740	'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐵 ≤ (𝐶 − 𝐴)))
Theorem	subled 11741	Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 − 𝐵) ≤ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐶) ≤ 𝐵)
Theorem	lesubd 11742	Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ (𝐵 − 𝐶))    ⇒   ⊢ (𝜑 → 𝐶 ≤ (𝐵 − 𝐴))
Theorem	ltsub23d 11743	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 − 𝐵) < 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐶) < 𝐵)
Theorem	ltsub13d 11744	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < (𝐵 − 𝐶))    ⇒   ⊢ (𝜑 → 𝐶 < (𝐵 − 𝐴))
Theorem	lesub1d 11745	Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝐶) ≤ (𝐵 − 𝐶)))
Theorem	lesub2d 11746	Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 − 𝐵) ≤ (𝐶 − 𝐴)))
Theorem	ltsub1d 11747	Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 − 𝐶) < (𝐵 − 𝐶)))
Theorem	ltsub2d 11748	Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 − 𝐵) < (𝐶 − 𝐴)))
Theorem	ltadd1dd 11749	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐶) < (𝐵 + 𝐶))
Theorem	ltsub1dd 11750	Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐶) < (𝐵 − 𝐶))
Theorem	ltsub2dd 11751	Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 − 𝐵) < (𝐶 − 𝐴))
Theorem	leadd1dd 11752	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))
Theorem	leadd2dd 11753	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))
Theorem	lesub1dd 11754	Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐶) ≤ (𝐵 − 𝐶))
Theorem	lesub2dd 11755	Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 − 𝐵) ≤ (𝐶 − 𝐴))
Theorem	lesub3d 11756	The result of subtracting a number less than or equal to an intermediate number from a number greater than or equal to a third number increased by the intermediate number is greater than or equal to the third number. (Contributed by AV, 13-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → (𝑋 + 𝐶) ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ≤ 𝑋)    ⇒   ⊢ (𝜑 → 𝐶 ≤ (𝐴 − 𝐵))
Theorem	le2addd 11757	Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.) (Proof shortened by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))
Theorem	le2subd 11758	Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐷) ≤ (𝐶 − 𝐵))
Theorem	ltleaddd 11759	Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐶)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))
Theorem	leltaddd 11760	Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐵 < 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))
Theorem	lt2addd 11761	Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐶)    &   ⊢ (𝜑 → 𝐵 < 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))
Theorem	lt2subd 11762	Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐶)    &   ⊢ (𝜑 → 𝐵 < 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐷) < (𝐶 − 𝐵))
Theorem	possumd 11763	Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (0 < (𝐴 + 𝐵) ↔ -𝐵 < 𝐴))
Theorem	sublt0d 11764	When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) < 0 ↔ 𝐴 < 𝐵))
Theorem	ltaddsublt 11765	Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐶 ↔ ((𝐴 + 𝐵) − 𝐶) < 𝐴))
Theorem	1le1 11766	One is less than or equal to one. (Contributed by David A. Wheeler, 16-Jul-2016.)
⊢ 1 ≤ 1
5.3.5  Reciprocals
Theorem	ixi 11767	i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ (i · i) = -1
Theorem	recextlem1 11768	Lemma for recex 11770. (Contributed by Eric Schmidt, 23-May-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (i · 𝐵)) · (𝐴 − (i · 𝐵))) = ((𝐴 · 𝐴) + (𝐵 · 𝐵)))
Theorem	recextlem2 11769	Lemma for recex 11770. (Contributed by Eric Schmidt, 23-May-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + (i · 𝐵)) ≠ 0) → ((𝐴 · 𝐴) + (𝐵 · 𝐵)) ≠ 0)
Theorem	recex 11770*	Existence of reciprocal of nonzero complex number. (Contributed by Eric Schmidt, 22-May-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1)
Theorem	mulcand 11771	Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
Theorem	mulcan2d 11772	Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
Theorem	mulcanad 11773	Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcand 11771. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    &   ⊢ (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	mulcan2ad 11774	Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcan2d 11772. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    &   ⊢ (𝜑 → (𝐴 · 𝐶) = (𝐵 · 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	mulcan 11775	Cancellation law for multiplication (full theorem form). Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
Theorem	mulcan2 11776	Cancellation law for multiplication. (Contributed by NM, 21-Jan-2005.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
Theorem	mulcani 11777	Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐶 ≠ 0    ⇒   ⊢ ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)
Theorem	mul0or 11778	If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0)))
Theorem	mulne0b 11779	The product of two nonzero numbers is nonzero. (Contributed by NM, 1-Aug-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) ↔ (𝐴 · 𝐵) ≠ 0))
Theorem	mulne0 11780	The product of two nonzero numbers is nonzero. (Contributed by NM, 30-Dec-2007.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 · 𝐵) ≠ 0)
Theorem	mulne0i 11781	The product of two nonzero numbers is nonzero. (Contributed by NM, 15-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ (𝐴 · 𝐵) ≠ 0
Theorem	muleqadd 11782	Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = (𝐴 + 𝐵) ↔ ((𝐴 − 1) · (𝐵 − 1)) = 1))
Theorem	receu 11783*	Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)
Theorem	mulnzcnf 11784	Multiplication maps nonzero complex numbers to nonzero complex numbers. (Contributed by Steve Rodriguez, 23-Feb-2007.)
⊢ ( · ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))):((ℂ ∖ {0}) × (ℂ ∖ {0}))⟶(ℂ ∖ {0})
Theorem	mul0ori 11785	If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 7-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0))
Theorem	mul0ord 11786	If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0)))
Theorem	msq0i 11787	A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ ((𝐴 · 𝐴) = 0 ↔ 𝐴 = 0)
Theorem	msq0d 11788	A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐴) = 0 ↔ 𝐴 = 0))
Theorem	mulne0bd 11789	The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) ↔ (𝐴 · 𝐵) ≠ 0))
Theorem	mulne0d 11790	The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ≠ 0)
Theorem	mulcan1g 11791	A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) = (𝐴 · 𝐶) ↔ (𝐴 = 0 ∨ 𝐵 = 𝐶)))
Theorem	mulcan2g 11792	A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ (𝐴 = 𝐵 ∨ 𝐶 = 0)))
Theorem	mulne0bad 11793	A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 11790 and consequence of mulne0bd 11789. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) ≠ 0)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0)
Theorem	mulne0bbd 11794	A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 11790 and consequence of mulne0bd 11789. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) ≠ 0)    ⇒   ⊢ (𝜑 → 𝐵 ≠ 0)
5.3.6  Division
Syntax	cdiv 11795	Extend class notation to include division.
class /
Definition	df-div 11796*	Define division. Theorem divmuli 11896 relates it to multiplication, and divcli 11884 and redivcli 11909 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divval 11799 instead. (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
Theorem	1div0 11797	You can't divide by zero, because division explicitly excludes zero from the domain of the function. Thus, by the definition of function value, it evaluates to the empty set. (This theorem is for information only and normally is not referenced by other proofs. To be meaningful, it assumes that ∅ is not a complex number, which depends on the particular complex number construction that is used.) (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof shortened by SN, 5-Jun-2025.) (New usage is discouraged.)
⊢ (1 / 0) = ∅
Theorem	1div0OLD 11798	Obsolete version of 1div0 11797 as of 5-Jun-2025. (Contributed by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (1 / 0) = ∅
Theorem	divval 11799*	Value of division: if 𝐴 and 𝐵 are complex numbers with 𝐵 nonzero, then (𝐴 / 𝐵) is the (unique) complex number such that (𝐵 · 𝑥) = 𝐴. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
Theorem	divmul 11800	Relationship between division and multiplication. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 17-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ (𝐶 · 𝐵) = 𝐴))