P. 127 of Theorem List - Metamath Proof Explorer

Home	Metamath Proof Explorer Theorem List (p. 127 of 498)	< Previous Next >
		Bad symbols? Try the GIF version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List

Color key:

Metamath Proof Explorer
(1-30880)

Hilbert Space Explorer
(30881-32403)

Users' Mathboxes
(32404-49791)

**Theorem List for Metamath Proof Explorer -** 12601-12700 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	peano2zd 12601	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ)

Theorem	zaddcld 12602	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ)

Theorem	zsubcld 12603	Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ)

Theorem	zmulcld 12604	Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ)

Theorem	znnn0nn 12605	The negative of a negative integer, is a natural number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝑁 ∈ ℤ ∧ ¬ 𝑁 ∈ ℕ₀) → -𝑁 ∈ ℕ)

Theorem	zadd2cl 12606	Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
⊢ (𝑁 ∈ ℤ → (𝑁 + 2) ∈ ℤ)

Theorem	zriotaneg 12607*	The negative of the unique integer such that 𝜑. (Contributed by AV, 1-Dec-2018.)
⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ ℤ 𝜑 → (℩𝑥 ∈ ℤ 𝜑) = -(℩𝑦 ∈ ℤ 𝜓))

Theorem	suprfinzcl 12608	The supremum of a nonempty finite set of integers is a member of the set. (Contributed by AV, 1-Oct-2019.)
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → sup(𝐴, ℝ, < ) ∈ 𝐴)

5.4.10 Decimal arithmetic

Syntax	cdc 12609	Constant used for decimal constructor.
class ;𝐴𝐵

Definition	df-dec 12610	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 30408. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)

Theorem	9p1e10 12611	9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.)
⊢ (9 + 1) = ;10

Theorem	dfdec10 12612	Version of the definition of the "decimal constructor" using ;10 instead of the symbol 10. Of course, this statement cannot be used as definition, because it uses the "decimal constructor". (Contributed by AV, 1-Aug-2021.)
⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)

Theorem	decex 12613	A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ ;𝐴𝐵 ∈ V

Theorem	deceq1 12614	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶)

Theorem	deceq2 12615	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵)

Theorem	deceq1i 12616	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵 ⇒ ⊢ ;𝐴𝐶 = ;𝐵𝐶

Theorem	deceq2i 12617	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵 ⇒ ⊢ ;𝐶𝐴 = ;𝐶𝐵

Theorem	deceq12i 12618	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ ;𝐴𝐶 = ;𝐵𝐷

Theorem	numnncl 12619	Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ

Theorem	num0u 12620	Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ ⇒ ⊢ (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0)

Theorem	num0h 12621	Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ ⇒ ⊢ 𝐴 = ((𝑇 · 0) + 𝐴)

Theorem	numcl 12622	Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ ⇒ ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ₀

Theorem	numsuc 12623	The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ (𝐵 + 1) = 𝐶 & ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) ⇒ ⊢ (𝑁 + 1) = ((𝑇 · 𝐴) + 𝐶)

Theorem	deccl 12624	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ ⇒ ⊢ ;𝐴𝐵 ∈ ℕ₀

Theorem	10nn 12625	10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ

Theorem	10pos 12626	The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ 0 < ;10

Theorem	10nn0 12627	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ₀

Theorem	10re 12628	The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.)
⊢ ;10 ∈ ℝ

Theorem	decnncl 12629	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ ;𝐴𝐵 ∈ ℕ

Theorem	dec0u 12630	Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ ⇒ ⊢ (;10 · 𝐴) = ;𝐴0

Theorem	dec0h 12631	Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ ⇒ ⊢ 𝐴 = ;0𝐴

Theorem	numnncl2 12632	Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝑇 ∈ ℕ & ⊢ 𝐴 ∈ ℕ ⇒ ⊢ ((𝑇 · 𝐴) + 0) ∈ ℕ

Theorem	decnncl2 12633	Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ ⇒ ⊢ ;𝐴0 ∈ ℕ

Theorem	numlt 12634	Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ & ⊢ 𝐵 < 𝐶 ⇒ ⊢ ((𝑇 · 𝐴) + 𝐵) < ((𝑇 · 𝐴) + 𝐶)

Theorem	numltc 12635	Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐶 < 𝑇 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷)

Theorem	le9lt10 12636	A "decimal digit" (i.e. a nonnegative integer less than or equal to 9) is less than 10. (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐴 ≤ 9 ⇒ ⊢ 𝐴 < ;10

Theorem	declt 12637	Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ & ⊢ 𝐵 < 𝐶 ⇒ ⊢ ;𝐴𝐵 < ;𝐴𝐶

Theorem	decltc 12638	Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐶 < ;10 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 < ;𝐵𝐷

Theorem	declth 12639	Comparing two decimal integers (unequal higher places). (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐶 ≤ 9 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 < ;𝐵𝐷

Theorem	decsuc 12640	The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ (𝐵 + 1) = 𝐶 & ⊢ 𝑁 = ;𝐴𝐵 ⇒ ⊢ (𝑁 + 1) = ;𝐴𝐶

Theorem	3declth 12641	Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐸 ∈ ℕ₀ & ⊢ 𝐹 ∈ ℕ₀ & ⊢ 𝐴 < 𝐵 & ⊢ 𝐶 ≤ 9 & ⊢ 𝐸 ≤ 9 ⇒ ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹

Theorem	3decltc 12642	Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 15-Jun-2021.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐸 ∈ ℕ₀ & ⊢ 𝐹 ∈ ℕ₀ & ⊢ 𝐴 < 𝐵 & ⊢ 𝐶 < ;10 & ⊢ 𝐸 < ;10 ⇒ ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹

Theorem	decle 12643	Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐵 ≤ 𝐶 ⇒ ⊢ ;𝐴𝐵 ≤ ;𝐴𝐶

Theorem	decleh 12644	Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐶 ≤ 9 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 ≤ ;𝐵𝐷

Theorem	declei 12645	Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐶 ≤ 9 ⇒ ⊢ 𝐶 ≤ ;𝐴𝐵

Theorem	numlti 12646	Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ & ⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐶 < 𝑇 ⇒ ⊢ 𝐶 < ((𝑇 · 𝐴) + 𝐵)

Theorem	declti 12647	Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐶 < ;10 ⇒ ⊢ 𝐶 < ;𝐴𝐵

Theorem	decltdi 12648	Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐶 ≤ 9 ⇒ ⊢ 𝐶 < ;𝐴𝐵

Theorem	numsucc 12649	The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑌 ∈ ℕ₀ & ⊢ 𝑇 = (𝑌 + 1) & ⊢ 𝐴 ∈ ℕ₀ & ⊢ (𝐴 + 1) = 𝐵 & ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝑌) ⇒ ⊢ (𝑁 + 1) = ((𝑇 · 𝐵) + 0)

Theorem	decsucc 12650	The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ (𝐴 + 1) = 𝐵 & ⊢ 𝑁 = ;𝐴9 ⇒ ⊢ (𝑁 + 1) = ;𝐵0

Theorem	1e0p1 12651	The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 1 = (0 + 1)

Theorem	dec10p 12652	Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (;10 + 𝐴) = ;1𝐴

Theorem	numma 12653	Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) & ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) & ⊢ 𝑃 ∈ ℕ₀ & ⊢ ((𝐴 · 𝑃) + 𝐶) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝐷) = 𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹)

Theorem	nummac 12654	Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) & ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) & ⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐹 ∈ ℕ₀ & ⊢ 𝐺 ∈ ℕ₀ & ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹)

Theorem	numma2c 12655	Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) & ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) & ⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐹 ∈ ℕ₀ & ⊢ 𝐺 ∈ ℕ₀ & ⊢ ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸 & ⊢ ((𝑃 · 𝐵) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) ⇒ ⊢ ((𝑃 · 𝑀) + 𝑁) = ((𝑇 · 𝐸) + 𝐹)

Theorem	numadd 12656	Add two decimal integers 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) & ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) & ⊢ (𝐴 + 𝐶) = 𝐸 & ⊢ (𝐵 + 𝐷) = 𝐹 ⇒ ⊢ (𝑀 + 𝑁) = ((𝑇 · 𝐸) + 𝐹)

Theorem	numaddc 12657	Add two decimal integers 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) & ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) & ⊢ 𝐹 ∈ ℕ₀ & ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 & ⊢ (𝐵 + 𝐷) = ((𝑇 · 1) + 𝐹) ⇒ ⊢ (𝑀 + 𝑁) = ((𝑇 · 𝐸) + 𝐹)

Theorem	nummul1c 12658	The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐸 ∈ ℕ₀ & ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 & ⊢ (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷) ⇒ ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷)

Theorem	nummul2c 12659	The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐸 ∈ ℕ₀ & ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 & ⊢ (𝑃 · 𝐵) = ((𝑇 · 𝐸) + 𝐷) ⇒ ⊢ (𝑃 · 𝑁) = ((𝑇 · 𝐶) + 𝐷)

Theorem	decma 12660	Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ 𝑃 ∈ ℕ₀ & ⊢ ((𝐴 · 𝑃) + 𝐶) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝐷) = 𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹

Theorem	decmac 12661	Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐹 ∈ ℕ₀ & ⊢ 𝐺 ∈ ℕ₀ & ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝐷) = ;𝐺𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹

Theorem	decma2c 12662	Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplier 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐹 ∈ ℕ₀ & ⊢ 𝐺 ∈ ℕ₀ & ⊢ ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸 & ⊢ ((𝑃 · 𝐵) + 𝐷) = ;𝐺𝐹 ⇒ ⊢ ((𝑃 · 𝑀) + 𝑁) = ;𝐸𝐹

Theorem	decadd 12663	Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ (𝐴 + 𝐶) = 𝐸 & ⊢ (𝐵 + 𝐷) = 𝐹 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸𝐹

Theorem	decaddc 12664	Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 & ⊢ 𝐹 ∈ ℕ₀ & ⊢ (𝐵 + 𝐷) = ;1𝐹 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸𝐹

Theorem	decaddc2 12665	Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 & ⊢ (𝐵 + 𝐷) = ;10 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸0

Theorem	decrmanc 12666	Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by AV, 16-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑃 ∈ ℕ₀ & ⊢ (𝐴 · 𝑃) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝑁) = 𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹

Theorem	decrmac 12667	Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by AV, 16-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐹 ∈ ℕ₀ & ⊢ 𝐺 ∈ ℕ₀ & ⊢ ((𝐴 · 𝑃) + 𝐺) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝑁) = ;𝐺𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹

Theorem	decaddm10 12668	The sum of two multiples of 10 is a multiple of 10. (Contributed by AV, 30-Jul-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ ⇒ ⊢ (;𝐴0 + ;𝐵0) = ;(𝐴 + 𝐵)0

Theorem	decaddi 12669	Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐵 + 𝑁) = 𝐶 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐴𝐶

Theorem	decaddci 12670	Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐴 + 1) = 𝐷 & ⊢ 𝐶 ∈ ℕ₀ & ⊢ (𝐵 + 𝑁) = ;1𝐶 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐷𝐶

Theorem	decaddci2 12671	Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐴 + 1) = 𝐷 & ⊢ (𝐵 + 𝑁) = ;10 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐷0

Theorem	decsubi 12672	Difference between a numeral 𝑀 and a nonnegative integer 𝑁 (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐴 + 1) = 𝐷 & ⊢ (𝐵 − 𝑁) = 𝐶 ⇒ ⊢ (𝑀 − 𝑁) = ;𝐴𝐶

Theorem	decmul1 12673	The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) Remove hypothesis 𝐷 ∈ ℕ₀. (Revised by Steven Nguyen, 7-Dec-2022.)
⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ (𝐴 · 𝑃) = 𝐶 & ⊢ (𝐵 · 𝑃) = 𝐷 ⇒ ⊢ (𝑁 · 𝑃) = ;𝐶𝐷

Theorem	decmul1c 12674	The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐸 ∈ ℕ₀ & ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 & ⊢ (𝐵 · 𝑃) = ;𝐸𝐷 ⇒ ⊢ (𝑁 · 𝑃) = ;𝐶𝐷

Theorem	decmul2c 12675	The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐸 ∈ ℕ₀ & ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 & ⊢ (𝑃 · 𝐵) = ;𝐸𝐷 ⇒ ⊢ (𝑃 · 𝑁) = ;𝐶𝐷

Theorem	decmulnc 12676	The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.)
⊢ 𝑁 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ ⇒ ⊢ (𝑁 · ;𝐴𝐵) = ;(𝑁 · 𝐴)(𝑁 · 𝐵)

Theorem	11multnc 12677	The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
⊢ 𝑁 ∈ ℕ₀ ⇒ ⊢ (𝑁 · ;11) = ;𝑁𝑁

Theorem	decmul10add 12678	A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑀 ∈ ℕ₀ & ⊢ 𝐸 = (𝑀 · 𝐴) & ⊢ 𝐹 = (𝑀 · 𝐵) ⇒ ⊢ (𝑀 · ;𝐴𝐵) = (;𝐸0 + 𝐹)

Theorem	6p5lem 12679	Lemma for 6p5e11 12682 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐸 ∈ ℕ₀ & ⊢ 𝐵 = (𝐷 + 1) & ⊢ 𝐶 = (𝐸 + 1) & ⊢ (𝐴 + 𝐷) = ;1𝐸 ⇒ ⊢ (𝐴 + 𝐵) = ;1𝐶

Theorem	5p5e10 12680	5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
⊢ (5 + 5) = ;10

Theorem	6p4e10 12681	6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
⊢ (6 + 4) = ;10

Theorem	6p5e11 12682	6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (6 + 5) = ;11

Theorem	6p6e12 12683	6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (6 + 6) = ;12

Theorem	7p3e10 12684	7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
⊢ (7 + 3) = ;10

Theorem	7p4e11 12685	7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (7 + 4) = ;11

Theorem	7p5e12 12686	7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 + 5) = ;12

Theorem	7p6e13 12687	7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 + 6) = ;13

Theorem	7p7e14 12688	7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 + 7) = ;14

Theorem	8p2e10 12689	8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
⊢ (8 + 2) = ;10

Theorem	8p3e11 12690	8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 + 3) = ;11

Theorem	8p4e12 12691	8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 4) = ;12

Theorem	8p5e13 12692	8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 5) = ;13

Theorem	8p6e14 12693	8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 6) = ;14

Theorem	8p7e15 12694	8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 7) = ;15

Theorem	8p8e16 12695	8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 8) = ;16

Theorem	9p2e11 12696	9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (9 + 2) = ;11

Theorem	9p3e12 12697	9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 3) = ;12

Theorem	9p4e13 12698	9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 4) = ;13

Theorem	9p5e14 12699	9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 5) = ;14

Theorem	9p6e15 12700	9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 6) = ;15

< Previous Next >

Page List Jump to page: Contents 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49100 492 49101-49200 493 49201-49300 494 49301-49400 495 49401-49500 496 49501-49600 497 49601-49700 498 49701-49791

< Previous Next >