P. 434 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43301-43400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	onsucf1lem 43301*	For ordinals, the successor operation is injective, so there is at most one ordinal that a given ordinal could be the successor of. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
⊢ (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
Theorem	onsucf1olem 43302*	The successor operation is bijective between the ordinals and the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃!𝑏 ∈ On 𝐴 = suc 𝑏)
Theorem	onsucrn 43303*	The successor operation is surjective onto its range, the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥)    ⇒   ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
Theorem	onsucf1o 43304*	The successor operation is a bijective function between the ordinals and the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥)    ⇒   ⊢ 𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
Theorem	dflim7 43305*	A limit ordinal is a non-zero ordinal that contains all the successors of its elements. Lemma 1.18 of [Schloeder] p. 2. Closely related to dflim4 7778. (Contributed by RP, 17-Jan-2025.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∀𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅))
Theorem	onov0suclim 43306	Compactly express rules for binary operations on ordinals. (Contributed by RP, 18-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 ⊗ ∅) = 𝐷)    &   ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊗ suc 𝐶) = 𝐸)    &   ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴 ⊗ 𝐵) = 𝐹)    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹)))
Theorem	oa0suclim 43307*	Closed form expression of the value of ordinal addition for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.3 of [Schloeder] p. 4. See oa0 8431, oasuc 8439, and oalim 8447. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 +o 𝐵) = 𝐴) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 +o 𝐵) = suc (𝐴 +o 𝐶)) ∧ (Lim 𝐵 → (𝐴 +o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 +o 𝑐))))
Theorem	om0suclim 43308*	Closed form expression of the value of ordinal multiplication for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.5 of [Schloeder] p. 4. See om0 8432, omsuc 8441, and omlim 8448. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ·o 𝐵) = ((𝐴 ·o 𝐶) +o 𝐴)) ∧ (Lim 𝐵 → (𝐴 ·o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 ·o 𝑐))))
Theorem	oe0suclim 43309*	Closed form expression of the value of ordinal exponentiation for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.6 of [Schloeder] p. 4. See oe0 8437, oesuc 8442, oe0m1 8436, and oelim 8449. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ↑o 𝐵) = 1o) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐵) = ((𝐴 ↑o 𝐶) ·o 𝐴)) ∧ (Lim 𝐵 → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))))
Theorem	oaomoecl 43310	The operations of addition, multiplication, and exponentiation are closed. Remark 2.8 of [Schloeder] p. 5. See oacl 8450, omcl 8451, oecl 8452. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ On ∧ (𝐴 ·o 𝐵) ∈ On ∧ (𝐴 ↑o 𝐵) ∈ On))
Theorem	onsupsucismax 43311*	If the union of a set of ordinals is a successor ordinal, then that union is the maximum element of the set. This is not a bijection because sets where the maximum element is zero or a limit ordinal exist. Lemma 2.11 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏 → ∪ 𝐴 ∈ 𝐴))
Theorem	onsssupeqcond 43312*	If for every element of a set of ordinals there is an element of a subset which is at least as large, then the union of the set and the subset is the same. Lemma 2.12 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((𝐵 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏) → ∪ 𝐴 = ∪ 𝐵))
Theorem	limexissup 43313	An ordinal which is a limit ordinal is equal to its supremum. Lemma 2.13 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.)
⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = sup(𝐴, On, E ))
Theorem	limiun 43314*	A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6368. (Contributed by RP, 27-Jan-2025.)
⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥)
Theorem	limexissupab 43315*	An ordinal which is a limit ordinal is equal to the supremum of the class of all its elements. Lemma 2.13 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.)
⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E ))
Theorem	om1om1r 43316	Ordinal one is both a left and right identity of ordinal multiplication. Lemma 2.15 of [Schloeder] p. 5. See om1 8457 and om1r 8458 for individual statements. (Contributed by RP, 29-Jan-2025.)
⊢ (𝐴 ∈ On → ((1o ·o 𝐴) = (𝐴 ·o 1o) ∧ (𝐴 ·o 1o) = 𝐴))
Theorem	oe0rif 43317	Ordinal zero raised to any non-zero ordinal power is zero and zero to the zeroth power is one. Lemma 2.18 of [Schloeder] p. 6. (Contributed by RP, 29-Jan-2025.)
⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ ∈ 𝐴, ∅, 1o))
Theorem	oasubex 43318*	While subtraction can't be a binary operation on ordinals, for any pair of ordinals there exists an ordinal that can be added to the lessor (or equal) one which will sum to the greater. Theorem 2.19 of [Schloeder] p. 6. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → ∃𝑐 ∈ On (𝑐 ⊆ 𝐴 ∧ (𝐵 +o 𝑐) = 𝐴))
Theorem	nnamecl 43319	Natural numbers are closed under ordinal addition, multiplication, and exponentiation. Theorem 2.20 of [Schloeder] p. 6. See nnacl 8526, nnmcl 8527, nnecl 8528. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ↑o 𝐵) ∈ ω))
Theorem	onsucwordi 43320	The successor operation preserves the less-than-or-equal relationship between ordinals. Lemma 3.1 of [Schloeder] p. 7. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → suc 𝐴 ⊆ suc 𝐵))
Theorem	oalim2cl 43321	The ordinal sum of any ordinal with a limit ordinal on the right is a limit ordinal. (Contributed by RP, 6-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ Lim 𝐵 ∧ 𝐵 ∈ 𝑉) → Lim (𝐴 +o 𝐵))
Theorem	oaltublim 43322	Given 𝐶 is a limit ordinal, the sum of any ordinal with an ordinal less than 𝐶 is less than the sum of the first ordinal with 𝐶. Lemma 3.5 of [Schloeder] p. 7. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐶 ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶))
Theorem	oaordi3 43323	Ordinal addition of the same number on the left preserves the ordering of the numbers on the right. Lemma 3.6 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶)))
Theorem	oaord3 43324	When the same ordinal is added on the left, ordering of the sums is equivalent to the ordering of the ordinals on the right. Theorem 3.7 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶)))
Theorem	1oaomeqom 43325	Ordinal one plus omega is equal to omega. See oaabs 8563 for the sum of any natural number on the left and ordinal at least as large as omega on the right. Lemma 3.8 of [Schloeder] p. 8. See oaabs2 8564 where a power of omega is the upper bound of the left and a lower bound on the right. (Contributed by RP, 29-Jan-2025.)
⊢ (1o +o ω) = ω
Theorem	oaabsb 43326	The right addend absorbs the sum with an ordinal iff that ordinal times omega is less than or equal to the right addend. (Contributed by RP, 19-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ (𝐴 +o 𝐵) = 𝐵))
Theorem	oaordnrex 43327	When omega is added on the right to ordinals zero and one, ordering of the sums is not equivalent to the ordering of the ordinals on the left. Remark 3.9 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.)
⊢ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))
Theorem	oaordnr 43328*	When the same ordinal is added on the right, ordering of the sums is not equivalent to the ordering of the ordinals on the left. Remark 3.9 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.)
⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐))
Theorem	omge1 43329	Any non-zero ordinal product is greater-than-or-equal to the term on the left. Lemma 3.11 of [Schloeder] p. 8. See omword1 8488. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ·o 𝐵))
Theorem	omge2 43330	Any non-zero ordinal product is greater-than-or-equal to the term on the right. Lemma 3.12 of [Schloeder] p. 9. See omword2 8489. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ⊆ (𝐴 ·o 𝐵))
Theorem	omlim2 43331	The non-zero product with an limit ordinal on the right is a limit ordinal. Lemma 3.13 of [Schloeder] p. 9. (Contributed by RP, 29-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (Lim 𝐵 ∧ 𝐵 ∈ 𝑉)) → Lim (𝐴 ·o 𝐵))
Theorem	omord2lim 43332	Given a limit ordinal, the product of any non-zero ordinal with an ordinal less than that limit ordinal is less than the product of the non-zero ordinal with the limit ordinal . Lemma 3.14 of [Schloeder] p. 9. (Contributed by RP, 29-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶)))
Theorem	omord2i 43333	Ordinal multiplication of the same non-zero number on the left preserves the ordering of the numbers on the right. Lemma 3.15 of [Schloeder] p. 9. (Contributed by RP, 29-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶)))
Theorem	omord2com 43334	When the same non-zero ordinal is multiplied on the left, ordering of the products is equivalent to the ordering of the ordinals on the right. Theorem 3.16 of [Schloeder] p. 9. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ∈ 𝐶 ∧ ∅ ∈ 𝐴) ↔ (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶)))
Theorem	2omomeqom 43335	Ordinal two times omega is omega. Lemma 3.17 of [Schloeder] p. 10. (Contributed by RP, 30-Jan-2025.)
⊢ (2o ·o ω) = ω
Theorem	omnord1ex 43336	When omega is multiplied on the right to ordinals one and two, ordering of the products is not equivalent to the ordering of the ordinals on the left. Remark 3.18 of [Schloeder] p. 10. (Contributed by RP, 29-Jan-2025.)
⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))
Theorem	omnord1 43337*	When the same non-zero ordinal is multiplied on the right, ordering of the products is not equivalent to the ordering of the ordinals on the left. Remark 3.18 of [Schloeder] p. 10. (Contributed by RP, 4-Feb-2025.)
⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐))
Theorem	oege1 43338	Any non-zero ordinal power is greater-than-or-equal to the term on the left. Lemma 3.19 of [Schloeder] p. 10. See oewordi 8506. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ↑o 𝐵))
Theorem	oege2 43339	Any power of an ordinal at least as large as two is greater-than-or-equal to the term on the right. Lemma 3.20 of [Schloeder] p. 10. See oeworde 8508. (Contributed by RP, 29-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))
Theorem	rp-oelim2 43340	The power of an ordinal at least as large as two with a limit ordinal on thr right is a limit ordinal. Lemma 3.21 of [Schloeder] p. 10. See oelimcl 8515. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐵 ∧ 𝐵 ∈ 𝑉)) → Lim (𝐴 ↑o 𝐵))
Theorem	oeord2lim 43341	Given a limit ordinal, the power of any base at least as large as two raised to an ordinal less than that limit ordinal is less than the power of that base raised to the limit ordinal . Lemma 3.22 of [Schloeder] p. 10. See oeordi 8502. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
Theorem	oeord2i 43342	Ordinal exponentiation of the same base at least as large as two preserves the ordering of the exponents. Lemma 3.23 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
Theorem	oeord2com 43343	When the same base at least as large as two is raised to ordinal powers, , ordering of the power is equivalent to the ordering of the exponents. Theorem 3.24 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
Theorem	nnoeomeqom 43344	Any natural number at least as large as two raised to the power of omega is omega. Lemma 3.25 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝐴 ↑o ω) = ω)
Theorem	df3o2 43345	Ordinal 3 is the unordered triple containing ordinals 0, 1, and 2. (Contributed by RP, 8-Jul-2021.)
⊢ 3o = {∅, 1o, 2o}
Theorem	df3o3 43346	Ordinal 3, fully expanded. (Contributed by RP, 8-Jul-2021.)
⊢ 3o = {∅, {∅}, {∅, {∅}}}
Theorem	oenord1ex 43347	When ordinals two and three are both raised to the power of omega, ordering of the powers is not equivalent to the ordering of the bases. Remark 3.26 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))
Theorem	oenord1 43348*	When two ordinals (both at least as large as two) are raised to the same power, ordering of the powers is not equivalent to the ordering of the bases. Remark 3.26 of [Schloeder] p. 11. (Contributed by RP, 4-Feb-2025.)
⊢ ∃𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))
Theorem	oaomoencom 43349*	Ordinal addition, multiplication, and exponentiation do not generally commute. Theorem 4.1 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ (∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎))
Theorem	oenassex 43350	Ordinal two raised to two to the zeroth power is not the same as two squared then raised to the zeroth power. (Contributed by RP, 30-Jan-2025.)
⊢ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)
Theorem	oenass 43351*	Ordinal exponentiation is not associative. Remark 4.6 of [Schloeder] p. 14. (Contributed by RP, 30-Jan-2025.)
⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐)
Theorem	cantnftermord 43352	For terms of the form of a power of omega times a non-zero natural number, ordering of the exponents implies ordering of the terms. Lemma 5.1 of [Schloeder] p. 15. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ (ω ∖ 1o) ∧ 𝐷 ∈ (ω ∖ 1o))) → (𝐴 ∈ 𝐵 → ((ω ↑o 𝐴) ·o 𝐶) ∈ ((ω ↑o 𝐵) ·o 𝐷)))
Theorem	cantnfub 43353*	Given a finite number of terms of the form ((ω ↑o (𝐴‘𝑛)) ·o (𝑀‘𝑛)) with distinct exponents, we may order them from largest to smallest and find the sum is less than (ω ↑o 𝑋) when (𝐴‘𝑛) is less than 𝑋 and (𝑀‘𝑛) is less than ω. Lemma 5.2 of [Schloeder] p. 15. (Contributed by RP, 31-Jan-2025.)
⊢ (𝜑 → 𝑋 ∈ On)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝐴:𝑁–1-1→𝑋)    &   ⊢ (𝜑 → 𝑀:𝑁⟶ω)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑥)), ∅))    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom (ω CNF 𝑋) ∧ ((ω CNF 𝑋)‘𝐹) ∈ (ω ↑o 𝑋)))
Theorem	cantnfub2 43354*	Given a finite number of terms of the form ((ω ↑o (𝐴‘𝑛)) ·o (𝑀‘𝑛)) with distinct exponents, we may order them from largest to smallest and find the sum is less than (ω ↑o suc ∪ ran 𝐴) when (𝑀‘𝑛) is less than ω. Lemma 5.2 of [Schloeder] p. 15. (Contributed by RP, 9-Feb-2025.)
⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝐴:𝑁–1-1→On)    &   ⊢ (𝜑 → 𝑀:𝑁⟶ω)    &   ⊢ 𝐹 = (𝑥 ∈ suc ∪ ran 𝐴 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑥)), ∅))    ⇒   ⊢ (𝜑 → (suc ∪ ran 𝐴 ∈ On ∧ 𝐹 ∈ dom (ω CNF suc ∪ ran 𝐴) ∧ ((ω CNF suc ∪ ran 𝐴)‘𝐹) ∈ (ω ↑o suc ∪ ran 𝐴)))
Theorem	bropabg 43355*	Equivalence for two classes related by an ordered-pair class abstraction. A generalization of brsslt 27723. (Contributed by RP, 26-Sep-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))
Theorem	cantnfresb 43356*	A Cantor normal form which sums to less than a certain power has only zeros for larger components. (Contributed by RP, 3-Feb-2025.)
⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶) ↔ ∀𝑥 ∈ (𝐵 ∖ 𝐶)(𝐹‘𝑥) = ∅))
Theorem	cantnf2 43357*	For every ordinal, 𝐴, there is a an ordinal exponent 𝑏 such that 𝐴 is less than (ω ↑o 𝑏) and for every ordinal at least as large as 𝑏 there is a unique Cantor normal form, 𝑓, with zeros for all the unnecessary higher terms, that sums to 𝐴. Theorem 5.3 of [Schloeder] p. 16. (Contributed by RP, 3-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑏 ∈ On ∀𝑐 ∈ (On ∖ 𝑏)∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))
21.36.2  Natural addition of Cantor normal forms
Theorem	oawordex2 43358*	If 𝐶 is between 𝐴 (inclusive) and (𝐴 +o 𝐵) (exclusive), there is an ordinal which equals 𝐶 when summed to 𝐴. This is a slightly different statement than oawordex 8472 or oawordeu 8470. (Contributed by RP, 7-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥 ∈ 𝐵 (𝐴 +o 𝑥) = 𝐶)
Theorem	nnawordexg 43359*	If an ordinal, 𝐵, is in a half-open interval between some 𝐴 and the next limit ordinal, 𝐵 is the sum of the 𝐴 and some natural number. This weakens the antecedent of nnawordex 8552. (Contributed by RP, 7-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +o ω)) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
Theorem	succlg 43360	Closure law for ordinal successor. (Contributed by RP, 8-Jan-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = ∅ ∨ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o)))) → suc 𝐴 ∈ 𝐵)
Theorem	dflim5 43361*	A limit ordinal is either the proper class of ordinals or some nonzero product with omega. (Contributed by RP, 8-Jan-2025.)
⊢ (Lim 𝐴 ↔ (𝐴 = On ∨ ∃𝑥 ∈ (On ∖ 1o)𝐴 = (ω ·o 𝑥)))
Theorem	oacl2g 43362	Closure law for ordinal addition. Here we show that ordinal addition is closed within the empty set or any ordinal power of omega. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On))) → (𝐴 +o 𝐵) ∈ 𝐶)
Theorem	onmcl 43363	If an ordinal is less than a power of omega, the product with a natural number is also less than that power of omega. (Contributed by RP, 19-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
Theorem	omabs2 43364	Ordinal multiplication by a larger ordinal is absorbed when the larger ordinal is either 2 or ω raised to some power of ω. (Contributed by RP, 12-Jan-2025.)
⊢ (((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ 𝐵 = 2o ∨ (𝐵 = (ω ↑o (ω ↑o 𝐶)) ∧ 𝐶 ∈ On))) → (𝐴 ·o 𝐵) = 𝐵)
Theorem	omcl2 43365	Closure law for ordinal multiplication. (Contributed by RP, 12-Jan-2025.)
⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
Theorem	omcl3g 43366	Closure law for ordinal multiplication. (Contributed by RP, 14-Jan-2025.)
⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
Theorem	ordsssucb 43367	An ordinal number is less than or equal to the successor of an ordinal class iff the ordinal number is either less than or equal to the ordinal class or the ordinal number is equal to the successor of the ordinal class. See also ordsssucim 43434, limsssuc 7780. (Contributed by RP, 22-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))
Theorem	tfsconcatlem 43368*	Lemma for tfsconcatun 43369. (Contributed by RP, 23-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑥∃𝑦 ∈ 𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))
Theorem	tfsconcatun 43369*	The concatenation of two transfinite series is a union of functions. (Contributed by RP, 23-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) = (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}))
Theorem	tfsconcatfn 43370*	The concatenation of two transfinite series is a transfinite series. (Contributed by RP, 22-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
Theorem	tfsconcatfv1 43371*	An early value of the concatenation of two transfinite series. (Contributed by RP, 23-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐶) → ((𝐴 + 𝐵)‘𝑋) = (𝐴‘𝑋))
Theorem	tfsconcatfv2 43372*	A latter value of the concatenation of two transfinite series. (Contributed by RP, 23-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → ((𝐴 + 𝐵)‘(𝐶 +o 𝑋)) = (𝐵‘𝑋))
Theorem	tfsconcatfv 43373*	The value of the concatenation of two transfinite series. (Contributed by RP, 24-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → ((𝐴 + 𝐵)‘𝑋) = if(𝑋 ∈ 𝐶, (𝐴‘𝑋), (𝐵‘(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋))))
Theorem	tfsconcatrn 43374*	The range of the concatenation of two transfinite series. (Contributed by RP, 24-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))
Theorem	tfsconcatfo 43375*	The concatenation of two transfinite series is onto the union of the ranges. (Contributed by RP, 24-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵):(𝐶 +o 𝐷)–onto→(ran 𝐴 ∪ ran 𝐵))
Theorem	tfsconcatb0 43376*	The concatentation with the empty series leaves the series unchanged. (Contributed by RP, 25-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵 = ∅ ↔ (𝐴 + 𝐵) = 𝐴))
Theorem	tfsconcat0i 43377*	The concatentation with the empty series leaves the series unchanged. (Contributed by RP, 28-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 = ∅ → (𝐴 + 𝐵) = 𝐵))
Theorem	tfsconcat0b 43378*	The concatentation with the empty series leaves the finite series unchanged. (Contributed by RP, 1-Mar-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ ↔ (𝐴 + 𝐵) = 𝐵))
Theorem	tfsconcat00 43379*	The concatentation of two empty series results in an empty series. (Contributed by RP, 25-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 + 𝐵) = ∅))
Theorem	tfsconcatrev 43380*	If the domain of a transfinite sequence is an ordinal sum, the sequence can be decomposed into two sequences with domains corresponding to the addends. Theorem 2 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ∃𝑢 ∈ (ran 𝐹 ↑m 𝐶)∃𝑣 ∈ (ran 𝐹 ↑m 𝐷)((𝑢 + 𝑣) = 𝐹 ∧ dom 𝑢 = 𝐶 ∧ dom 𝑣 = 𝐷))
Theorem	tfsconcatrnss12 43381*	The range of the concatenation of transfinite sequences is a superset of the ranges of both sequences. Theorem 3 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran 𝐴 ⊆ ran (𝐴 + 𝐵) ∧ ran 𝐵 ⊆ ran (𝐴 + 𝐵)))
Theorem	tfsconcatrnss 43382*	The concatenation of transfinite sequences yields elements from a class iff both sequences yield elements from that class. (Contributed by RP, 2-Mar-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran (𝐴 + 𝐵) ⊆ 𝑋 ↔ (ran 𝐴 ⊆ 𝑋 ∧ ran 𝐵 ⊆ 𝑋)))
Theorem	tfsconcatrnsson 43383*	The concatenation of transfinite sequences yields ordinals iff both sequences yield ordinals. Theorem 4 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran (𝐴 + 𝐵) ⊆ On ↔ (ran 𝐴 ⊆ On ∧ ran 𝐵 ⊆ On)))
Theorem	tfsnfin 43384	A transfinite sequence is infinite iff its domain is greater than or equal to omega. Theorem 5 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 1-Mar-2025.)
⊢ ((𝐴 Fn 𝐵 ∧ 𝐵 ∈ On) → (¬ 𝐴 ∈ Fin ↔ ω ⊆ 𝐵))
Theorem	rp-tfslim 43385*	The limit of a sequence of ordinals is the union of its range. (Contributed by RP, 1-Mar-2025.)
⊢ (𝐴 Fn 𝐵 → ∪ 𝑥 ∈ 𝐵 (𝐴‘𝑥) = ∪ ran 𝐴)
Theorem	ofoafg 43386*	Addition operator for functions from sets into ordinals results in a function from the intersection of sets into an ordinal. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → ( ∘f +o ↾ ((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))):((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))⟶(𝐹 ↑m 𝐶))
Theorem	ofoaf 43387	Addition operator for functions from sets into power of omega results in a function from the intersection of sets to that power of omega. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷))) → ( ∘f +o ↾ ((𝐸 ↑m 𝐴) × (𝐸 ↑m 𝐵))):((𝐸 ↑m 𝐴) × (𝐸 ↑m 𝐵))⟶(𝐸 ↑m 𝐶))
Theorem	ofoafo 43388	Addition operator for functions from a set into a power of omega is an onto binary operator. (Contributed by RP, 5-Jan-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))):((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))–onto→(𝐶 ↑m 𝐴))
Theorem	ofoacl 43389	Closure law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) ∈ (𝐶 ↑m 𝐴))
Theorem	ofoaid1 43390	Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → (𝐹 ∘f +o (𝐴 × {∅})) = 𝐹)
Theorem	ofoaid2 43391	Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → ((𝐴 × {∅}) ∘f +o 𝐹) = 𝐹)
Theorem	ofoaass 43392	Component-wise addition of ordinal-yielding functions is associative. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))
Theorem	ofoacom 43393	Component-wise addition of natural numnber-yielding functions commutes. (Contributed by RP, 5-Jan-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))
Theorem	naddcnff 43394	Addition operator for Cantor normal forms is a function into Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)
Theorem	naddcnffn 43395	Addition operator for Cantor normal forms is a function. (Contributed by RP, 2-Jan-2025.)
⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
Theorem	naddcnffo 43396	Addition of Cantor normal forms is a function onto Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto→𝑆)
Theorem	naddcnfcl 43397	Closure law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) ∈ 𝑆)
Theorem	naddcnfcom 43398	Component-wise ordinal addition of Cantor normal forms commutes. (Contributed by RP, 2-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))
Theorem	naddcnfid1 43399	Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)
Theorem	naddcnfid2 43400	Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = 𝐹)