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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Syntax | ccnf 9701 | Extend class notation with the Cantor normal form function. |
| class CNF | ||
| Definition | df-cnf 9702* | Define the Cantor normal form function, which takes as input a finitely supported function from 𝑦 to 𝑥 and outputs the corresponding member of the ordinal exponential 𝑥 ↑o 𝑦. The content of the original Cantor Normal Form theorem is that for 𝑥 = ω this function is a bijection onto ω ↑o 𝑦 for any ordinal 𝑦 (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to On). More can be said about the function, however, and in particular it is an order isomorphism for a certain easily defined well-ordering of the finitely supported functions, which gives an alternate definition cantnffval2 9735 of this function in terms of df-oi 9550. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
| ⊢ CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥 ↑m 𝑦) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ))) | ||
| Theorem | cantnffval 9703* | The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
| ⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) ⇒ ⊢ (𝜑 → (𝐴 CNF 𝐵) = (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ))) | ||
| Theorem | cantnfdm 9704* | The domain of the Cantor normal form function (in later lemmas we will use dom (𝐴 CNF 𝐵) to abbreviate "the set of finitely supported functions from 𝐵 to 𝐴"). (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
| ⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) ⇒ ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆) | ||
| Theorem | cantnfvalf 9705* | Lemma for cantnf 9733. The function appearing in cantnfval 9708 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.) |
| ⊢ 𝐹 = seqω((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)), ∅) ⇒ ⊢ 𝐹:ω⟶On | ||
| Theorem | cantnfs 9706 | Elementhood in the set of finitely supported functions from 𝐵 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) ⇒ ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))) | ||
| Theorem | cantnfcl 9707 | Basic properties of the order isomorphism 𝐺 used later. The support of an 𝐹 ∈ 𝑆 is a finite subset of 𝐴, so it is well-ordered by E and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) ⇒ ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω)) | ||
| Theorem | cantnfval 9708* | The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺)) | ||
| Theorem | cantnfval2 9709* | Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)‘dom 𝐺)) | ||
| Theorem | cantnfsuc 9710* | The value of the recursive function 𝐻 at a successor. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅) ⇒ ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾))) | ||
| Theorem | cantnfle 9711* | A lower bound on the CNF function. Since ((𝐴 CNF 𝐵)‘𝐹) is defined as the sum of (𝐴 ↑o 𝑥) ·o (𝐹‘𝑥) over all 𝑥 in the support of 𝐹, it is larger than any of these terms (and all other terms are zero, so we can extend the statement to all 𝐶 ∈ 𝐵 instead of just those 𝐶 in the support). (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 28-Jun-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐴 ↑o 𝐶) ·o (𝐹‘𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹)) | ||
| Theorem | cantnflt 9712* | An upper bound on the partial sums of the CNF function. Since each term dominates all previous terms, by induction we can bound the whole sum with any exponent 𝐴 ↑o 𝐶 where 𝐶 is larger than any exponent (𝐺‘𝑥), 𝑥 ∈ 𝐾 which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅) & ⊢ (𝜑 → ∅ ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ suc dom 𝐺) & ⊢ (𝜑 → 𝐶 ∈ On) & ⊢ (𝜑 → (𝐺 “ 𝐾) ⊆ 𝐶) ⇒ ⊢ (𝜑 → (𝐻‘𝐾) ∈ (𝐴 ↑o 𝐶)) | ||
| Theorem | cantnflt2 9713 | An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → ∅ ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ On) & ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐶) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶)) | ||
| Theorem | cantnff 9714 | The CNF function is a function from finitely supported functions from 𝐵 to 𝐴, to the ordinal exponential 𝐴 ↑o 𝐵. (Contributed by Mario Carneiro, 28-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) ⇒ ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵)) | ||
| Theorem | cantnf0 9715 | The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → ∅ ∈ 𝐴) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅) | ||
| Theorem | cantnfrescl 9716* | A function is finitely supported from 𝐵 to 𝐴 iff the extended function is finitely supported from 𝐷 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐷 ∈ On) & ⊢ (𝜑 → 𝐵 ⊆ 𝐷) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅) & ⊢ (𝜑 → ∅ ∈ 𝐴) & ⊢ 𝑇 = dom (𝐴 CNF 𝐷) ⇒ ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇)) | ||
| Theorem | cantnfres 9717* | The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐷 ∈ On) & ⊢ (𝜑 → 𝐵 ⊆ 𝐷) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅) & ⊢ (𝜑 → ∅ ∈ 𝐴) & ⊢ 𝑇 = dom (𝐴 CNF 𝐷) & ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋))) | ||
| Theorem | cantnfp1lem1 9718* | Lemma for cantnfp1 9721. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by AV, 30-Jun-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋) & ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡))) ⇒ ⊢ (𝜑 → 𝐹 ∈ 𝑆) | ||
| Theorem | cantnfp1lem2 9719* | Lemma for cantnfp1 9721. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 30-Jun-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋) & ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡))) & ⊢ (𝜑 → ∅ ∈ 𝑌) & ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅)) ⇒ ⊢ (𝜑 → dom 𝑂 = suc ∪ dom 𝑂) | ||
| Theorem | cantnfp1lem3 9720* | Lemma for cantnfp1 9721. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋) & ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡))) & ⊢ (𝜑 → ∅ ∈ 𝑌) & ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐹‘(𝑂‘𝑘))) +o 𝑧)), ∅) & ⊢ 𝐾 = OrdIso( E , (𝐺 supp ∅)) & ⊢ 𝑀 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐾‘𝑘)) ·o (𝐺‘(𝐾‘𝑘))) +o 𝑧)), ∅) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))) | ||
| Theorem | cantnfp1 9721* | If 𝐹 is created by adding a single term (𝐹‘𝑋) = 𝑌 to 𝐺, where 𝑋 is larger than any element of the support of 𝐺, then 𝐹 is also a finitely supported function and it is assigned the value ((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑧 where 𝑧 is the value of 𝐺. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋) & ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡))) ⇒ ⊢ (𝜑 → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))) | ||
| Theorem | oemapso 9722* | The relation 𝑇 is a strict order on 𝑆 (a corollary of wemapso2 9593). (Contributed by Mario Carneiro, 28-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ (𝜑 → 𝑇 Or 𝑆) | ||
| Theorem | oemapval 9723* | Value of the relation 𝑇. (Contributed by Mario Carneiro, 28-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) | ||
| Theorem | oemapvali 9724* | If 𝐹 < 𝐺, then there is some 𝑧 witnessing this, but we can say more and in fact there is a definable expression 𝑋 that also witnesses 𝐹 < 𝐺. (Contributed by Mario Carneiro, 25-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐹𝑇𝐺) & ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))) | ||
| Theorem | cantnflem1a 9725* | Lemma for cantnf 9733. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐹𝑇𝐺) & ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅)) | ||
| Theorem | cantnflem1b 9726* | Lemma for cantnf 9733. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐹𝑇𝐺) & ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} & ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅)) ⇒ ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂‘𝑢)) | ||
| Theorem | cantnflem1c 9727* | Lemma for cantnf 9733. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) (Proof shortened by AV, 4-Apr-2020.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐹𝑇𝐺) & ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} & ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅)) ⇒ ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅)) | ||
| Theorem | cantnflem1d 9728* | Lemma for cantnf 9733. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐹𝑇𝐺) & ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} & ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐺‘(𝑂‘𝑘))) +o 𝑧)), ∅) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ (𝐻‘suc (◡𝑂‘𝑋))) | ||
| Theorem | cantnflem1 9729* | Lemma for cantnf 9733. This part of the proof is showing uniqueness of the Cantor normal form. We already know that the relation 𝑇 is a strict order, but we haven't shown it is a well-order yet. But being a strict order is enough to show that two distinct 𝐹, 𝐺 are 𝑇 -related as 𝐹 < 𝐺 or 𝐺 < 𝐹, and WLOG assuming that 𝐹 < 𝐺, we show that CNF respects this order and maps these two to different ordinals. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 2-Jul-2019.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐹𝑇𝐺) & ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} & ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐺‘(𝑂‘𝑘))) +o 𝑧)), ∅) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘𝐺)) | ||
| Theorem | cantnflem2 9730* | Lemma for cantnf 9733. (Contributed by Mario Carneiro, 28-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵)) & ⊢ (𝜑 → ∅ ∈ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o))) | ||
| Theorem | cantnflem3 9731* | Lemma for cantnf 9733. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than 𝐶 has a normal form, we can use oeeu 8641 to factor 𝐶 into the form ((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍 where 0 < 𝑌 < 𝐴 and 𝑍 < (𝐴 ↑o 𝑋) (and a fortiori 𝑋 < 𝐵). Then since 𝑍 < (𝐴 ↑o 𝑋) ≤ (𝐴 ↑o 𝑋) ·o 𝑌 ≤ 𝐶, 𝑍 has a normal form, and by appending the term (𝐴 ↑o 𝑋) ·o 𝑌 using cantnfp1 9721 we get a normal form for 𝐶. (Contributed by Mario Carneiro, 28-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵)) & ⊢ (𝜑 → ∅ ∈ 𝐶) & ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)} & ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = 〈𝑎, 𝑏〉 ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶)) & ⊢ 𝑌 = (1st ‘𝑃) & ⊢ 𝑍 = (2nd ‘𝑃) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍) & ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡))) ⇒ ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵)) | ||
| Theorem | cantnflem4 9732* | Lemma for cantnf 9733. Complete the induction step of cantnflem3 9731. (Contributed by Mario Carneiro, 25-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵)) & ⊢ (𝜑 → ∅ ∈ 𝐶) & ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)} & ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = 〈𝑎, 𝑏〉 ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶)) & ⊢ 𝑌 = (1st ‘𝑃) & ⊢ 𝑍 = (2nd ‘𝑃) ⇒ ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵)) | ||
| Theorem | cantnf 9733* | The Cantor Normal Form theorem. The function (𝐴 CNF 𝐵), which maps a finitely supported function from 𝐵 to 𝐴 to the sum ((𝐴 ↑o 𝑓(𝑎1)) ∘ 𝑎1) +o ((𝐴 ↑o 𝑓(𝑎2)) ∘ 𝑎2) +o ... over all indices 𝑎 < 𝐵 such that 𝑓(𝑎) is nonzero, is an order isomorphism from the ordering 𝑇 of finitely supported functions to the set (𝐴 ↑o 𝐵) under the natural order. Setting 𝐴 = ω and letting 𝐵 be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres 9717, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵))) | ||
| Theorem | oemapwe 9734* | The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternate definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵))) | ||
| Theorem | cantnffval2 9735* | An alternate definition of df-cnf 9702 which relies on cantnf 9733. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9704 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ (𝜑 → (𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆)) | ||
| Theorem | cantnff1o 9736 | Simplify the isomorphism of cantnf 9733 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.) |
| ⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) ⇒ ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆–1-1-onto→(𝐴 ↑o 𝐵)) | ||
| Theorem | wemapwe 9737* | Construct lexicographic order on a function space based on a reverse well-ordering of the indices and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by AV, 3-Jul-2019.) |
| ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑆 We 𝐵) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ 𝐹 = OrdIso(𝑅, 𝐴) & ⊢ 𝐺 = OrdIso(𝑆, 𝐵) & ⊢ 𝑍 = (𝐺‘∅) ⇒ ⊢ (𝜑 → 𝑇 We 𝑈) | ||
| Theorem | oef1o 9738* | A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7322.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
| ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶) & ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐴 ∈ (On ∖ 1o)) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐶 ∈ On) & ⊢ (𝜑 → 𝐷 ∈ On) & ⊢ (𝜑 → (𝐹‘∅) = ∅) & ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))) & ⊢ 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)) ⇒ ⊢ (𝜑 → 𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷)) | ||
| Theorem | cnfcomlem 9739* | Lemma for cnfcom 9740. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
| ⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴)) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ (𝜑 → 𝐼 ∈ dom 𝐺) & ⊢ (𝜑 → 𝑂 ∈ (ω ↑o (𝐺‘𝐼))) & ⊢ (𝜑 → (𝑇‘𝐼):(𝐻‘𝐼)–1-1-onto→𝑂) ⇒ ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))) | ||
| Theorem | cnfcom 9740* | Any ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
| ⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴)) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ (𝜑 → 𝐼 ∈ dom 𝐺) ⇒ ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))) | ||
| Theorem | cnfcom2lem 9741* | Lemma for cnfcom2 9742. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
| ⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴)) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ 𝑊 = (𝐺‘∪ dom 𝐺) & ⊢ (𝜑 → ∅ ∈ 𝐵) ⇒ ⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺) | ||
| Theorem | cnfcom2 9742* | Any nonzero ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
| ⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴)) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ 𝑊 = (𝐺‘∪ dom 𝐺) & ⊢ (𝜑 → ∅ ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊))) | ||
| Theorem | cnfcom3lem 9743* | Lemma for cnfcom3 9744. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.) |
| ⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴)) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ 𝑊 = (𝐺‘∪ dom 𝐺) & ⊢ (𝜑 → ω ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1o)) | ||
| Theorem | cnfcom3 9744* | Any infinite ordinal 𝐵 is equinumerous to a power of ω. (We are being careful here to show explicit bijections rather than simple equinumerosity because we want a uniform construction for cnfcom3c 9746.) (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 4-Jul-2019.) |
| ⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴)) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ 𝑊 = (𝐺‘∪ dom 𝐺) & ⊢ (𝜑 → ω ⊆ 𝐵) & ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹‘𝑊) ·o 𝑣) +o 𝑢)) & ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣)) & ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺)) ⇒ ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→(ω ↑o 𝑊)) | ||
| Theorem | cnfcom3clem 9745* | Lemma for cnfcom3c 9746. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.) |
| ⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝑏) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ 𝑊 = (𝐺‘∪ dom 𝐺) & ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹‘𝑊) ·o 𝑣) +o 𝑢)) & ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣)) & ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺)) & ⊢ 𝐿 = (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁) ⇒ ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) | ||
| Theorem | cnfcom3c 9746* | Wrap the construction of cnfcom3 9744 into an existential quantifier. For any ω ⊆ 𝑏, there is a bijection from 𝑏 to some power of ω. Furthermore, this bijection is canonical , which means that we can find a single function 𝑔 which will give such bijections for every 𝑏 less than some arbitrarily large bound 𝐴. (Contributed by Mario Carneiro, 30-May-2015.) |
| ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) | ||
| Syntax | cttrcl 9747 | Declare the syntax for the transitive closure of a class. |
| class t++𝑅 | ||
| Definition | df-ttrcl 9748* | Define the transitive closure of a class. This is the smallest relation containing 𝑅 (or more precisely, the relation (𝑅 ↾ V) induced by 𝑅) and having the transitive property. Definition from [Levy] p. 59, who denotes it as 𝑅∗ and calls it the "ancestral" of 𝑅. (Contributed by Scott Fenton, 17-Oct-2024.) |
| ⊢ t++𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚))} | ||
| Theorem | ttrcleq 9749 | Equality theorem for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.) |
| ⊢ (𝑅 = 𝑆 → t++𝑅 = t++𝑆) | ||
| Theorem | nfttrcld 9750 | Bound variable hypothesis builder for transitive closure. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.) |
| ⊢ (𝜑 → Ⅎ𝑥𝑅) ⇒ ⊢ (𝜑 → Ⅎ𝑥t++𝑅) | ||
| Theorem | nfttrcl 9751 | Bound variable hypothesis builder for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.) |
| ⊢ Ⅎ𝑥𝑅 ⇒ ⊢ Ⅎ𝑥t++𝑅 | ||
| Theorem | relttrcl 9752 | The transitive closure of a class is a relation. (Contributed by Scott Fenton, 17-Oct-2024.) |
| ⊢ Rel t++𝑅 | ||
| Theorem | brttrcl 9753* | Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 18-Aug-2024.) |
| ⊢ (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) | ||
| Theorem | brttrcl2 9754* | Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 24-Aug-2024.) |
| ⊢ (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ ω ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) | ||
| Theorem | ssttrcl 9755 | If 𝑅 is a relation, then it is a subclass of its transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.) |
| ⊢ (Rel 𝑅 → 𝑅 ⊆ t++𝑅) | ||
| Theorem | ttrcltr 9756 | The transitive closure of a class is transitive. (Contributed by Scott Fenton, 17-Oct-2024.) |
| ⊢ (t++𝑅 ∘ t++𝑅) ⊆ t++𝑅 | ||
| Theorem | ttrclresv 9757 | The transitive closure of 𝑅 restricted to V is the same as the transitive closure of 𝑅 itself. (Contributed by Scott Fenton, 20-Oct-2024.) |
| ⊢ t++(𝑅 ↾ V) = t++𝑅 | ||
| Theorem | ttrclco 9758 | Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.) |
| ⊢ (t++𝑅 ∘ 𝑅) ⊆ t++𝑅 | ||
| Theorem | cottrcl 9759 | Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.) |
| ⊢ (𝑅 ∘ t++𝑅) ⊆ t++𝑅 | ||
| Theorem | ttrclss 9760 | If 𝑅 is a subclass of 𝑆 and 𝑆 is transitive, then the transitive closure of 𝑅 is a subclass of 𝑆. (Contributed by Scott Fenton, 20-Oct-2024.) |
| ⊢ ((𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → t++𝑅 ⊆ 𝑆) | ||
| Theorem | dmttrcl 9761 | The domain of a transitive closure is the same as the domain of the original class. (Contributed by Scott Fenton, 26-Oct-2024.) |
| ⊢ dom t++𝑅 = dom 𝑅 | ||
| Theorem | rnttrcl 9762 | The range of a transitive closure is the same as the range of the original class. (Contributed by Scott Fenton, 26-Oct-2024.) |
| ⊢ ran t++𝑅 = ran 𝑅 | ||
| Theorem | ttrclexg 9763 | If 𝑅 is a set, then so is t++𝑅. (Contributed by Scott Fenton, 26-Oct-2024.) |
| ⊢ (𝑅 ∈ 𝑉 → t++𝑅 ∈ V) | ||
| Theorem | dfttrcl2 9764* | When 𝑅 is a set and a relation, then its transitive closure can be defined by an intersection. (Contributed by Scott Fenton, 26-Oct-2024.) |
| ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → t++𝑅 = ∩ {𝑧 ∣ (𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) | ||
| Theorem | ttrclselem1 9765* | Lemma for ttrclse 9767. Show that all finite ordinal function values of 𝐹 are subsets of 𝐴. (Contributed by Scott Fenton, 31-Oct-2024.) |
| ⊢ 𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋)) ⇒ ⊢ (𝑁 ∈ ω → (𝐹‘𝑁) ⊆ 𝐴) | ||
| Theorem | ttrclselem2 9766* | Lemma for ttrclse 9767. Show that a suc 𝑁 element long chain gives membership in the 𝑁-th predecessor class and vice-versa. (Contributed by Scott Fenton, 31-Oct-2024.) |
| ⊢ 𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋)) ⇒ ⊢ ((𝑁 ∈ ω ∧ 𝑅 Se 𝐴 ∧ 𝑋 ∈ 𝐴) → (∃𝑓(𝑓 Fn suc suc 𝑁 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘suc 𝑁) = 𝑋) ∧ ∀𝑎 ∈ suc 𝑁(𝑓‘𝑎)(𝑅 ↾ 𝐴)(𝑓‘suc 𝑎)) ↔ 𝑦 ∈ (𝐹‘𝑁))) | ||
| Theorem | ttrclse 9767 |
If 𝑅 is set-like over 𝐴, then
the transitive closure of the
restriction of 𝑅 to 𝐴 is set-like over 𝐴.
This theorem requires the axioms of infinity and replacement for its proof. (Contributed by Scott Fenton, 31-Oct-2024.) |
| ⊢ (𝑅 Se 𝐴 → t++(𝑅 ↾ 𝐴) Se 𝐴) | ||
| Theorem | trcl 9768* | For any set 𝐴, show the properties of its transitive closure 𝐶. Similar to Theorem 9.1 of [TakeutiZaring] p. 73 except that we show an explicit expression for the transitive closure rather than just its existence. See tz9.1 9769 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω) & ⊢ 𝐶 = ∪ 𝑦 ∈ ω (𝐹‘𝑦) ⇒ ⊢ (𝐴 ⊆ 𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥)) | ||
| Theorem | tz9.1 9769* |
Every set has a transitive closure (the smallest transitive extension).
Theorem 9.1 of [TakeutiZaring] p.
73. See trcl 9768 for an explicit
expression for the transitive closure. Apparently open problems are
whether this theorem can be proved without the Axiom of Infinity; if
not, then whether it implies Infinity; and if not, what is the
"property" that Infinity has that the other axioms don't have
that is
weaker than Infinity itself?
(Added 22-Mar-2011) The following article seems to answer the first question, that it can't be proved without Infinity, in the affirmative: Mancini, Antonella and Zambella, Domenico (2001). "A note on recursive models of set theories." Notre Dame Journal of Formal Logic, 42(2):109-115. (Thanks to Scott Fenton.) (Contributed by NM, 15-Sep-2003.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) | ||
| Theorem | tz9.1c 9770* | Alternate expression for the existence of transitive closures tz9.1 9769: the intersection of all transitive sets containing 𝐴 is a set. (Contributed by Mario Carneiro, 22-Mar-2013.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V | ||
| Theorem | epfrs 9771* | The strong form of the Axiom of Regularity (no sethood requirement on 𝐴), with the axiom itself present as an antecedent. See also zfregs 9772. (Contributed by Mario Carneiro, 22-Mar-2013.) |
| ⊢ (( E Fr 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅) | ||
| Theorem | zfregs 9772* | The strong form of the Axiom of Regularity, which does not require that 𝐴 be a set. Axiom 6' of [TakeutiZaring] p. 21. See also epfrs 9771. (Contributed by NM, 17-Sep-2003.) |
| ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅) | ||
| Theorem | zfregs2 9773* | Alternate strong form of the Axiom of Regularity. Not every element of a nonempty class contains some element of that class. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) |
| ⊢ (𝐴 ≠ ∅ → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) | ||
| Theorem | setind 9774* | Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
| ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V) | ||
| Theorem | setind2 9775 | Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.) |
| ⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V) | ||
| Syntax | ctc 9776 | Extend class notation to include the transitive closure function. |
| class TC | ||
| Definition | df-tc 9777* | The transitive closure function. (Contributed by Mario Carneiro, 23-Jun-2013.) |
| ⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)}) | ||
| Theorem | tcvalg 9778* | Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9678; see tz9.1 9769.) (Contributed by Mario Carneiro, 23-Jun-2013.) |
| ⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}) | ||
| Theorem | tcid 9779 | Defining property of the transitive closure function: it contains its argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013.) |
| ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (TC‘𝐴)) | ||
| Theorem | tctr 9780 | Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.) |
| ⊢ Tr (TC‘𝐴) | ||
| Theorem | tcmin 9781 | Defining property of the transitive closure function: it is a subset of any transitive class containing 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.) |
| ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵)) | ||
| Theorem | tc2 9782* | A variant of the definition of the transitive closure function, using instead the smallest transitive set containing 𝐴 as a member, gives almost the same set, except that 𝐴 itself must be added because it is not usually a member of (TC‘𝐴) (and it is never a member if 𝐴 is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ((TC‘𝐴) ∪ {𝐴}) = ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} | ||
| Theorem | tcsni 9783 | The transitive closure of a singleton. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (TC‘{𝐴}) = ((TC‘𝐴) ∪ {𝐴}) | ||
| Theorem | tcss 9784 | The transitive closure function inherits the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ⊆ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴)) | ||
| Theorem | tcel 9785 | The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴)) | ||
| Theorem | tcidm 9786 | The transitive closure function is idempotent. (Contributed by Mario Carneiro, 23-Jun-2013.) |
| ⊢ (TC‘(TC‘𝐴)) = (TC‘𝐴) | ||
| Theorem | tc0 9787 | The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015.) |
| ⊢ (TC‘∅) = ∅ | ||
| Theorem | tc00 9788 | The transitive closure is empty iff its argument is. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.) |
| ⊢ (𝐴 ∈ 𝑉 → ((TC‘𝐴) = ∅ ↔ 𝐴 = ∅)) | ||
| Theorem | frmin 9789* | Every (possibly proper) subclass of a class 𝐴 with a well-founded set-like relation 𝑅 has a minimal element. This is a very strong generalization of tz6.26 6368 and tz7.5 6405. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 27-Nov-2024.) |
| ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) | ||
| Theorem | frind 9790* | A subclass of a well-founded class 𝐴 with the property that whenever it contains all predecessors of an element of 𝐴 it also contains that element, is equal to 𝐴. Compare wfi 6371 and tfi 7874, which are special cases of this theorem that do not require the axiom of infinity. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
| ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵) | ||
| Theorem | frinsg 9791* | Well-Founded Induction Schema. If a property passes from all elements less than 𝑦 of a well-founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. Theorem 5.6(ii) of [Levy] p. 64. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
| ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑)) ⇒ ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | ||
| Theorem | frins 9792* | Well-Founded Induction Schema. If a property passes from all elements less than 𝑦 of a well-founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
| ⊢ 𝑅 Fr 𝐴 & ⊢ 𝑅 Se 𝐴 & ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑)) ⇒ ⊢ (𝑦 ∈ 𝐴 → 𝜑) | ||
| Theorem | frins2f 9793* | Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.) |
| ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) & ⊢ Ⅎ𝑦𝜓 & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | ||
| Theorem | frins2 9794* | Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 8-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
| ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | ||
| Theorem | frins3 9795* | Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
| ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) ⇒ ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝐵 ∈ 𝐴) → 𝜒) | ||
| Theorem | frr3g 9796* | Functions defined by well-founded recursion are identical up to relation, domain, and characteristic function. General version of frr3 9801. (Contributed by Scott Fenton, 10-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
| ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺) | ||
| Theorem | frrlem15 9797* | Lemma for general well-founded recursion. Two acceptable functions are compatible. (Contributed by Scott Fenton, 11-Sep-2023.) |
| ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} & ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) | ||
| Theorem | frrlem16 9798* | Lemma for general well-founded recursion. Establish a subset relation. (Contributed by Scott Fenton, 11-Sep-2023.) Revised notion of transitive closure. (Revised by Scott Fenton, 1-Dec-2024.) |
| ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ Pred (t++(𝑅 ↾ 𝐴), 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧)) | ||
| Theorem | frr1 9799 | Law of general well-founded recursion, part one. This theorem and the following two drop the partial order requirement from fpr1 8328, fpr2 8329, and fpr3 8330, which requires using the axiom of infinity (Contributed by Scott Fenton, 11-Sep-2023.) |
| ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn 𝐴) | ||
| Theorem | frr2 9800 | Law of general well-founded recursion, part two. Now we establish the value of 𝐹 within 𝐴. (Contributed by Scott Fenton, 11-Sep-2023.) |
| ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) | ||
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