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Theorem precsex 28604

Description: Every positive surreal has a reciprocal. Theorem 10(iv) of [Conway] p. 21. (Contributed by Scott Fenton, 15-Mar-2025.)

**Assertion**
Ref	Expression
precsex	⊢ ((𝐴 ∈ No ∧ 0_s <s 𝐴) → ∃𝑦 ∈ No (𝐴 ·_s 𝑦) = 1_s )

Distinct variable group: 𝑦,𝐴

Proof of Theorem precsex Dummy variables 𝑎 𝑏 𝑥 𝑥_𝑂 𝑥_𝐿 𝑥_𝑅 𝑦_𝐿 𝑦_𝑅 𝑧 𝑧_𝐿 𝑧_𝑅 𝑙 𝑚 𝑝 𝑞 𝑟 𝑠 𝑡 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5106	. . . 4 ⊢ (𝑧 = 𝑥_𝑂 → ( 0_s <s 𝑧 ↔ 0_s <s 𝑥_𝑂))
2		oveq1 7427	. . . . . 6 ⊢ (𝑧 = 𝑥_𝑂 → (𝑧 ·_s 𝑦) = (𝑥_𝑂 ·_s 𝑦))
3	2	eqeq1d 2762	. . . . 5 ⊢ (𝑧 = 𝑥_𝑂 → ((𝑧 ·_s 𝑦) = 1_s ↔ (𝑥_𝑂 ·_s 𝑦) = 1_s ))
4	3	rexbidv 3186	. . . 4 ⊢ (𝑧 = 𝑥_𝑂 → (∃𝑦 ∈ No (𝑧 ·_s 𝑦) = 1_s ↔ ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
5	1, 4	imbi12d 347	. . 3 ⊢ (𝑧 = 𝑥_𝑂 → (( 0_s <s 𝑧 → ∃𝑦 ∈ No (𝑧 ·_s 𝑦) = 1_s ) ↔ ( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s )))
6		breq2 5106	. . . 4 ⊢ (𝑧 = 𝐴 → ( 0_s <s 𝑧 ↔ 0_s <s 𝐴))
7		oveq1 7427	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ·_s 𝑦) = (𝐴 ·_s 𝑦))
8	7	eqeq1d 2762	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑧 ·_s 𝑦) = 1_s ↔ (𝐴 ·_s 𝑦) = 1_s ))
9	8	rexbidv 3186	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ No (𝑧 ·_s 𝑦) = 1_s ↔ ∃𝑦 ∈ No (𝐴 ·_s 𝑦) = 1_s ))
10	6, 9	imbi12d 347	. . 3 ⊢ (𝑧 = 𝐴 → (( 0_s <s 𝑧 → ∃𝑦 ∈ No (𝑧 ·_s 𝑦) = 1_s ) ↔ ( 0_s <s 𝐴 → ∃𝑦 ∈ No (𝐴 ·_s 𝑦) = 1_s )))
11		eqid 2760	. . . . . . . . 9 ⊢ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩) = rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)
12	11	precsexlemcbv 28592	. . . . . . . 8 ⊢ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩) = rec((𝑝 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑙⦌⦋(2^nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ (R‘𝑧)∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝑅 −_s 𝑧) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ (L‘𝑧) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝐿 −_s 𝑧) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ (L‘𝑧) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝐿 −_s 𝑧) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ (R‘𝑧)∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝑅 −_s 𝑧) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)
13		eqid 2760	. . . . . . . 8 ⊢ (1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) = (1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩))
14		eqid 2760	. . . . . . . 8 ⊢ (2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) = (2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩))
15		simp1 1154	. . . . . . . 8 ⊢ ((𝑧 ∈ No ∧ 0_s <s 𝑧 ∧ ∀𝑥_𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s )) → 𝑧 ∈ No)
16		simp2 1155	. . . . . . . 8 ⊢ ((𝑧 ∈ No ∧ 0_s <s 𝑧 ∧ ∀𝑥_𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s )) → 0_s <s 𝑧)
17		simp3 1156	. . . . . . . 8 ⊢ ((𝑧 ∈ No ∧ 0_s <s 𝑧 ∧ ∀𝑥_𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s )) → ∀𝑥_𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
18	12, 13, 14, 15, 16, 17	precsexlem10 28602	. . . . . . 7 ⊢ ((𝑧 ∈ No ∧ 0_s <s 𝑧 ∧ ∀𝑥_𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s )) → ∪ ((1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω) <<s ∪ ((2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω))
19	18	cutscld 28169	. . . . . 6 ⊢ ((𝑧 ∈ No ∧ 0_s <s 𝑧 ∧ ∀𝑥_𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s )) → (∪ ((1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω) \|s ∪ ((2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω)) ∈ No)
20		eqid 2760	. . . . . . 7 ⊢ (∪ ((1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω) \|s ∪ ((2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω)) = (∪ ((1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω) \|s ∪ ((2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω))
21	12, 13, 14, 15, 16, 17, 20	precsexlem11 28603	. . . . . 6 ⊢ ((𝑧 ∈ No ∧ 0_s <s 𝑧 ∧ ∀𝑥_𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s )) → (𝑧 ·_s (∪ ((1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω) \|s ∪ ((2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω))) = 1_s )
22		oveq2 7428	. . . . . . . 8 ⊢ (𝑦 = (∪ ((1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω) \|s ∪ ((2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω)) → (𝑧 ·_s 𝑦) = (𝑧 ·_s (∪ ((1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω) \|s ∪ ((2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω))))
23	22	eqeq1d 2762	. . . . . . 7 ⊢ (𝑦 = (∪ ((1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω) \|s ∪ ((2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω)) → ((𝑧 ·_s 𝑦) = 1_s ↔ (𝑧 ·_s (∪ ((1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω) \|s ∪ ((2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω))) = 1_s ))
24	23	rspcev 3576	. . . . . 6 ⊢ (((∪ ((1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω) \|s ∪ ((2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω)) ∈ No ∧ (𝑧 ·_s (∪ ((1^st ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω) \|s ∪ ((2^nd ∘ rec((𝑞 ∈ V ↦ ⦋(1^st ‘𝑞) / 𝑚⦌⦋(2^nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝑅)} ∪ {𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧_𝐿 ∈ {𝑢 ∈ (L‘𝑧) ∣ 0_s <s 𝑢}∃𝑤 ∈ 𝑚 𝑏 = (( 1_s +_s ((𝑧_𝐿 −_s 𝑧) ·_s 𝑤)) /_su 𝑧_𝐿)} ∪ {𝑏 ∣ ∃𝑧_𝑅 ∈ (R‘𝑧)∃𝑡 ∈ 𝑠 𝑏 = (( 1_s +_s ((𝑧_𝑅 −_s 𝑧) ·_s 𝑡)) /_su 𝑧_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)) “ ω))) = 1_s ) → ∃𝑦 ∈ No (𝑧 ·_s 𝑦) = 1_s )
25	19, 21, 24	syl2anc 596	. . . . 5 ⊢ ((𝑧 ∈ No ∧ 0_s <s 𝑧 ∧ ∀𝑥_𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s )) → ∃𝑦 ∈ No (𝑧 ·_s 𝑦) = 1_s )
26	25	3exp 1137	. . . 4 ⊢ (𝑧 ∈ No → ( 0_s <s 𝑧 → (∀𝑥_𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ) → ∃𝑦 ∈ No (𝑧 ·_s 𝑦) = 1_s )))
27	26	com23 87	. . 3 ⊢ (𝑧 ∈ No → (∀𝑥_𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ) → ( 0_s <s 𝑧 → ∃𝑦 ∈ No (𝑧 ·_s 𝑦) = 1_s )))
28	5, 10, 27	noinds 28331	. 2 ⊢ (𝐴 ∈ No → ( 0_s <s 𝐴 → ∃𝑦 ∈ No (𝐴 ·_s 𝑦) = 1_s ))
29	28	imp 412	1 ⊢ ((𝐴 ∈ No ∧ 0_s <s 𝐴) → ∃𝑦 ∈ No (𝐴 ·_s 𝑦) = 1_s )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 {cab 2738 ∀wral 3076 ∃wrex 3086 {crab 3412 Vcvv 3450 ⦋csb 3846 ∪ cun 4149 ∅c0 4278 {csn 4583 ⟨cop 4589 ∪ cuni 4866 class class class wbr 5102 ↦ cmpt 5185 “ cima 5653 ∘ ccom 5654 ‘cfv 6538 (class class class)co 7420 ωcom 7877 1^st c1st 7999 2^nd c2nd 8000 reccrdg 8417 Nocsur 27997 <s clts 27998 |s ccuts 28145 0_s c0s 28191 1_s c1s 28192 Lcleft 28211 Rcright 28212 +_s cadds 28345 −_s csubs 28406 ·_s cmuls 28492 /_su cdivs 28573

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-dc 10524

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-ot 4592 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-nadd 8675 df-no 28000 df-lts 28001 df-bday 28002 df-les 28102 df-slts 28144 df-cuts 28146 df-0s 28193 df-1s 28194 df-made 28213 df-old 28214 df-left 28216 df-right 28217 df-norec 28324 df-norec2 28335 df-adds 28346 df-negs 28407 df-subs 28408 df-muls 28493 df-divs 28574

This theorem is used by: recsex 28605

W3C validator