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Theorem precsex 28631

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 5110	. . . 4 ⊢ (𝑧 = 𝑥_𝑂 → ( 0_s <s 𝑧 ↔ 0_s <s 𝑥_𝑂))
2		oveq1 7447	. . . . . 6 ⊢ (𝑧 = 𝑥_𝑂 → (𝑧 ·_s 𝑦) = (𝑥_𝑂 ·_s 𝑦))
3	2	eqeq1d 2765	. . . . 5 ⊢ (𝑧 = 𝑥_𝑂 → ((𝑧 ·_s 𝑦) = 1_s ↔ (𝑥_𝑂 ·_s 𝑦) = 1_s ))
4	3	rexbidv 3190	. . . 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 5110	. . . 4 ⊢ (𝑧 = 𝐴 → ( 0_s <s 𝑧 ↔ 0_s <s 𝐴))
7		oveq1 7447	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ·_s 𝑦) = (𝐴 ·_s 𝑦))
8	7	eqeq1d 2765	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑧 ·_s 𝑦) = 1_s ↔ (𝐴 ·_s 𝑦) = 1_s ))
9	8	rexbidv 3190	. . . 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 2763	. . . . . . . . 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 28619	. . . . . . . 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 2763	. . . . . . . 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 2763	. . . . . . . 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 28629	. . . . . . 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 28196	. . . . . 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 2763	. . . . . . 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 28630	. . . . . 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 7448	. . . . . . . 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 2765	. . . . . . 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 3580	. . . . . 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 28358	. 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 1570 ∈ wcel 2147 {cab 2741 ∀wral 3080 ∃wrex 3090 {crab 3416 Vcvv 3454 ⦋csb 3851 ∪ cun 4152 ∅c0 4281 {csn 4588 ⟨cop 4838 ∪ cuni 4870 class class class wbr 5106 ↦ cmpt 5189 “ cima 5658 ∘ ccom 5659 ‘cfv 6555 (class class class)co 7440 ωcom 7898 1^st c1st 8020 2^nd c2nd 8021 reccrdg 8442 Nocsur 28024 <s clts 28025 |s ccuts 28172 0_s c0s 28218 1_s c1s 28219 Lcleft 28238 Rcright 28239 +_s cadds 28372 −_s csubs 28433 ·_s cmuls 28519 /_su cdivs 28600

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-dc 10548

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-ot 4862 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-oadd 8505 df-nadd 8700 df-no 28027 df-lts 28028 df-bday 28029 df-les 28129 df-slts 28171 df-cuts 28173 df-0s 28220 df-1s 28221 df-made 28240 df-old 28241 df-left 28243 df-right 28244 df-norec 28351 df-norec2 28362 df-adds 28373 df-negs 28434 df-subs 28435 df-muls 28520 df-divs 28601

This theorem is used by: recsex 28632

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