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Theorem precsex 28242

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 5147	. . . 4 ⊢ (𝑧 = 𝑥_𝑂 → ( 0_s <s 𝑧 ↔ 0_s <s 𝑥_𝑂))
2		oveq1 7438	. . . . . 6 ⊢ (𝑧 = 𝑥_𝑂 → (𝑧 ·_s 𝑦) = (𝑥_𝑂 ·_s 𝑦))
3	2	eqeq1d 2739	. . . . 5 ⊢ (𝑧 = 𝑥_𝑂 → ((𝑧 ·_s 𝑦) = 1_s ↔ (𝑥_𝑂 ·_s 𝑦) = 1_s ))
4	3	rexbidv 3179	. . . 4 ⊢ (𝑧 = 𝑥_𝑂 → (∃𝑦 ∈ No (𝑧 ·_s 𝑦) = 1_s ↔ ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
5	1, 4	imbi12d 344	. . 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 5147	. . . 4 ⊢ (𝑧 = 𝐴 → ( 0_s <s 𝑧 ↔ 0_s <s 𝐴))
7		oveq1 7438	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ·_s 𝑦) = (𝐴 ·_s 𝑦))
8	7	eqeq1d 2739	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑧 ·_s 𝑦) = 1_s ↔ (𝐴 ·_s 𝑦) = 1_s ))
9	8	rexbidv 3179	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ No (𝑧 ·_s 𝑦) = 1_s ↔ ∃𝑦 ∈ No (𝐴 ·_s 𝑦) = 1_s ))
10	6, 9	imbi12d 344	. . 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 2737	. . . . . . . . 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 28230	. . . . . . . 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 2737	. . . . . . . 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 2737	. . . . . . . 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 1137	. . . . . . . 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 1138	. . . . . . . 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 1139	. . . . . . . 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 28240	. . . . . . 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	scutcld 27848	. . . . . 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 2737	. . . . . . 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 28241	. . . . . 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 7439	. . . . . . . 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 2739	. . . . . . 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 3622	. . . . . 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 584	. . . . 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 1120	. . . 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 86	. . 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 27978	. 2 ⊢ (𝐴 ∈ No → ( 0_s <s 𝐴 → ∃𝑦 ∈ No (𝐴 ·_s 𝑦) = 1_s ))
29	28	imp 406	1 ⊢ ((𝐴 ∈ No ∧ 0_s <s 𝐴) → ∃𝑦 ∈ No (𝐴 ·_s 𝑦) = 1_s )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ⦋csb 3899 ∪ cun 3949 ∅c0 4333 {csn 4626 ⟨cop 4632 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 “ cima 5688 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ωcom 7887 1^st c1st 8012 2^nd c2nd 8013 reccrdg 8449 No csur 27684 <s cslt 27685 |s cscut 27827 0_s c0s 27867 1_s c1s 27868 L cleft 27884 R cright 27885 +_s cadds 27992 -_s csubs 28052 ·_s cmuls 28132 /_su cdivs 28213

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-dc 10486

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-nadd 8704 df-no 27687 df-slt 27688 df-bday 27689 df-sle 27790 df-sslt 27826 df-scut 27828 df-0s 27869 df-1s 27870 df-made 27886 df-old 27887 df-left 27889 df-right 27890 df-norec 27971 df-norec2 27982 df-adds 27993 df-negs 28053 df-subs 28054 df-muls 28133 df-divs 28214

This theorem is referenced by: recsex 28243

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