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Theorem precsex 28243

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 5146	. . . 4 ⊢ (𝑧 = 𝑥_𝑂 → ( 0_s <s 𝑧 ↔ 0_s <s 𝑥_𝑂))
2		oveq1 7439	. . . . . 6 ⊢ (𝑧 = 𝑥_𝑂 → (𝑧 ·_s 𝑦) = (𝑥_𝑂 ·_s 𝑦))
3	2	eqeq1d 2738	. . . . 5 ⊢ (𝑧 = 𝑥_𝑂 → ((𝑧 ·_s 𝑦) = 1_s ↔ (𝑥_𝑂 ·_s 𝑦) = 1_s ))
4	3	rexbidv 3178	. . . 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 5146	. . . 4 ⊢ (𝑧 = 𝐴 → ( 0_s <s 𝑧 ↔ 0_s <s 𝐴))
7		oveq1 7439	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ·_s 𝑦) = (𝐴 ·_s 𝑦))
8	7	eqeq1d 2738	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑧 ·_s 𝑦) = 1_s ↔ (𝐴 ·_s 𝑦) = 1_s ))
9	8	rexbidv 3178	. . . 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 2736	. . . . . . . . 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 28231	. . . . . . . 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 2736	. . . . . . . 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 2736	. . . . . . . 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 1136	. . . . . . . 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 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 )) → 0_s <s 𝑧)
17		simp3 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 )) → ∀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 28241	. . . . . . 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 27849	. . . . . 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 2736	. . . . . . 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 28242	. . . . . 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 7440	. . . . . . . 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 2738	. . . . . . 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 3621	. . . . . 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 1119	. . . 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 27979	. 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 1086 = wceq 1539 ∈ wcel 2107 {cab 2713 ∀wral 3060 ∃wrex 3069 {crab 3435 Vcvv 3479 ⦋csb 3898 ∪ cun 3948 ∅c0 4332 {csn 4625 ⟨cop 4631 ∪ cuni 4906 class class class wbr 5142 ↦ cmpt 5224 “ cima 5687 ∘ ccom 5688 ‘cfv 6560 (class class class)co 7432 ωcom 7888 1^st c1st 8013 2^nd c2nd 8014 reccrdg 8450 No csur 27685 <s cslt 27686 |s cscut 27828 0_s c0s 27868 1_s c1s 27869 L cleft 27885 R cright 27886 +_s cadds 27993 -_s csubs 28053 ·_s cmuls 28133 /_su cdivs 28214

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-dc 10487

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-nadd 8705 df-no 27688 df-slt 27689 df-bday 27690 df-sle 27791 df-sslt 27827 df-scut 27829 df-0s 27870 df-1s 27871 df-made 27887 df-old 27888 df-left 27890 df-right 27891 df-norec 27972 df-norec2 27983 df-adds 27994 df-negs 28054 df-subs 28055 df-muls 28134 df-divs 28215

This theorem is referenced by: recsex 28244

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