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Theorem precsex 28218

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 5103	. . . 4 ⊢ (𝑧 = 𝑥_𝑂 → ( 0_s <s 𝑧 ↔ 0_s <s 𝑥_𝑂))
2		oveq1 7367	. . . . . 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 3161	. . . 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 5103	. . . 4 ⊢ (𝑧 = 𝐴 → ( 0_s <s 𝑧 ↔ 0_s <s 𝐴))
7		oveq1 7367	. . . . . 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 3161	. . . 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 28206	. . . . . . . 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 28216	. . . . . . 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 27783	. . . . . 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 28217	. . . . . 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 7368	. . . . . . . 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 3577	. . . . . 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 585	. . . . 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 27945	. 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 1542 ∈ wcel 2114 {cab 2715 ∀wral 3052 ∃wrex 3061 {crab 3400 Vcvv 3441 ⦋csb 3850 ∪ cun 3900 ∅c0 4286 {csn 4581 ⟨cop 4587 ∪ cuni 4864 class class class wbr 5099 ↦ cmpt 5180 “ cima 5628 ∘ ccom 5629 ‘cfv 6493 (class class class)co 7360 ωcom 7810 1^st c1st 7933 2^nd c2nd 7934 reccrdg 8342 No csur 27611 <s clts 27612 |s ccuts 27759 0_s c0s 27805 1_s c1s 27806 L cleft 27825 R cright 27826 +_s cadds 27959 -_s csubs 28020 ·_s cmuls 28106 /_su cdivs 28187

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-dc 10360

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-nadd 8596 df-no 27614 df-lts 27615 df-bday 27616 df-les 27717 df-slts 27758 df-cuts 27760 df-0s 27807 df-1s 27808 df-made 27827 df-old 27828 df-left 27830 df-right 27831 df-norec 27938 df-norec2 27949 df-adds 27960 df-negs 28021 df-subs 28022 df-muls 28107 df-divs 28188

This theorem is referenced by: recsex 28219

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