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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 ∧ 0s <s 𝐴) → ∃𝑦 No (𝐴 ·s 𝑦) = 1s )
Distinct variable group:   𝑦,𝐴

Proof of Theorem precsex
Dummy variables 𝑎 𝑏 𝑥 𝑥𝑂 𝑥𝐿 𝑥𝑅 𝑦𝐿 𝑦𝑅 𝑧 𝑧𝐿 𝑧𝑅 𝑙 𝑚 𝑝 𝑞 𝑟 𝑠 𝑡 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 5147 . . . 4 (𝑧 = 𝑥𝑂 → ( 0s <s 𝑧 ↔ 0s <s 𝑥𝑂))
2 oveq1 7438 . . . . . 6 (𝑧 = 𝑥𝑂 → (𝑧 ·s 𝑦) = (𝑥𝑂 ·s 𝑦))
32eqeq1d 2739 . . . . 5 (𝑧 = 𝑥𝑂 → ((𝑧 ·s 𝑦) = 1s ↔ (𝑥𝑂 ·s 𝑦) = 1s ))
43rexbidv 3179 . . . 4 (𝑧 = 𝑥𝑂 → (∃𝑦 No (𝑧 ·s 𝑦) = 1s ↔ ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s ))
51, 4imbi12d 344 . . 3 (𝑧 = 𝑥𝑂 → (( 0s <s 𝑧 → ∃𝑦 No (𝑧 ·s 𝑦) = 1s ) ↔ ( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s )))
6 breq2 5147 . . . 4 (𝑧 = 𝐴 → ( 0s <s 𝑧 ↔ 0s <s 𝐴))
7 oveq1 7438 . . . . . 6 (𝑧 = 𝐴 → (𝑧 ·s 𝑦) = (𝐴 ·s 𝑦))
87eqeq1d 2739 . . . . 5 (𝑧 = 𝐴 → ((𝑧 ·s 𝑦) = 1s ↔ (𝐴 ·s 𝑦) = 1s ))
98rexbidv 3179 . . . 4 (𝑧 = 𝐴 → (∃𝑦 No (𝑧 ·s 𝑦) = 1s ↔ ∃𝑦 No (𝐴 ·s 𝑦) = 1s ))
106, 9imbi12d 344 . . 3 (𝑧 = 𝐴 → (( 0s <s 𝑧 → ∃𝑦 No (𝑧 ·s 𝑦) = 1s ) ↔ ( 0s <s 𝐴 → ∃𝑦 No (𝐴 ·s 𝑦) = 1s )))
11 eqid 2737 . . . . . . . . 9 rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩) = rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)
1211precsexlemcbv 28230 . . . . . . . 8 rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩) = rec((𝑝 ∈ V ↦ (1st𝑝) / 𝑙(2nd𝑝) / 𝑟⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑧)∃𝑦𝐿𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝑧) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝑧) ∣ 0s <s 𝑥}∃𝑦𝑅𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝑧) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝑧) ∣ 0s <s 𝑥}∃𝑦𝐿𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝑧) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑧)∃𝑦𝑅𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝑧) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)
13 eqid 2737 . . . . . . . 8 (1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) = (1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩))
14 eqid 2737 . . . . . . . 8 (2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) = (2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩))
15 simp1 1137 . . . . . . . 8 ((𝑧 No ∧ 0s <s 𝑧 ∧ ∀𝑥𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s )) → 𝑧 No )
16 simp2 1138 . . . . . . . 8 ((𝑧 No ∧ 0s <s 𝑧 ∧ ∀𝑥𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s )) → 0s <s 𝑧)
17 simp3 1139 . . . . . . . 8 ((𝑧 No ∧ 0s <s 𝑧 ∧ ∀𝑥𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s )) → ∀𝑥𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s ))
1812, 13, 14, 15, 16, 17precsexlem10 28240 . . . . . . 7 ((𝑧 No ∧ 0s <s 𝑧 ∧ ∀𝑥𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s )) → ((1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω) <<s ((2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω))
1918scutcld 27848 . . . . . 6 ((𝑧 No ∧ 0s <s 𝑧 ∧ ∀𝑥𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s )) → ( ((1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω) |s ((2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω)) ∈ No )
20 eqid 2737 . . . . . . 7 ( ((1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω) |s ((2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω)) = ( ((1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω) |s ((2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω))
2112, 13, 14, 15, 16, 17, 20precsexlem11 28241 . . . . . 6 ((𝑧 No ∧ 0s <s 𝑧 ∧ ∀𝑥𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s )) → (𝑧 ·s ( ((1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω) |s ((2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω))) = 1s )
22 oveq2 7439 . . . . . . . 8 (𝑦 = ( ((1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω) |s ((2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω)) → (𝑧 ·s 𝑦) = (𝑧 ·s ( ((1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω) |s ((2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω))))
2322eqeq1d 2739 . . . . . . 7 (𝑦 = ( ((1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω) |s ((2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω)) → ((𝑧 ·s 𝑦) = 1s ↔ (𝑧 ·s ( ((1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω) |s ((2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω))) = 1s ))
2423rspcev 3622 . . . . . 6 ((( ((1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω) |s ((2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω)) ∈ No ∧ (𝑧 ·s ( ((1st ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω) |s ((2nd ∘ rec((𝑞 ∈ V ↦ (1st𝑞) / 𝑚(2nd𝑞) / 𝑠⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑢 ∈ ( L ‘𝑧) ∣ 0s <s 𝑢}∃𝑤𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝑧) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑡𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝑧) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)) “ ω))) = 1s ) → ∃𝑦 No (𝑧 ·s 𝑦) = 1s )
2519, 21, 24syl2anc 584 . . . . 5 ((𝑧 No ∧ 0s <s 𝑧 ∧ ∀𝑥𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s )) → ∃𝑦 No (𝑧 ·s 𝑦) = 1s )
26253exp 1120 . . . 4 (𝑧 No → ( 0s <s 𝑧 → (∀𝑥𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s ) → ∃𝑦 No (𝑧 ·s 𝑦) = 1s )))
2726com23 86 . . 3 (𝑧 No → (∀𝑥𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s ) → ( 0s <s 𝑧 → ∃𝑦 No (𝑧 ·s 𝑦) = 1s )))
285, 10, 27noinds 27978 . 2 (𝐴 No → ( 0s <s 𝐴 → ∃𝑦 No (𝐴 ·s 𝑦) = 1s ))
2928imp 406 1 ((𝐴 No ∧ 0s <s 𝐴) → ∃𝑦 No (𝐴 ·s 𝑦) = 1s )
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  1st c1st 8012  2nd c2nd 8013  reccrdg 8449   No csur 27684   <s cslt 27685   |s cscut 27827   0s c0s 27867   1s 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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