MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  precsex Structured version   Visualization version   GIF version

Theorem precsex 28257
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 5152 . . . 4 (𝑧 = 𝑥𝑂 → ( 0s <s 𝑧 ↔ 0s <s 𝑥𝑂))
2 oveq1 7438 . . . . . 6 (𝑧 = 𝑥𝑂 → (𝑧 ·s 𝑦) = (𝑥𝑂 ·s 𝑦))
32eqeq1d 2737 . . . . 5 (𝑧 = 𝑥𝑂 → ((𝑧 ·s 𝑦) = 1s ↔ (𝑥𝑂 ·s 𝑦) = 1s ))
43rexbidv 3177 . . . 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 5152 . . . 4 (𝑧 = 𝐴 → ( 0s <s 𝑧 ↔ 0s <s 𝐴))
7 oveq1 7438 . . . . . 6 (𝑧 = 𝐴 → (𝑧 ·s 𝑦) = (𝐴 ·s 𝑦))
87eqeq1d 2737 . . . . 5 (𝑧 = 𝐴 → ((𝑧 ·s 𝑦) = 1s ↔ (𝐴 ·s 𝑦) = 1s ))
98rexbidv 3177 . . . 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 2735 . . . . . . . . 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 28245 . . . . . . . 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 2735 . . . . . . . 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 2735 . . . . . . . 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 1135 . . . . . . . 8 ((𝑧 No ∧ 0s <s 𝑧 ∧ ∀𝑥𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s )) → 𝑧 No )
16 simp2 1136 . . . . . . . 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 1137 . . . . . . . 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 28255 . . . . . . 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 27863 . . . . . 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 2735 . . . . . . 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 28256 . . . . . 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 2737 . . . . . . 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 1118 . . . 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 27993 . 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 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  csb 3908  cun 3961  c0 4339  {csn 4631  cop 4637   cuni 4912   class class class wbr 5148  cmpt 5231  cima 5692  ccom 5693  cfv 6563  (class class class)co 7431  ωcom 7887  1st c1st 8011  2nd c2nd 8012  reccrdg 8448   No csur 27699   <s cslt 27700   |s cscut 27842   0s c0s 27882   1s c1s 27883   L cleft 27899   R cright 27900   +s cadds 28007   -s csubs 28067   ·s cmuls 28147   /su cdivs 28228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-dc 10484
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148  df-divs 28229
This theorem is referenced by:  recsex  28258
  Copyright terms: Public domain W3C validator