Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iccelpart Structured version   Visualization version   GIF version

Theorem iccelpart 40664
Description: An element of any partitioned half opened interval of extended reals is an element of a part of this partition. (Contributed by AV, 18-Jul-2020.)
Assertion
Ref Expression
iccelpart (𝑀 ∈ ℕ → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
Distinct variable groups:   𝑖,𝑀,𝑝   𝑖,𝑋,𝑝

Proof of Theorem iccelpart
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6148 . . 3 (𝑥 = 1 → (RePart‘𝑥) = (RePart‘1))
2 fveq2 6148 . . . . . 6 (𝑥 = 1 → (𝑝𝑥) = (𝑝‘1))
32oveq2d 6620 . . . . 5 (𝑥 = 1 → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝‘1)))
43eleq2d 2684 . . . 4 (𝑥 = 1 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
5 oveq2 6612 . . . . . 6 (𝑥 = 1 → (0..^𝑥) = (0..^1))
6 fzo01 12491 . . . . . 6 (0..^1) = {0}
75, 6syl6eq 2671 . . . . 5 (𝑥 = 1 → (0..^𝑥) = {0})
87rexeqdv 3134 . . . 4 (𝑥 = 1 → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
94, 8imbi12d 334 . . 3 (𝑥 = 1 → ((𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
101, 9raleqbidv 3141 . 2 (𝑥 = 1 → (∀𝑝 ∈ (RePart‘𝑥)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ ∀𝑝 ∈ (RePart‘1)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
11 fveq2 6148 . . 3 (𝑥 = 𝑦 → (RePart‘𝑥) = (RePart‘𝑦))
12 fveq2 6148 . . . . . 6 (𝑥 = 𝑦 → (𝑝𝑥) = (𝑝𝑦))
1312oveq2d 6620 . . . . 5 (𝑥 = 𝑦 → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝𝑦)))
1413eleq2d 2684 . . . 4 (𝑥 = 𝑦 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))))
15 oveq2 6612 . . . . 5 (𝑥 = 𝑦 → (0..^𝑥) = (0..^𝑦))
1615rexeqdv 3134 . . . 4 (𝑥 = 𝑦 → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
1714, 16imbi12d 334 . . 3 (𝑥 = 𝑦 → ((𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
1811, 17raleqbidv 3141 . 2 (𝑥 = 𝑦 → (∀𝑝 ∈ (RePart‘𝑥)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
19 fveq2 6148 . . 3 (𝑥 = (𝑦 + 1) → (RePart‘𝑥) = (RePart‘(𝑦 + 1)))
20 fveq2 6148 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝑝𝑥) = (𝑝‘(𝑦 + 1)))
2120oveq2d 6620 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))
2221eleq2d 2684 . . . 4 (𝑥 = (𝑦 + 1) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1)))))
23 oveq2 6612 . . . . 5 (𝑥 = (𝑦 + 1) → (0..^𝑥) = (0..^(𝑦 + 1)))
2423rexeqdv 3134 . . . 4 (𝑥 = (𝑦 + 1) → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
2522, 24imbi12d 334 . . 3 (𝑥 = (𝑦 + 1) → ((𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
2619, 25raleqbidv 3141 . 2 (𝑥 = (𝑦 + 1) → (∀𝑝 ∈ (RePart‘𝑥)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ ∀𝑝 ∈ (RePart‘(𝑦 + 1))(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
27 fveq2 6148 . . 3 (𝑥 = 𝑀 → (RePart‘𝑥) = (RePart‘𝑀))
28 fveq2 6148 . . . . . 6 (𝑥 = 𝑀 → (𝑝𝑥) = (𝑝𝑀))
2928oveq2d 6620 . . . . 5 (𝑥 = 𝑀 → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝𝑀)))
3029eleq2d 2684 . . . 4 (𝑥 = 𝑀 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀))))
31 oveq2 6612 . . . . 5 (𝑥 = 𝑀 → (0..^𝑥) = (0..^𝑀))
3231rexeqdv 3134 . . . 4 (𝑥 = 𝑀 → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
3330, 32imbi12d 334 . . 3 (𝑥 = 𝑀 → ((𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
3427, 33raleqbidv 3141 . 2 (𝑥 = 𝑀 → (∀𝑝 ∈ (RePart‘𝑥)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
35 0nn0 11251 . . . . 5 0 ∈ ℕ0
36 fveq2 6148 . . . . . . . 8 (𝑖 = 0 → (𝑝𝑖) = (𝑝‘0))
37 oveq1 6611 . . . . . . . . . 10 (𝑖 = 0 → (𝑖 + 1) = (0 + 1))
38 0p1e1 11076 . . . . . . . . . 10 (0 + 1) = 1
3937, 38syl6eq 2671 . . . . . . . . 9 (𝑖 = 0 → (𝑖 + 1) = 1)
4039fveq2d 6152 . . . . . . . 8 (𝑖 = 0 → (𝑝‘(𝑖 + 1)) = (𝑝‘1))
4136, 40oveq12d 6622 . . . . . . 7 (𝑖 = 0 → ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑝‘0)[,)(𝑝‘1)))
4241eleq2d 2684 . . . . . 6 (𝑖 = 0 → (𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
4342rexsng 4190 . . . . 5 (0 ∈ ℕ0 → (∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
4435, 43ax-mp 5 . . . 4 (∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)))
4544biimpri 218 . . 3 (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
4645rgenw 2919 . 2 𝑝 ∈ (RePart‘1)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
47 nfv 1840 . . . . 5 𝑝 𝑦 ∈ ℕ
48 nfra1 2936 . . . . 5 𝑝𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
4947, 48nfan 1825 . . . 4 𝑝(𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
50 nnnn0 11243 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
51 fzonn0p1 12485 . . . . . . . . . 10 (𝑦 ∈ ℕ0𝑦 ∈ (0..^(𝑦 + 1)))
5250, 51syl 17 . . . . . . . . 9 (𝑦 ∈ ℕ → 𝑦 ∈ (0..^(𝑦 + 1)))
5352ad2antrr 761 . . . . . . . 8 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → 𝑦 ∈ (0..^(𝑦 + 1)))
54 fveq2 6148 . . . . . . . . . . 11 (𝑖 = 𝑦 → (𝑝𝑖) = (𝑝𝑦))
55 oveq1 6611 . . . . . . . . . . . 12 (𝑖 = 𝑦 → (𝑖 + 1) = (𝑦 + 1))
5655fveq2d 6152 . . . . . . . . . . 11 (𝑖 = 𝑦 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑦 + 1)))
5754, 56oveq12d 6622 . . . . . . . . . 10 (𝑖 = 𝑦 → ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1))))
5857eleq2d 2684 . . . . . . . . 9 (𝑖 = 𝑦 → (𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1)))))
5958adantl 482 . . . . . . . 8 ((((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) ∧ 𝑖 = 𝑦) → (𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1)))))
60 peano2nn 10976 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
6160adantr 481 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑦 + 1) ∈ ℕ)
62 simpr 477 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 𝑝 ∈ (RePart‘(𝑦 + 1)))
6360nnnn0d 11295 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ0)
64 0elfz 12377 . . . . . . . . . . . . . . . . 17 ((𝑦 + 1) ∈ ℕ0 → 0 ∈ (0...(𝑦 + 1)))
6563, 64syl 17 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ → 0 ∈ (0...(𝑦 + 1)))
6665adantr 481 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 0 ∈ (0...(𝑦 + 1)))
6761, 62, 66iccpartxr 40650 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘0) ∈ ℝ*)
68 nn0fz0 12378 . . . . . . . . . . . . . . . . 17 ((𝑦 + 1) ∈ ℕ0 ↔ (𝑦 + 1) ∈ (0...(𝑦 + 1)))
6963, 68sylib 208 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ (0...(𝑦 + 1)))
7069adantr 481 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑦 + 1) ∈ (0...(𝑦 + 1)))
7161, 62, 70iccpartxr 40650 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘(𝑦 + 1)) ∈ ℝ*)
7267, 71jca 554 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
7372adantlr 750 . . . . . . . . . . . 12 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
74 elico1 12160 . . . . . . . . . . . 12 (((𝑝‘0) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
7573, 74syl 17 . . . . . . . . . . 11 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
76 simp1 1059 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → 𝑋 ∈ ℝ*)
7776adantl 482 . . . . . . . . . . . . . . 15 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 ∈ ℝ*)
78 simpl 473 . . . . . . . . . . . . . . 15 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑝𝑦) ≤ 𝑋)
79 simpr3 1067 . . . . . . . . . . . . . . 15 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 < (𝑝‘(𝑦 + 1)))
8077, 78, 793jca 1240 . . . . . . . . . . . . . 14 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))))
8180ex 450 . . . . . . . . . . . . 13 ((𝑝𝑦) ≤ 𝑋 → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
8281adantl 482 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
8382adantr 481 . . . . . . . . . . 11 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
8475, 83sylbid 230 . . . . . . . . . 10 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
8584impr 648 . . . . . . . . 9 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))))
86 nn0fz0 12378 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0𝑦 ∈ (0...𝑦))
8750, 86sylib 208 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ → 𝑦 ∈ (0...𝑦))
88 fzelp1 12335 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...𝑦) → 𝑦 ∈ (0...(𝑦 + 1)))
8987, 88syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ ℕ → 𝑦 ∈ (0...(𝑦 + 1)))
9089adantr 481 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 𝑦 ∈ (0...(𝑦 + 1)))
9161, 62, 90iccpartxr 40650 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝𝑦) ∈ ℝ*)
9291, 71jca 554 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝𝑦) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
9392ad2ant2r 782 . . . . . . . . . 10 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → ((𝑝𝑦) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
94 elico1 12160 . . . . . . . . . 10 (((𝑝𝑦) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*) → (𝑋 ∈ ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
9593, 94syl 17 . . . . . . . . 9 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → (𝑋 ∈ ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
9685, 95mpbird 247 . . . . . . . 8 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → 𝑋 ∈ ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1))))
9753, 59, 96rspcedvd 3302 . . . . . . 7 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
9897exp43 639 . . . . . 6 (𝑦 ∈ ℕ → ((𝑝𝑦) ≤ 𝑋 → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))))
9998adantr 481 . . . . 5 ((𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))) → ((𝑝𝑦) ≤ 𝑋 → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))))
100 iccpartres 40649 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦))
101 rspsbca 3500 . . . . . . . . . . . 12 (((𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦) ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))) → [(𝑝 ↾ (0...𝑦)) / 𝑝](𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
102 vex 3189 . . . . . . . . . . . . . . 15 𝑝 ∈ V
103102resex 5402 . . . . . . . . . . . . . 14 (𝑝 ↾ (0...𝑦)) ∈ V
104 sbcimg 3459 . . . . . . . . . . . . . . 15 ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝](𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → [(𝑝 ↾ (0...𝑦)) / 𝑝]𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
105 sbcel2 3961 . . . . . . . . . . . . . . . . 17 ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)))
106 csbov12g 6642 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)) = ((𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘0)[,)(𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑦)))
107 csbfv12 6188 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘0) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝0)
108 csbvarg 3975 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝𝑝 = (𝑝 ↾ (0...𝑦)))
109 csbconstg 3527 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝0 = 0)
110108, 109fveq12d 6154 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝0) = ((𝑝 ↾ (0...𝑦))‘0))
111107, 110syl5eq 2667 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘0) = ((𝑝 ↾ (0...𝑦))‘0))
112 csbfv12 6188 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑦) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑦)
113 csbconstg 3527 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝𝑦 = 𝑦)
114108, 113fveq12d 6154 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
115112, 114syl5eq 2667 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
116111, 115oveq12d 6622 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘0)[,)(𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
117106, 116eqtrd 2655 . . . . . . . . . . . . . . . . . 18 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
118117eleq2d 2684 . . . . . . . . . . . . . . . . 17 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
119105, 118syl5bb 272 . . . . . . . . . . . . . . . 16 ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
120 sbcrex 3496 . . . . . . . . . . . . . . . . 17 ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑦)[(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
121 sbcel2 3961 . . . . . . . . . . . . . . . . . . 19 ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
122 csbov12g 6642 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑖)[,)(𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘(𝑖 + 1))))
123 csbfv12 6188 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑖) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑖)
124 csbconstg 3527 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝𝑖 = 𝑖)
125108, 124fveq12d 6154 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑖) = ((𝑝 ↾ (0...𝑦))‘𝑖))
126123, 125syl5eq 2667 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑖) = ((𝑝 ↾ (0...𝑦))‘𝑖))
127 csbfv12 6188 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝(𝑖 + 1))
128 csbconstg 3527 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑖 + 1) = (𝑖 + 1))
129108, 128fveq12d 6154 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))
130127, 129syl5eq 2667 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))
131126, 130oveq12d 6622 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑖)[,)(𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘(𝑖 + 1))) = (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))))
132122, 131eqtrd 2655 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))))
133132eleq2d 2684 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))))
134121, 133syl5bb 272 . . . . . . . . . . . . . . . . . 18 ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))))
135134rexbidv 3045 . . . . . . . . . . . . . . . . 17 ((𝑝 ↾ (0...𝑦)) ∈ V → (∃𝑖 ∈ (0..^𝑦)[(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))))
136120, 135syl5bb 272 . . . . . . . . . . . . . . . 16 ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))))
137119, 136imbi12d 334 . . . . . . . . . . . . . . 15 ((𝑝 ↾ (0...𝑦)) ∈ V → (([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → [(𝑝 ↾ (0...𝑦)) / 𝑝]𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))))))
138104, 137bitrd 268 . . . . . . . . . . . . . 14 ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝](𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))))))
139103, 138ax-mp 5 . . . . . . . . . . . . 13 ([(𝑝 ↾ (0...𝑦)) / 𝑝](𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))))
14072, 74syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
141140adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
14276adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 ∈ ℝ*)
143 simpr2 1066 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑝‘0) ≤ 𝑋)
144 xrltnle 10049 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑋 ∈ ℝ* ∧ (𝑝𝑦) ∈ ℝ*) → (𝑋 < (𝑝𝑦) ↔ ¬ (𝑝𝑦) ≤ 𝑋))
14576, 91, 144syl2anr 495 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 < (𝑝𝑦) ↔ ¬ (𝑝𝑦) ≤ 𝑋))
146145exbiri 651 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → (¬ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝𝑦))))
147146com23 86 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (¬ (𝑝𝑦) ≤ 𝑋 → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → 𝑋 < (𝑝𝑦))))
148147imp31 448 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 < (𝑝𝑦))
149142, 143, 1483jca 1240 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝𝑦)))
15067, 91jca 554 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝𝑦) ∈ ℝ*))
151150ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝𝑦) ∈ ℝ*))
152 elico1 12160 . . . . . . . . . . . . . . . . . . . . 21 (((𝑝‘0) ∈ ℝ* ∧ (𝑝𝑦) ∈ ℝ*) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝𝑦))))
153151, 152syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝𝑦))))
154149, 153mpbird 247 . . . . . . . . . . . . . . . . . . 19 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)))
155154ex 450 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))))
156141, 155sylbid 230 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))))
157 0elfz 12377 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ ℕ0 → 0 ∈ (0...𝑦))
15850, 157syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ℕ → 0 ∈ (0...𝑦))
159158adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 0 ∈ (0...𝑦))
160 fvres 6164 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ (0...𝑦) → ((𝑝 ↾ (0...𝑦))‘0) = (𝑝‘0))
161159, 160syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝 ↾ (0...𝑦))‘0) = (𝑝‘0))
162161eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘0) = ((𝑝 ↾ (0...𝑦))‘0))
16387adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 𝑦 ∈ (0...𝑦))
164 fvres 6164 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (0...𝑦) → ((𝑝 ↾ (0...𝑦))‘𝑦) = (𝑝𝑦))
165163, 164syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝 ↾ (0...𝑦))‘𝑦) = (𝑝𝑦))
166165eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
167162, 166oveq12d 6622 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0)[,)(𝑝𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
168167eleq2d 2684 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
169168biimpa 501 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) → 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
170 elfzofz 12426 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 ∈ (0..^𝑦) → 𝑖 ∈ (0...𝑦))
171170adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑖 ∈ (0...𝑦))
172 fvres 6164 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (0...𝑦) → ((𝑝 ↾ (0...𝑦))‘𝑖) = (𝑝𝑖))
173171, 172syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑝 ↾ (0...𝑦))‘𝑖) = (𝑝𝑖))
174 fzofzp1 12506 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (0..^𝑦) → (𝑖 + 1) ∈ (0...𝑦))
175174adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑖 + 1) ∈ (0...𝑦))
176 fvres 6164 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 + 1) ∈ (0...𝑦) → ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)) = (𝑝‘(𝑖 + 1)))
177175, 176syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)) = (𝑝‘(𝑖 + 1)))
178177adantlr 750 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)) = (𝑝‘(𝑖 + 1)))
179173, 178oveq12d 6622 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))) = ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
180179eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
181180rexbidva 3042 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) → (∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
182 nnz 11343 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
183 uzid 11646 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℤ → 𝑦 ∈ (ℤ𝑦))
184182, 183syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℕ → 𝑦 ∈ (ℤ𝑦))
185 peano2uz 11685 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (ℤ𝑦) → (𝑦 + 1) ∈ (ℤ𝑦))
186 fzoss2 12437 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 + 1) ∈ (ℤ𝑦) → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
187184, 185, 1863syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ℕ → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
188187ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
189 ssrexv 3646 . . . . . . . . . . . . . . . . . . . . . 22 ((0..^𝑦) ⊆ (0..^(𝑦 + 1)) → (∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
190188, 189syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) → (∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
191181, 190sylbid 230 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) → (∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
192169, 191embantd 59 . . . . . . . . . . . . . . . . . . 19 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) → ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
193192ex 450 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
194193adantr 481 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
195156, 194syld 47 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
196195ex 450 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (¬ (𝑝𝑦) ≤ 𝑋 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))))
197196com34 91 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (¬ (𝑝𝑦) ≤ 𝑋 → ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))))
198197com13 88 . . . . . . . . . . . . 13 ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → (¬ (𝑝𝑦) ≤ 𝑋 → ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))))
199139, 198sylbi 207 . . . . . . . . . . . 12 ([(𝑝 ↾ (0...𝑦)) / 𝑝](𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) → (¬ (𝑝𝑦) ≤ 𝑋 → ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))))
200101, 199syl 17 . . . . . . . . . . 11 (((𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦) ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))) → (¬ (𝑝𝑦) ≤ 𝑋 → ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))))
201200ex 450 . . . . . . . . . 10 ((𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦) → (∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) → (¬ (𝑝𝑦) ≤ 𝑋 → ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))))
202201com24 95 . . . . . . . . 9 ((𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦) → ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (¬ (𝑝𝑦) ≤ 𝑋 → (∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))))
203100, 202mpcom 38 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (¬ (𝑝𝑦) ≤ 𝑋 → (∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))))
204203ex 450 . . . . . . 7 (𝑦 ∈ ℕ → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (¬ (𝑝𝑦) ≤ 𝑋 → (∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))))
205204com24 95 . . . . . 6 (𝑦 ∈ ℕ → (∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) → (¬ (𝑝𝑦) ≤ 𝑋 → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))))
206205imp 445 . . . . 5 ((𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))) → (¬ (𝑝𝑦) ≤ 𝑋 → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))))
20799, 206pm2.61d 170 . . . 4 ((𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))) → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
20849, 207ralrimi 2951 . . 3 ((𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))) → ∀𝑝 ∈ (RePart‘(𝑦 + 1))(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
209208ex 450 . 2 (𝑦 ∈ ℕ → (∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) → ∀𝑝 ∈ (RePart‘(𝑦 + 1))(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))))
21010, 18, 26, 34, 46, 209nnind 10982 1 (𝑀 ∈ ℕ → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  [wsbc 3417  csb 3514  wss 3555  {csn 4148   class class class wbr 4613  cres 5076  cfv 5847  (class class class)co 6604  0cc0 9880  1c1 9881   + caddc 9883  *cxr 10017   < clt 10018  cle 10019  cn 10964  0cn0 11236  cz 11321  cuz 11631  [,)cico 12119  ...cfz 12268  ..^cfzo 12406  RePartciccp 40644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-ico 12123  df-fz 12269  df-fzo 12407  df-iccp 40645
This theorem is referenced by:  iccpartiun  40665  icceuelpart  40667  bgoldbtbnd  40983
  Copyright terms: Public domain W3C validator