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Theorem iccelpart 43444
Description: An element of any partitioned half-open 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 6669 . . 3 (𝑥 = 1 → (RePart‘𝑥) = (RePart‘1))
2 fveq2 6669 . . . . . 6 (𝑥 = 1 → (𝑝𝑥) = (𝑝‘1))
32oveq2d 7166 . . . . 5 (𝑥 = 1 → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝‘1)))
43eleq2d 2903 . . . 4 (𝑥 = 1 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
5 oveq2 7158 . . . . . 6 (𝑥 = 1 → (0..^𝑥) = (0..^1))
6 fzo01 13114 . . . . . 6 (0..^1) = {0}
75, 6syl6eq 2877 . . . . 5 (𝑥 = 1 → (0..^𝑥) = {0})
87rexeqdv 3422 . . . 4 (𝑥 = 1 → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
94, 8imbi12d 346 . . 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 3407 . 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 6669 . . 3 (𝑥 = 𝑦 → (RePart‘𝑥) = (RePart‘𝑦))
12 fveq2 6669 . . . . . 6 (𝑥 = 𝑦 → (𝑝𝑥) = (𝑝𝑦))
1312oveq2d 7166 . . . . 5 (𝑥 = 𝑦 → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝𝑦)))
1413eleq2d 2903 . . . 4 (𝑥 = 𝑦 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))))
15 oveq2 7158 . . . . 5 (𝑥 = 𝑦 → (0..^𝑥) = (0..^𝑦))
1615rexeqdv 3422 . . . 4 (𝑥 = 𝑦 → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
1714, 16imbi12d 346 . . 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 3407 . 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 6669 . . 3 (𝑥 = (𝑦 + 1) → (RePart‘𝑥) = (RePart‘(𝑦 + 1)))
20 fveq2 6669 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝑝𝑥) = (𝑝‘(𝑦 + 1)))
2120oveq2d 7166 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))
2221eleq2d 2903 . . . 4 (𝑥 = (𝑦 + 1) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1)))))
23 oveq2 7158 . . . . 5 (𝑥 = (𝑦 + 1) → (0..^𝑥) = (0..^(𝑦 + 1)))
2423rexeqdv 3422 . . . 4 (𝑥 = (𝑦 + 1) → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
2522, 24imbi12d 346 . . 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 3407 . 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 6669 . . 3 (𝑥 = 𝑀 → (RePart‘𝑥) = (RePart‘𝑀))
28 fveq2 6669 . . . . . 6 (𝑥 = 𝑀 → (𝑝𝑥) = (𝑝𝑀))
2928oveq2d 7166 . . . . 5 (𝑥 = 𝑀 → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝𝑀)))
3029eleq2d 2903 . . . 4 (𝑥 = 𝑀 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀))))
31 oveq2 7158 . . . . 5 (𝑥 = 𝑀 → (0..^𝑥) = (0..^𝑀))
3231rexeqdv 3422 . . . 4 (𝑥 = 𝑀 → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
3330, 32imbi12d 346 . . 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 3407 . 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 11906 . . . . 5 0 ∈ ℕ0
36 fveq2 6669 . . . . . . . 8 (𝑖 = 0 → (𝑝𝑖) = (𝑝‘0))
37 fv0p1e1 11754 . . . . . . . 8 (𝑖 = 0 → (𝑝‘(𝑖 + 1)) = (𝑝‘1))
3836, 37oveq12d 7168 . . . . . . 7 (𝑖 = 0 → ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑝‘0)[,)(𝑝‘1)))
3938eleq2d 2903 . . . . . 6 (𝑖 = 0 → (𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
4039rexsng 4613 . . . . 5 (0 ∈ ℕ0 → (∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
4135, 40ax-mp 5 . . . 4 (∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)))
4241biimpri 229 . . 3 (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
4342rgenw 3155 . 2 𝑝 ∈ (RePart‘1)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
44 nfv 1908 . . . . 5 𝑝 𝑦 ∈ ℕ
45 nfra1 3224 . . . . 5 𝑝𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
4644, 45nfan 1893 . . . 4 𝑝(𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
47 nnnn0 11898 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
48 fzonn0p1 13109 . . . . . . . . . 10 (𝑦 ∈ ℕ0𝑦 ∈ (0..^(𝑦 + 1)))
4947, 48syl 17 . . . . . . . . 9 (𝑦 ∈ ℕ → 𝑦 ∈ (0..^(𝑦 + 1)))
5049ad2antrr 722 . . . . . . . 8 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → 𝑦 ∈ (0..^(𝑦 + 1)))
51 fveq2 6669 . . . . . . . . . . 11 (𝑖 = 𝑦 → (𝑝𝑖) = (𝑝𝑦))
52 fvoveq1 7173 . . . . . . . . . . 11 (𝑖 = 𝑦 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑦 + 1)))
5351, 52oveq12d 7168 . . . . . . . . . 10 (𝑖 = 𝑦 → ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1))))
5453eleq2d 2903 . . . . . . . . 9 (𝑖 = 𝑦 → (𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1)))))
5554adantl 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)))))
56 peano2nn 11644 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
5756adantr 481 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑦 + 1) ∈ ℕ)
58 simpr 485 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 𝑝 ∈ (RePart‘(𝑦 + 1)))
5956nnnn0d 11949 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ0)
60 0elfz 12999 . . . . . . . . . . . . . . . . 17 ((𝑦 + 1) ∈ ℕ0 → 0 ∈ (0...(𝑦 + 1)))
6159, 60syl 17 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ → 0 ∈ (0...(𝑦 + 1)))
6261adantr 481 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 0 ∈ (0...(𝑦 + 1)))
6357, 58, 62iccpartxr 43430 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘0) ∈ ℝ*)
64 nn0fz0 13000 . . . . . . . . . . . . . . . . 17 ((𝑦 + 1) ∈ ℕ0 ↔ (𝑦 + 1) ∈ (0...(𝑦 + 1)))
6559, 64sylib 219 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ (0...(𝑦 + 1)))
6665adantr 481 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑦 + 1) ∈ (0...(𝑦 + 1)))
6757, 58, 66iccpartxr 43430 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘(𝑦 + 1)) ∈ ℝ*)
6863, 67jca 512 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
6968adantlr 711 . . . . . . . . . . . 12 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
70 elico1 12776 . . . . . . . . . . . 12 (((𝑝‘0) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
7169, 70syl 17 . . . . . . . . . . 11 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
72 simp1 1130 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → 𝑋 ∈ ℝ*)
7372adantl 482 . . . . . . . . . . . . . . 15 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 ∈ ℝ*)
74 simpl 483 . . . . . . . . . . . . . . 15 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑝𝑦) ≤ 𝑋)
75 simpr3 1190 . . . . . . . . . . . . . . 15 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 < (𝑝‘(𝑦 + 1)))
7673, 74, 753jca 1122 . . . . . . . . . . . . . 14 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))))
7776ex 413 . . . . . . . . . . . . 13 ((𝑝𝑦) ≤ 𝑋 → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
7877adantl 482 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
7978adantr 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)))))
8071, 79sylbid 241 . . . . . . . . . 10 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
8180impr 455 . . . . . . . . 9 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))))
82 nn0fz0 13000 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0𝑦 ∈ (0...𝑦))
8347, 82sylib 219 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ → 𝑦 ∈ (0...𝑦))
84 fzelp1 12954 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...𝑦) → 𝑦 ∈ (0...(𝑦 + 1)))
8583, 84syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ ℕ → 𝑦 ∈ (0...(𝑦 + 1)))
8685adantr 481 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 𝑦 ∈ (0...(𝑦 + 1)))
8757, 58, 86iccpartxr 43430 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝𝑦) ∈ ℝ*)
8887, 67jca 512 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝𝑦) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
8988ad2ant2r 743 . . . . . . . . . 10 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → ((𝑝𝑦) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
90 elico1 12776 . . . . . . . . . 10 (((𝑝𝑦) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*) → (𝑋 ∈ ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
9189, 90syl 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)))))
9281, 91mpbird 258 . . . . . . . 8 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → 𝑋 ∈ ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1))))
9350, 55, 92rspcedvd 3630 . . . . . . 7 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
9493exp43 437 . . . . . 6 (𝑦 ∈ ℕ → ((𝑝𝑦) ≤ 𝑋 → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))))
9594adantr 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)))))))
96 iccpartres 43429 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦))
97 rspsbca 3867 . . . . . . . . . . . 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)))))
98 vex 3503 . . . . . . . . . . . . . . 15 𝑝 ∈ V
9998resex 5898 . . . . . . . . . . . . . 14 (𝑝 ↾ (0...𝑦)) ∈ V
100 sbcimg 3824 . . . . . . . . . . . . . . 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))))))
101 sbcel2 4371 . . . . . . . . . . . . . . . . 17 ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)))
102 csbov12g 7194 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)) = ((𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘0)[,)(𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑦)))
103 csbfv12 6712 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘0) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝0)
104 csbvarg 4387 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝𝑝 = (𝑝 ↾ (0...𝑦)))
105 csbconstg 3906 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝0 = 0)
106104, 105fveq12d 6676 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝0) = ((𝑝 ↾ (0...𝑦))‘0))
107103, 106syl5eq 2873 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘0) = ((𝑝 ↾ (0...𝑦))‘0))
108 csbfv12 6712 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑦) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑦)
109 csbconstg 3906 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝𝑦 = 𝑦)
110104, 109fveq12d 6676 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
111108, 110syl5eq 2873 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
112107, 111oveq12d 7168 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘0)[,)(𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
113102, 112eqtrd 2861 . . . . . . . . . . . . . . . . . 18 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
114113eleq2d 2903 . . . . . . . . . . . . . . . . 17 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
115101, 114syl5bb 284 . . . . . . . . . . . . . . . 16 ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
116 sbcrex 3862 . . . . . . . . . . . . . . . . 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))))
117 sbcel2 4371 . . . . . . . . . . . . . . . . . . 19 ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
118 csbov12g 7194 . . . . . . . . . . . . . . . . . . . . 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))))
119 csbfv12 6712 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑖) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑖)
120 csbconstg 3906 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝𝑖 = 𝑖)
121104, 120fveq12d 6676 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑖) = ((𝑝 ↾ (0...𝑦))‘𝑖))
122119, 121syl5eq 2873 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑖) = ((𝑝 ↾ (0...𝑦))‘𝑖))
123 csbfv12 6712 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝(𝑖 + 1))
124 csbconstg 3906 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑖 + 1) = (𝑖 + 1))
125104, 124fveq12d 6676 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))
126123, 125syl5eq 2873 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))
127122, 126oveq12d 7168 . . . . . . . . . . . . . . . . . . . . 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))))
128118, 127eqtrd 2861 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))))
129128eleq2d 2903 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))))
130117, 129syl5bb 284 . . . . . . . . . . . . . . . . . 18 ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))))
131130rexbidv 3302 . . . . . . . . . . . . . . . . 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)))))
132116, 131syl5bb 284 . . . . . . . . . . . . . . . 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)))))
133115, 132imbi12d 346 . . . . . . . . . . . . . . 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))))))
134100, 133bitrd 280 . . . . . . . . . . . . . 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))))))
13599, 134ax-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)))))
13668, 70syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
137136adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
13872adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 ∈ ℝ*)
139 simpr2 1189 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑝‘0) ≤ 𝑋)
140 xrltnle 10702 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑋 ∈ ℝ* ∧ (𝑝𝑦) ∈ ℝ*) → (𝑋 < (𝑝𝑦) ↔ ¬ (𝑝𝑦) ≤ 𝑋))
14172, 87, 140syl2anr 596 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 < (𝑝𝑦) ↔ ¬ (𝑝𝑦) ≤ 𝑋))
142141exbiri 807 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → (¬ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝𝑦))))
143142com23 86 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (¬ (𝑝𝑦) ≤ 𝑋 → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → 𝑋 < (𝑝𝑦))))
144143imp31 418 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 < (𝑝𝑦))
145138, 139, 1443jca 1122 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝𝑦)))
14663, 87jca 512 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝𝑦) ∈ ℝ*))
147146ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝𝑦) ∈ ℝ*))
148 elico1 12776 . . . . . . . . . . . . . . . . . . . . 21 (((𝑝‘0) ∈ ℝ* ∧ (𝑝𝑦) ∈ ℝ*) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝𝑦))))
149147, 148syl 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))))
150145, 149mpbird 258 . . . . . . . . . . . . . . . . . . 19 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)))
151150ex 413 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))))
152137, 151sylbid 241 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))))
153 0elfz 12999 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ ℕ0 → 0 ∈ (0...𝑦))
15447, 153syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ℕ → 0 ∈ (0...𝑦))
155154adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 0 ∈ (0...𝑦))
156 fvres 6688 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ (0...𝑦) → ((𝑝 ↾ (0...𝑦))‘0) = (𝑝‘0))
157155, 156syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝 ↾ (0...𝑦))‘0) = (𝑝‘0))
158157eqcomd 2832 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘0) = ((𝑝 ↾ (0...𝑦))‘0))
15983adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 𝑦 ∈ (0...𝑦))
160 fvres 6688 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (0...𝑦) → ((𝑝 ↾ (0...𝑦))‘𝑦) = (𝑝𝑦))
161159, 160syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝 ↾ (0...𝑦))‘𝑦) = (𝑝𝑦))
162161eqcomd 2832 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
163158, 162oveq12d 7168 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0)[,)(𝑝𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
164163eleq2d 2903 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
165164biimpa 477 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) → 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
166 elfzofz 13048 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 ∈ (0..^𝑦) → 𝑖 ∈ (0...𝑦))
167166adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑖 ∈ (0...𝑦))
168 fvres 6688 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (0...𝑦) → ((𝑝 ↾ (0...𝑦))‘𝑖) = (𝑝𝑖))
169167, 168syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑝 ↾ (0...𝑦))‘𝑖) = (𝑝𝑖))
170 fzofzp1 13129 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (0..^𝑦) → (𝑖 + 1) ∈ (0...𝑦))
171170adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑖 + 1) ∈ (0...𝑦))
172 fvres 6688 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 + 1) ∈ (0...𝑦) → ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)) = (𝑝‘(𝑖 + 1)))
173171, 172syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)) = (𝑝‘(𝑖 + 1)))
174173adantlr 711 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)) = (𝑝‘(𝑖 + 1)))
175169, 174oveq12d 7168 . . . . . . . . . . . . . . . . . . . . . . 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))))
176175eleq2d 2903 . . . . . . . . . . . . . . . . . . . . . 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)))))
177176rexbidva 3301 . . . . . . . . . . . . . . . . . . . . 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)))))
178 nnz 11998 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
179 uzid 12252 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℤ → 𝑦 ∈ (ℤ𝑦))
180178, 179syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℕ → 𝑦 ∈ (ℤ𝑦))
181 peano2uz 12295 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (ℤ𝑦) → (𝑦 + 1) ∈ (ℤ𝑦))
182 fzoss2 13060 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 + 1) ∈ (ℤ𝑦) → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
183180, 181, 1823syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ℕ → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
184183ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
185 ssrexv 4038 . . . . . . . . . . . . . . . . . . . . . 22 ((0..^𝑦) ⊆ (0..^(𝑦 + 1)) → (∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
186184, 185syl 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)))))
187177, 186sylbid 241 . . . . . . . . . . . . . . . . . . . 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)))))
188165, 187embantd 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)))))
189188ex 413 . . . . . . . . . . . . . . . . . 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))))))
190189adantr 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))))))
191152, 190syld 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))))))
192191ex 413 . . . . . . . . . . . . . . 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)))))))
193192com34 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)))))))
194193com13 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)))))))
195135, 194sylbi 218 . . . . . . . . . . . 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)))))))
19697, 195syl 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)))))))
197196ex 413 . . . . . . . . . 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))))))))
198197com24 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))))))))
19996, 198mpcom 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)))))))
200199ex 413 . . . . . . 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))))))))
201200com24 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))))))))
202201imp 407 . . . . 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)))))))
20395, 202pm2.61d 180 . . . 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))))))
20446, 203ralrimi 3221 . . 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)))))
205204ex 413 . 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))))))
20610, 18, 26, 34, 43, 205nnind 11650 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  wrex 3144  Vcvv 3500  [wsbc 3776  csb 3887  wss 3940  {csn 4564   class class class wbr 5063  cres 5556  cfv 6354  (class class class)co 7150  0cc0 10531  1c1 10532   + caddc 10534  *cxr 10668   < clt 10669  cle 10670  cn 11632  0cn0 11891  cz 11975  cuz 12237  [,)cico 12735  ...cfz 12887  ..^cfzo 13028  RePartciccp 43424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-ico 12739  df-fz 12888  df-fzo 13029  df-iccp 43425
This theorem is referenced by:  iccpartiun  43445  icceuelpart  43447  bgoldbtbnd  43825
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