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Theorem iccelpart 47358
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 6907 . . 3 (𝑥 = 1 → (RePart‘𝑥) = (RePart‘1))
2 fveq2 6907 . . . . . 6 (𝑥 = 1 → (𝑝𝑥) = (𝑝‘1))
32oveq2d 7447 . . . . 5 (𝑥 = 1 → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝‘1)))
43eleq2d 2825 . . . 4 (𝑥 = 1 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
5 oveq2 7439 . . . . . 6 (𝑥 = 1 → (0..^𝑥) = (0..^1))
6 fzo01 13783 . . . . . 6 (0..^1) = {0}
75, 6eqtrdi 2791 . . . . 5 (𝑥 = 1 → (0..^𝑥) = {0})
87rexeqdv 3325 . . . 4 (𝑥 = 1 → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
94, 8imbi12d 344 . . 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 3344 . 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 6907 . . 3 (𝑥 = 𝑦 → (RePart‘𝑥) = (RePart‘𝑦))
12 fveq2 6907 . . . . . 6 (𝑥 = 𝑦 → (𝑝𝑥) = (𝑝𝑦))
1312oveq2d 7447 . . . . 5 (𝑥 = 𝑦 → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝𝑦)))
1413eleq2d 2825 . . . 4 (𝑥 = 𝑦 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))))
15 oveq2 7439 . . . . 5 (𝑥 = 𝑦 → (0..^𝑥) = (0..^𝑦))
1615rexeqdv 3325 . . . 4 (𝑥 = 𝑦 → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
1714, 16imbi12d 344 . . 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 3344 . 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 6907 . . 3 (𝑥 = (𝑦 + 1) → (RePart‘𝑥) = (RePart‘(𝑦 + 1)))
20 fveq2 6907 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝑝𝑥) = (𝑝‘(𝑦 + 1)))
2120oveq2d 7447 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))
2221eleq2d 2825 . . . 4 (𝑥 = (𝑦 + 1) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1)))))
23 oveq2 7439 . . . . 5 (𝑥 = (𝑦 + 1) → (0..^𝑥) = (0..^(𝑦 + 1)))
2423rexeqdv 3325 . . . 4 (𝑥 = (𝑦 + 1) → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
2522, 24imbi12d 344 . . 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 3344 . 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 6907 . . 3 (𝑥 = 𝑀 → (RePart‘𝑥) = (RePart‘𝑀))
28 fveq2 6907 . . . . . 6 (𝑥 = 𝑀 → (𝑝𝑥) = (𝑝𝑀))
2928oveq2d 7447 . . . . 5 (𝑥 = 𝑀 → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝𝑀)))
3029eleq2d 2825 . . . 4 (𝑥 = 𝑀 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀))))
31 oveq2 7439 . . . . 5 (𝑥 = 𝑀 → (0..^𝑥) = (0..^𝑀))
3231rexeqdv 3325 . . . 4 (𝑥 = 𝑀 → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
3330, 32imbi12d 344 . . 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 3344 . 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 12539 . . . . 5 0 ∈ ℕ0
36 fveq2 6907 . . . . . . . 8 (𝑖 = 0 → (𝑝𝑖) = (𝑝‘0))
37 fv0p1e1 12387 . . . . . . . 8 (𝑖 = 0 → (𝑝‘(𝑖 + 1)) = (𝑝‘1))
3836, 37oveq12d 7449 . . . . . . 7 (𝑖 = 0 → ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑝‘0)[,)(𝑝‘1)))
3938eleq2d 2825 . . . . . 6 (𝑖 = 0 → (𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
4039rexsng 4681 . . . . 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 228 . . 3 (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
4342rgenw 3063 . 2 𝑝 ∈ (RePart‘1)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
44 nfv 1912 . . . . 5 𝑝 𝑦 ∈ ℕ
45 nfra1 3282 . . . . 5 𝑝𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
4644, 45nfan 1897 . . . 4 𝑝(𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
47 nnnn0 12531 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
48 fzonn0p1 13778 . . . . . . . . . 10 (𝑦 ∈ ℕ0𝑦 ∈ (0..^(𝑦 + 1)))
4947, 48syl 17 . . . . . . . . 9 (𝑦 ∈ ℕ → 𝑦 ∈ (0..^(𝑦 + 1)))
5049ad2antrr 726 . . . . . . . 8 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → 𝑦 ∈ (0..^(𝑦 + 1)))
51 fveq2 6907 . . . . . . . . . . 11 (𝑖 = 𝑦 → (𝑝𝑖) = (𝑝𝑦))
52 fvoveq1 7454 . . . . . . . . . . 11 (𝑖 = 𝑦 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑦 + 1)))
5351, 52oveq12d 7449 . . . . . . . . . 10 (𝑖 = 𝑦 → ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1))))
5453eleq2d 2825 . . . . . . . . 9 (𝑖 = 𝑦 → (𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1)))))
5554adantl 481 . . . . . . . 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 12276 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
5756adantr 480 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑦 + 1) ∈ ℕ)
58 simpr 484 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 𝑝 ∈ (RePart‘(𝑦 + 1)))
5956nnnn0d 12585 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ0)
60 0elfz 13661 . . . . . . . . . . . . . . . . 17 ((𝑦 + 1) ∈ ℕ0 → 0 ∈ (0...(𝑦 + 1)))
6159, 60syl 17 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ → 0 ∈ (0...(𝑦 + 1)))
6261adantr 480 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 0 ∈ (0...(𝑦 + 1)))
6357, 58, 62iccpartxr 47344 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘0) ∈ ℝ*)
64 nn0fz0 13662 . . . . . . . . . . . . . . . . 17 ((𝑦 + 1) ∈ ℕ0 ↔ (𝑦 + 1) ∈ (0...(𝑦 + 1)))
6559, 64sylib 218 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ (0...(𝑦 + 1)))
6665adantr 480 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑦 + 1) ∈ (0...(𝑦 + 1)))
6757, 58, 66iccpartxr 47344 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘(𝑦 + 1)) ∈ ℝ*)
6863, 67jca 511 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
6968adantlr 715 . . . . . . . . . . . 12 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
70 elico1 13427 . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → 𝑋 ∈ ℝ*)
7372adantl 481 . . . . . . . . . . . . . . 15 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 ∈ ℝ*)
74 simpl 482 . . . . . . . . . . . . . . 15 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑝𝑦) ≤ 𝑋)
75 simpr3 1195 . . . . . . . . . . . . . . 15 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 < (𝑝‘(𝑦 + 1)))
7673, 74, 753jca 1127 . . . . . . . . . . . . . 14 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))))
7776ex 412 . . . . . . . . . . . . 13 ((𝑝𝑦) ≤ 𝑋 → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
7877adantl 481 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
7978adantr 480 . . . . . . . . . . 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 240 . . . . . . . . . 10 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
8180impr 454 . . . . . . . . 9 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → (𝑋 ∈ ℝ* ∧ (𝑝𝑦) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))))
82 nn0fz0 13662 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0𝑦 ∈ (0...𝑦))
8347, 82sylib 218 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ → 𝑦 ∈ (0...𝑦))
84 fzelp1 13613 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...𝑦) → 𝑦 ∈ (0...(𝑦 + 1)))
8583, 84syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ ℕ → 𝑦 ∈ (0...(𝑦 + 1)))
8685adantr 480 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 𝑦 ∈ (0...(𝑦 + 1)))
8757, 58, 86iccpartxr 47344 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝𝑦) ∈ ℝ*)
8887, 67jca 511 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝𝑦) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
8988ad2ant2r 747 . . . . . . . . . 10 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → ((𝑝𝑦) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
90 elico1 13427 . . . . . . . . . 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 257 . . . . . . . 8 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → 𝑋 ∈ ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1))))
9350, 55, 92rspcedvd 3624 . . . . . . 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 436 . . . . . 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 480 . . . . 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 47343 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦))
97 rspsbca 3889 . . . . . . . . . . . 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 3482 . . . . . . . . . . . . . . 15 𝑝 ∈ V
9998resex 6049 . . . . . . . . . . . . . 14 (𝑝 ↾ (0...𝑦)) ∈ V
100 sbcimg 3843 . . . . . . . . . . . . . . 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 4424 . . . . . . . . . . . . . . . . 17 ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)))
102 csbov12g 7477 . . . . . . . . . . . . . . . . . . 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 6955 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘0) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝0)
104 csbvarg 4440 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝𝑝 = (𝑝 ↾ (0...𝑦)))
105 csbconstg 3927 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝0 = 0)
106104, 105fveq12d 6914 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝0) = ((𝑝 ↾ (0...𝑦))‘0))
107103, 106eqtrid 2787 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘0) = ((𝑝 ↾ (0...𝑦))‘0))
108 csbfv12 6955 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑦) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑦)
109 csbconstg 3927 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝𝑦 = 𝑦)
110104, 109fveq12d 6914 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
111108, 110eqtrid 2787 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
112107, 111oveq12d 7449 . . . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . . . 18 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
114113eleq2d 2825 . . . . . . . . . . . . . . . . 17 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
115101, 114bitrid 283 . . . . . . . . . . . . . . . 16 ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
116 sbcrex 3884 . . . . . . . . . . . . . . . . 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 4424 . . . . . . . . . . . . . . . . . . 19 ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
118 csbov12g 7477 . . . . . . . . . . . . . . . . . . . . 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 6955 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑖) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑖)
120 csbconstg 3927 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝𝑖 = 𝑖)
121104, 120fveq12d 6914 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑖) = ((𝑝 ↾ (0...𝑦))‘𝑖))
122119, 121eqtrid 2787 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑖) = ((𝑝 ↾ (0...𝑦))‘𝑖))
123 csbfv12 6955 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝(𝑖 + 1))
124 csbconstg 3927 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑖 + 1) = (𝑖 + 1))
125104, 124fveq12d 6914 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))
126123, 125eqtrid 2787 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))
127122, 126oveq12d 7449 . . . . . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))))
129128eleq2d 2825 . . . . . . . . . . . . . . . . . . 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, 129bitrid 283 . . . . . . . . . . . . . . . . . 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 3177 . . . . . . . . . . . . . . . . 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, 131bitrid 283 . . . . . . . . . . . . . . . 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 344 . . . . . . . . . . . . . . 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 279 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))))
13872adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 ∈ ℝ*)
139 simpr2 1194 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑝‘0) ≤ 𝑋)
140 xrltnle 11326 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑋 ∈ ℝ* ∧ (𝑝𝑦) ∈ ℝ*) → (𝑋 < (𝑝𝑦) ↔ ¬ (𝑝𝑦) ≤ 𝑋))
14172, 87, 140syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 < (𝑝𝑦) ↔ ¬ (𝑝𝑦) ≤ 𝑋))
142141exbiri 811 . . . . . . . . . . . . . . . . . . . . . . 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 417 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 < (𝑝𝑦))
145138, 139, 1443jca 1127 . . . . . . . . . . . . . . . . . . . 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 511 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝𝑦) ∈ ℝ*))
147146ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝𝑦) ∈ ℝ*))
148 elico1 13427 . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . 19 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)))
151150ex 412 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) → ((𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))))
152137, 151sylbid 240 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))))
153 0elfz 13661 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ ℕ0 → 0 ∈ (0...𝑦))
15447, 153syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ℕ → 0 ∈ (0...𝑦))
155154adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 0 ∈ (0...𝑦))
156 fvres 6926 . . . . . . . . . . . . . . . . . . . . . . . . 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 2741 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘0) = ((𝑝 ↾ (0...𝑦))‘0))
15983adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 𝑦 ∈ (0...𝑦))
160 fvres 6926 . . . . . . . . . . . . . . . . . . . . . . . . 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 2741 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
163158, 162oveq12d 7449 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0)[,)(𝑝𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
164163eleq2d 2825 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
165164biimpa 476 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) → 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
166 elfzofz 13712 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 ∈ (0..^𝑦) → 𝑖 ∈ (0...𝑦))
167166adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑖 ∈ (0...𝑦))
168 fvres 6926 . . . . . . . . . . . . . . . . . . . . . . . . 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 13800 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (0..^𝑦) → (𝑖 + 1) ∈ (0...𝑦))
171170adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑖 + 1) ∈ (0...𝑦))
172 fvres 6926 . . . . . . . . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)) = (𝑝‘(𝑖 + 1)))
175169, 174oveq12d 7449 . . . . . . . . . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . . . . . . . . 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 3175 . . . . . . . . . . . . . . . . . . . . 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 12632 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
179 uzid 12891 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℤ → 𝑦 ∈ (ℤ𝑦))
180 peano2uz 12941 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (ℤ𝑦) → (𝑦 + 1) ∈ (ℤ𝑦))
181 fzoss2 13724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 + 1) ∈ (ℤ𝑦) → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
182178, 179, 180, 1814syl 19 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ℕ → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
183182ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
184 ssrexv 4065 . . . . . . . . . . . . . . . . . . . . . 22 ((0..^𝑦) ⊆ (0..^(𝑦 + 1)) → (∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
185183, 184syl 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)))))
186177, 185sylbid 240 . . . . . . . . . . . . . . . . . . . 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)))))
187165, 186embantd 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)))))
188187ex 412 . . . . . . . . . . . . . . . . . 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))))))
189188adantr 480 . . . . . . . . . . . . . . . . 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))))))
190152, 189syld 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))))))
191190ex 412 . . . . . . . . . . . . . . 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)))))))
192191com34 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)))))))
193192com13 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)))))))
194135, 193sylbi 217 . . . . . . . . . . . 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)))))))
19597, 194syl 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)))))))
196195ex 412 . . . . . . . . . 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))))))))
197196com24 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))))))))
19896, 197mpcom 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)))))))
199198ex 412 . . . . . . 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))))))))
200199com24 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))))))))
201200imp 406 . . . . 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)))))))
20295, 201pm2.61d 179 . . . 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))))))
20346, 202ralrimi 3255 . . 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)))))
204203ex 412 . 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))))))
20510, 18, 26, 34, 43, 204nnind 12282 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  [wsbc 3791  csb 3908  wss 3963  {csn 4631   class class class wbr 5148  cres 5691  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154   + caddc 11156  *cxr 11292   < clt 11293  cle 11294  cn 12264  0cn0 12524  cz 12611  cuz 12876  [,)cico 13386  ...cfz 13544  ..^cfzo 13691  RePartciccp 47338
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-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
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-nel 3045  df-ral 3060  df-rex 3069  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-op 4638  df-uni 4913  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-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-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-ico 13390  df-fz 13545  df-fzo 13692  df-iccp 47339
This theorem is referenced by:  iccpartiun  47359  icceuelpart  47361  bgoldbtbnd  47734
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