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Theorem iccelpart 45810
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 6873 . . 3 (𝑥 = 1 → (RePart‘𝑥) = (RePart‘1))
2 fveq2 6873 . . . . . 6 (𝑥 = 1 → (𝑝𝑥) = (𝑝‘1))
32oveq2d 7404 . . . . 5 (𝑥 = 1 → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝‘1)))
43eleq2d 2818 . . . 4 (𝑥 = 1 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
5 oveq2 7396 . . . . . 6 (𝑥 = 1 → (0..^𝑥) = (0..^1))
6 fzo01 13691 . . . . . 6 (0..^1) = {0}
75, 6eqtrdi 2787 . . . . 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 3341 . 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 6873 . . 3 (𝑥 = 𝑦 → (RePart‘𝑥) = (RePart‘𝑦))
12 fveq2 6873 . . . . . 6 (𝑥 = 𝑦 → (𝑝𝑥) = (𝑝𝑦))
1312oveq2d 7404 . . . . 5 (𝑥 = 𝑦 → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝𝑦)))
1413eleq2d 2818 . . . 4 (𝑥 = 𝑦 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))))
15 oveq2 7396 . . . . 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 3341 . 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 6873 . . 3 (𝑥 = (𝑦 + 1) → (RePart‘𝑥) = (RePart‘(𝑦 + 1)))
20 fveq2 6873 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝑝𝑥) = (𝑝‘(𝑦 + 1)))
2120oveq2d 7404 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))
2221eleq2d 2818 . . . 4 (𝑥 = (𝑦 + 1) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1)))))
23 oveq2 7396 . . . . 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 3341 . 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 6873 . . 3 (𝑥 = 𝑀 → (RePart‘𝑥) = (RePart‘𝑀))
28 fveq2 6873 . . . . . 6 (𝑥 = 𝑀 → (𝑝𝑥) = (𝑝𝑀))
2928oveq2d 7404 . . . . 5 (𝑥 = 𝑀 → ((𝑝‘0)[,)(𝑝𝑥)) = ((𝑝‘0)[,)(𝑝𝑀)))
3029eleq2d 2818 . . . 4 (𝑥 = 𝑀 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀))))
31 oveq2 7396 . . . . 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 3341 . 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 12464 . . . . 5 0 ∈ ℕ0
36 fveq2 6873 . . . . . . . 8 (𝑖 = 0 → (𝑝𝑖) = (𝑝‘0))
37 fv0p1e1 12312 . . . . . . . 8 (𝑖 = 0 → (𝑝‘(𝑖 + 1)) = (𝑝‘1))
3836, 37oveq12d 7406 . . . . . . 7 (𝑖 = 0 → ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑝‘0)[,)(𝑝‘1)))
3938eleq2d 2818 . . . . . 6 (𝑖 = 0 → (𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
4039rexsng 4666 . . . . 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 227 . . 3 (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
4342rgenw 3064 . 2 𝑝 ∈ (RePart‘1)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
44 nfv 1917 . . . . 5 𝑝 𝑦 ∈ ℕ
45 nfra1 3280 . . . . 5 𝑝𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
4644, 45nfan 1902 . . . 4 𝑝(𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
47 nnnn0 12456 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
48 fzonn0p1 13686 . . . . . . . . . 10 (𝑦 ∈ ℕ0𝑦 ∈ (0..^(𝑦 + 1)))
4947, 48syl 17 . . . . . . . . 9 (𝑦 ∈ ℕ → 𝑦 ∈ (0..^(𝑦 + 1)))
5049ad2antrr 724 . . . . . . . 8 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → 𝑦 ∈ (0..^(𝑦 + 1)))
51 fveq2 6873 . . . . . . . . . . 11 (𝑖 = 𝑦 → (𝑝𝑖) = (𝑝𝑦))
52 fvoveq1 7411 . . . . . . . . . . 11 (𝑖 = 𝑦 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑦 + 1)))
5351, 52oveq12d 7406 . . . . . . . . . 10 (𝑖 = 𝑦 → ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1))))
5453eleq2d 2818 . . . . . . . . 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 12201 . . . . . . . . . . . . . . . 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 12509 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ0)
60 0elfz 13575 . . . . . . . . . . . . . . . . 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 45796 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘0) ∈ ℝ*)
64 nn0fz0 13576 . . . . . . . . . . . . . . . . 17 ((𝑦 + 1) ∈ ℕ0 ↔ (𝑦 + 1) ∈ (0...(𝑦 + 1)))
6559, 64sylib 217 . . . . . . . . . . . . . . . 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 45796 . . . . . . . . . . . . . 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 713 . . . . . . . . . . . 12 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
70 elico1 13344 . . . . . . . . . . . 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 1136 . . . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . . . 15 (((𝑝𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 < (𝑝‘(𝑦 + 1)))
7673, 74, 753jca 1128 . . . . . . . . . . . . . 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 239 . . . . . . . . . 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 13576 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0𝑦 ∈ (0...𝑦))
8347, 82sylib 217 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ → 𝑦 ∈ (0...𝑦))
84 fzelp1 13530 . . . . . . . . . . . . . . 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 45796 . . . . . . . . . . . 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 745 . . . . . . . . . 10 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → ((𝑝𝑦) ∈ ℝ* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ*))
90 elico1 13344 . . . . . . . . . 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 256 . . . . . . . 8 (((𝑦 ∈ ℕ ∧ (𝑝𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → 𝑋 ∈ ((𝑝𝑦)[,)(𝑝‘(𝑦 + 1))))
9350, 55, 92rspcedvd 3607 . . . . . . 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 45795 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦))
97 rspsbca 3865 . . . . . . . . . . . 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 3473 . . . . . . . . . . . . . . 15 𝑝 ∈ V
9998resex 6016 . . . . . . . . . . . . . 14 (𝑝 ↾ (0...𝑦)) ∈ V
100 sbcimg 3819 . . . . . . . . . . . . . . 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 4406 . . . . . . . . . . . . . . . . 17 ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)))
102 csbov12g 7432 . . . . . . . . . . . . . . . . . . 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 6921 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘0) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝0)
104 csbvarg 4422 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝𝑝 = (𝑝 ↾ (0...𝑦)))
105 csbconstg 3903 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝0 = 0)
106104, 105fveq12d 6880 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝0) = ((𝑝 ↾ (0...𝑦))‘0))
107103, 106eqtrid 2783 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘0) = ((𝑝 ↾ (0...𝑦))‘0))
108 csbfv12 6921 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑦) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑦)
109 csbconstg 3903 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝𝑦 = 𝑦)
110104, 109fveq12d 6880 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
111108, 110eqtrid 2783 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
112107, 111oveq12d 7406 . . . . . . . . . . . . . . . . . . 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 2771 . . . . . . . . . . . . . . . . . 18 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
114113eleq2d 2818 . . . . . . . . . . . . . . . . 17 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
115101, 114bitrid 282 . . . . . . . . . . . . . . . 16 ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
116 sbcrex 3860 . . . . . . . . . . . . . . . . 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 4406 . . . . . . . . . . . . . . . . . . 19 ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋(𝑝 ↾ (0...𝑦)) / 𝑝((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))
118 csbov12g 7432 . . . . . . . . . . . . . . . . . . . . 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 6921 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑖) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑖)
120 csbconstg 3903 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝𝑖 = 𝑖)
121104, 120fveq12d 6880 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝𝑖) = ((𝑝 ↾ (0...𝑦))‘𝑖))
122119, 121eqtrid 2783 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝𝑖) = ((𝑝 ↾ (0...𝑦))‘𝑖))
123 csbfv12 6921 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝(𝑖 + 1))
124 csbconstg 3903 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑖 + 1) = (𝑖 + 1))
125104, 124fveq12d 6880 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑝 ↾ (0...𝑦)) ∈ V → ((𝑝 ↾ (0...𝑦)) / 𝑝𝑝(𝑝 ↾ (0...𝑦)) / 𝑝(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))
126123, 125eqtrid 2783 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝(𝑝‘(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))
127122, 126oveq12d 7406 . . . . . . . . . . . . . . . . . . . . 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 2771 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑝 ↾ (0...𝑦)) / 𝑝((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))))
129128eleq2d 2818 . . . . . . . . . . . . . . . . . . 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 282 . . . . . . . . . . . . . . . . . 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 282 . . . . . . . . . . . . . . . 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 278 . . . . . . . . . . . . . 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 1195 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑝‘0) ≤ 𝑋)
140 xrltnle 11258 . . . . . . . . . . . . . . . . . . . . . . . . 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 809 . . . . . . . . . . . . . . . . . . . . . . 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 1128 . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ* ∧ (𝑝‘0) ≤ 𝑋𝑋 < (𝑝‘(𝑦 + 1)))) → ((𝑝‘0) ∈ ℝ* ∧ (𝑝𝑦) ∈ ℝ*))
148 elico1 13344 . . . . . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . . . . 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 239 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))))
153 0elfz 13575 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
163158, 162oveq12d 7406 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0)[,)(𝑝𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
164163eleq2d 2818 . . . . . . . . . . . . . . . . . . . . 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 13625 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . . . . . . . . . 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 13706 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)) = (𝑝‘(𝑖 + 1)))
175169, 174oveq12d 7406 . . . . . . . . . . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . . . . . . . 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 12556 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
179 uzid 12814 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℤ → 𝑦 ∈ (ℤ𝑦))
180178, 179syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℕ → 𝑦 ∈ (ℤ𝑦))
181 peano2uz 12862 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (ℤ𝑦) → (𝑦 + 1) ∈ (ℤ𝑦))
182 fzoss2 13637 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 + 1) ∈ (ℤ𝑦) → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
183180, 181, 1823syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ℕ → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
184183ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑦))) → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
185 ssrexv 4042 . . . . . . . . . . . . . . . . . . . . . 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 239 . . . . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . 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 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))))))
20446, 203ralrimi 3253 . . 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 12207 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3060  wrex 3069  Vcvv 3469  [wsbc 3768  csb 3884  wss 3939  {csn 4617   class class class wbr 5136  cres 5666  cfv 6527  (class class class)co 7388  0cc0 11087  1c1 11088   + caddc 11090  *cxr 11224   < clt 11225  cle 11226  cn 12189  0cn0 12449  cz 12535  cuz 12799  [,)cico 13303  ...cfz 13461  ..^cfzo 13604  RePartciccp 45790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5287  ax-nul 5294  ax-pow 5351  ax-pr 5415  ax-un 7703  ax-cnex 11143  ax-resscn 11144  ax-1cn 11145  ax-icn 11146  ax-addcl 11147  ax-addrcl 11148  ax-mulcl 11149  ax-mulrcl 11150  ax-mulcom 11151  ax-addass 11152  ax-mulass 11153  ax-distr 11154  ax-i2m1 11155  ax-1ne0 11156  ax-1rid 11157  ax-rnegex 11158  ax-rrecex 11159  ax-cnre 11160  ax-pre-lttri 11161  ax-pre-lttrn 11162  ax-pre-ltadd 11163  ax-pre-mulgt0 11164
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3375  df-rab 3429  df-v 3471  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4314  df-if 4518  df-pw 4593  df-sn 4618  df-pr 4620  df-op 4624  df-uni 4897  df-iun 4987  df-br 5137  df-opab 5199  df-mpt 5220  df-tr 5254  df-id 5562  df-eprel 5568  df-po 5576  df-so 5577  df-fr 5619  df-we 5621  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6284  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7344  df-ov 7391  df-oprab 7392  df-mpo 7393  df-om 7834  df-1st 7952  df-2nd 7953  df-frecs 8243  df-wrecs 8274  df-recs 8348  df-rdg 8387  df-er 8681  df-map 8800  df-en 8918  df-dom 8919  df-sdom 8920  df-pnf 11227  df-mnf 11228  df-xr 11229  df-ltxr 11230  df-le 11231  df-sub 11423  df-neg 11424  df-nn 12190  df-n0 12450  df-z 12536  df-uz 12800  df-ico 13307  df-fz 13462  df-fzo 13605  df-iccp 45791
This theorem is referenced by:  iccpartiun  45811  icceuelpart  45813  bgoldbtbnd  46186
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