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Theorem fourierdlem12 43242
Description: A point of a partition is not an element of any open interval determined by the partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem12.1 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem12.2 (𝜑𝑀 ∈ ℕ)
fourierdlem12.3 (𝜑𝑄 ∈ (𝑃𝑀))
fourierdlem12.4 (𝜑𝑋 ∈ ran 𝑄)
Assertion
Ref Expression
fourierdlem12 ((𝜑𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
Distinct variable groups:   𝐴,𝑚,𝑝   𝐵,𝑚,𝑝   𝑖,𝑀,𝑚,𝑝   𝑄,𝑖,𝑝   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑚,𝑝)   𝐴(𝑖)   𝐵(𝑖)   𝑃(𝑖,𝑚,𝑝)   𝑄(𝑚)   𝑋(𝑖,𝑚,𝑝)

Proof of Theorem fourierdlem12
Dummy variables 𝑗 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem12.4 . . . 4 (𝜑𝑋 ∈ ran 𝑄)
2 fourierdlem12.3 . . . . . . . 8 (𝜑𝑄 ∈ (𝑃𝑀))
3 fourierdlem12.2 . . . . . . . . 9 (𝜑𝑀 ∈ ℕ)
4 fourierdlem12.1 . . . . . . . . . 10 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
54fourierdlem2 43232 . . . . . . . . 9 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
63, 5syl 17 . . . . . . . 8 (𝜑 → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
72, 6mpbid 235 . . . . . . 7 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
87simpld 498 . . . . . 6 (𝜑𝑄 ∈ (ℝ ↑m (0...𝑀)))
9 elmapi 8471 . . . . . 6 (𝑄 ∈ (ℝ ↑m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
10 ffn 6514 . . . . . 6 (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
118, 9, 103syl 18 . . . . 5 (𝜑𝑄 Fn (0...𝑀))
12 fvelrnb 6742 . . . . 5 (𝑄 Fn (0...𝑀) → (𝑋 ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄𝑗) = 𝑋))
1311, 12syl 17 . . . 4 (𝜑 → (𝑋 ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄𝑗) = 𝑋))
141, 13mpbid 235 . . 3 (𝜑 → ∃𝑗 ∈ (0...𝑀)(𝑄𝑗) = 𝑋)
1514adantr 484 . 2 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑗 ∈ (0...𝑀)(𝑄𝑗) = 𝑋)
168, 9syl 17 . . . . . . . . . . . 12 (𝜑𝑄:(0...𝑀)⟶ℝ)
1716adantr 484 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
18 fzofzp1 13237 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
1918adantl 485 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
2017, 19ffvelrnd 6874 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
2120adantr 484 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
22213ad2antl1 1186 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
23 frn 6521 . . . . . . . . . . . 12 (𝑄:(0...𝑀)⟶ℝ → ran 𝑄 ⊆ ℝ)
2416, 23syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑄 ⊆ ℝ)
2524, 1sseldd 3888 . . . . . . . . . 10 (𝜑𝑋 ∈ ℝ)
2625ad2antrr 726 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑖 < 𝑗) → 𝑋 ∈ ℝ)
27263ad2antl1 1186 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → 𝑋 ∈ ℝ)
2817ffvelrnda 6873 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑄𝑗) ∈ ℝ)
29283adant3 1133 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) ∈ ℝ)
3029adantr 484 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄𝑗) ∈ ℝ)
31 simpr 488 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
32 elfzoelz 13141 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
3332ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 ∈ ℤ)
34 elfzelz 13010 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
3534ad2antlr 727 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
36 zltp1le 12125 . . . . . . . . . . . . . . 15 ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
3733, 35, 36syl2anc 587 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
3831, 37mpbid 235 . . . . . . . . . . . . 13 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ≤ 𝑗)
3933peano2zd 12183 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ∈ ℤ)
40 eluz 12350 . . . . . . . . . . . . . 14 (((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ‘(𝑖 + 1)) ↔ (𝑖 + 1) ≤ 𝑗))
4139, 35, 40syl2anc 587 . . . . . . . . . . . . 13 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑗 ∈ (ℤ‘(𝑖 + 1)) ↔ (𝑖 + 1) ≤ 𝑗))
4238, 41mpbird 260 . . . . . . . . . . . 12 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (ℤ‘(𝑖 + 1)))
4342adantlll 718 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (ℤ‘(𝑖 + 1)))
4417ad2antrr 726 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑄:(0...𝑀)⟶ℝ)
45 0zd 12086 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ∈ ℤ)
46 elfzel2 13008 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
4746ad2antlr 727 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑀 ∈ ℤ)
48 elfzelz 13010 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ∈ ℤ)
4948adantl 485 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℤ)
50 0red 10734 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ∈ ℝ)
5148zred 12180 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ∈ ℝ)
5251adantl 485 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℝ)
5332peano2zd 12183 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ ℤ)
5453zred 12180 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ ℝ)
5554adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑖 + 1) ∈ ℝ)
5632zred 12180 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
5756adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑖 ∈ ℝ)
58 elfzole1 13149 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 0 ≤ 𝑖)
5958adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑖)
6057ltp1d 11660 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑖 < (𝑖 + 1))
6150, 57, 55, 59, 60lelttrd 10888 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 < (𝑖 + 1))
62 elfzle1 13013 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ((𝑖 + 1)...𝑗) → (𝑖 + 1) ≤ 𝑤)
6362adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑖 + 1) ≤ 𝑤)
6450, 55, 52, 61, 63ltletrd 10890 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 < 𝑤)
6550, 52, 64ltled 10878 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑤)
6665adantlr 715 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑤)
6751adantl 485 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℝ)
6834zred 12180 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
6968adantr 484 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑗 ∈ ℝ)
7046zred 12180 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
7170adantr 484 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑀 ∈ ℝ)
72 elfzle2 13014 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤𝑗)
7372adantl 485 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤𝑗)
74 elfzle2 13014 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → 𝑗𝑀)
7574adantr 484 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑗𝑀)
7667, 69, 71, 73, 75letrd 10887 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤𝑀)
7776adantll 714 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤𝑀)
7845, 47, 49, 66, 77elfzd 13001 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ (0...𝑀))
7978adantlll 718 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ (0...𝑀))
8044, 79ffvelrnd 6874 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑄𝑤) ∈ ℝ)
8180adantlr 715 . . . . . . . . . . 11 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑄𝑤) ∈ ℝ)
82 simp-4l 783 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝜑)
83 0red 10734 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℝ)
84 elfzelz 13010 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ∈ ℤ)
8584zred 12180 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ∈ ℝ)
8685adantl 485 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
87 0red 10734 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℝ)
8854adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑖 + 1) ∈ ℝ)
8985adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
90 0red 10734 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 0 ∈ ℝ)
9156ltp1d 11660 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 𝑖 < (𝑖 + 1))
9290, 56, 54, 58, 91lelttrd 10888 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑀) → 0 < (𝑖 + 1))
9392adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < (𝑖 + 1))
94 elfzle1 13013 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → (𝑖 + 1) ≤ 𝑤)
9594adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑖 + 1) ≤ 𝑤)
9687, 88, 89, 93, 95ltletrd 10890 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < 𝑤)
9796adantlr 715 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < 𝑤)
9883, 86, 97ltled 10878 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
9998adantlll 718 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
10099adantlr 715 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
10185adantl 485 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
102 peano2rem 11043 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℝ → (𝑗 − 1) ∈ ℝ)
10368, 102syl 17 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℝ)
104103adantr 484 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑗 − 1) ∈ ℝ)
10570adantr 484 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑀 ∈ ℝ)
106 elfzle2 13014 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ≤ (𝑗 − 1))
107106adantl 485 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ≤ (𝑗 − 1))
108 zlem1lt 12127 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗𝑀 ↔ (𝑗 − 1) < 𝑀))
10934, 46, 108syl2anc 587 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → (𝑗𝑀 ↔ (𝑗 − 1) < 𝑀))
11074, 109mpbid 235 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑀)
111110adantr 484 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑗 − 1) < 𝑀)
112101, 104, 105, 107, 111lelttrd 10888 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
113112adantlr 715 . . . . . . . . . . . . . 14 (((𝑗 ∈ (0...𝑀) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
114113adantlll 718 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
11584adantl 485 . . . . . . . . . . . . . 14 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℤ)
116 0zd 12086 . . . . . . . . . . . . . 14 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℤ)
11746ad3antlr 731 . . . . . . . . . . . . . 14 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑀 ∈ ℤ)
118 elfzo 13143 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤𝑤 < 𝑀)))
119115, 116, 117, 118syl3anc 1372 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤𝑤 < 𝑀)))
120100, 114, 119mpbir2and 713 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ (0..^𝑀))
12116adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
122 elfzofz 13156 . . . . . . . . . . . . . . 15 (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ (0...𝑀))
123122adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (0..^𝑀)) → 𝑤 ∈ (0...𝑀))
124121, 123ffvelrnd 6874 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄𝑤) ∈ ℝ)
125 fzofzp1 13237 . . . . . . . . . . . . . . 15 (𝑤 ∈ (0..^𝑀) → (𝑤 + 1) ∈ (0...𝑀))
126125adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑤 + 1) ∈ (0...𝑀))
127121, 126ffvelrnd 6874 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄‘(𝑤 + 1)) ∈ ℝ)
128 eleq1w 2816 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑤 → (𝑖 ∈ (0..^𝑀) ↔ 𝑤 ∈ (0..^𝑀)))
129128anbi2d 632 . . . . . . . . . . . . . . 15 (𝑖 = 𝑤 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑤 ∈ (0..^𝑀))))
130 fveq2 6686 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑤 → (𝑄𝑖) = (𝑄𝑤))
131 oveq1 7189 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑤 → (𝑖 + 1) = (𝑤 + 1))
132131fveq2d 6690 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑤 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑤 + 1)))
133130, 132breq12d 5053 . . . . . . . . . . . . . . 15 (𝑖 = 𝑤 → ((𝑄𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄𝑤) < (𝑄‘(𝑤 + 1))))
134129, 133imbi12d 348 . . . . . . . . . . . . . 14 (𝑖 = 𝑤 → (((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄𝑤) < (𝑄‘(𝑤 + 1)))))
1357simprrd 774 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
136135r19.21bi 3122 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
137134, 136chvarvv 2010 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄𝑤) < (𝑄‘(𝑤 + 1)))
138124, 127, 137ltled 10878 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄𝑤) ≤ (𝑄‘(𝑤 + 1)))
13982, 120, 138syl2anc 587 . . . . . . . . . . 11 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑄𝑤) ≤ (𝑄‘(𝑤 + 1)))
14043, 81, 139monoord 13504 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ (𝑄𝑗))
1411403adantl3 1169 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ (𝑄𝑗))
14216ffvelrnda 6873 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑄𝑗) ∈ ℝ)
1431423adant3 1133 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) ∈ ℝ)
144 simp3 1139 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) = 𝑋)
145143, 144eqled 10833 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) ≤ 𝑋)
1461453adant1r 1178 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) ≤ 𝑋)
147146adantr 484 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄𝑗) ≤ 𝑋)
14822, 30, 27, 141, 147letrd 10887 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ 𝑋)
14922, 27, 148lensymd 10881 . . . . . . 7 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → ¬ 𝑋 < (𝑄‘(𝑖 + 1)))
150149intnand 492 . . . . . 6 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → ¬ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1))))
15168ad2antlr 727 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ∈ ℝ)
15256ad3antlr 731 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑖 ∈ ℝ)
153 simpr 488 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → ¬ 𝑖 < 𝑗)
154151, 152, 153nltled 10880 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑗𝑖)
1551543adantl3 1169 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → 𝑗𝑖)
156 eqcom 2746 . . . . . . . . . . . . 13 ((𝑄𝑗) = 𝑋𝑋 = (𝑄𝑗))
157156biimpi 219 . . . . . . . . . . . 12 ((𝑄𝑗) = 𝑋𝑋 = (𝑄𝑗))
158157adantr 484 . . . . . . . . . . 11 (((𝑄𝑗) = 𝑋𝑗𝑖) → 𝑋 = (𝑄𝑗))
1591583ad2antl3 1188 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → 𝑋 = (𝑄𝑗))
16034ad2antlr 727 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑗 ∈ ℤ)
16132ad2antrr 726 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑖 ∈ ℤ)
162 simpr 488 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑗𝑖)
163 eluz2 12342 . . . . . . . . . . . . . 14 (𝑖 ∈ (ℤ𝑗) ↔ (𝑗 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗𝑖))
164160, 161, 162, 163syl3anbrc 1344 . . . . . . . . . . . . 13 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑖 ∈ (ℤ𝑗))
165164adantlll 718 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑖 ∈ (ℤ𝑗))
16617ad2antrr 726 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
167 0zd 12086 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ∈ ℤ)
16846ad2antlr 727 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑀 ∈ ℤ)
169 elfzelz 13010 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝑗...𝑖) → 𝑤 ∈ ℤ)
170169adantl 485 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℤ)
171167, 168, 1703jca 1129 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ))
172 0red 10734 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ∈ ℝ)
17368adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑗 ∈ ℝ)
174169zred 12180 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝑗...𝑖) → 𝑤 ∈ ℝ)
175174adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℝ)
176 elfzle1 13013 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...𝑀) → 0 ≤ 𝑗)
177176adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑗)
178 elfzle1 13013 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝑗...𝑖) → 𝑗𝑤)
179178adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑗𝑤)
180172, 173, 175, 177, 179letrd 10887 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑤)
181180adantll 714 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑤)
182174adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℝ)
183 elfzoel2 13140 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
184183zred 12180 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 𝑀 ∈ ℝ)
185184adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑀 ∈ ℝ)
18656adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑖 ∈ ℝ)
187 elfzle2 13014 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ (𝑗...𝑖) → 𝑤𝑖)
188187adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤𝑖)
189 elfzolt2 13150 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑀) → 𝑖 < 𝑀)
190189adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑖 < 𝑀)
191182, 186, 185, 188, 190lelttrd 10888 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 < 𝑀)
192182, 185, 191ltled 10878 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤𝑀)
193192adantlr 715 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤𝑀)
194171, 181, 193jca32 519 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤𝑤𝑀)))
195194adantlll 718 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤𝑤𝑀)))
196 elfz2 13000 . . . . . . . . . . . . . . 15 (𝑤 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤𝑤𝑀)))
197195, 196sylibr 237 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ (0...𝑀))
198166, 197ffvelrnd 6874 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → (𝑄𝑤) ∈ ℝ)
199198adantlr 715 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) ∧ 𝑤 ∈ (𝑗...𝑖)) → (𝑄𝑤) ∈ ℝ)
200 simplll 775 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝜑)
201 0red 10734 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ∈ ℝ)
20268ad2antlr 727 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑗 ∈ ℝ)
203 elfzelz 13010 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ∈ ℤ)
204203zred 12180 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ∈ ℝ)
205204adantl 485 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℝ)
206176ad2antlr 727 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ≤ 𝑗)
207 elfzle1 13013 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑗𝑤)
208207adantl 485 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑗𝑤)
209201, 202, 205, 206, 208letrd 10887 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ≤ 𝑤)
210204adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℝ)
21156adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑖 ∈ ℝ)
212184adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑀 ∈ ℝ)
213 peano2rem 11043 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
214211, 213syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
215 elfzle2 13014 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ≤ (𝑖 − 1))
216215adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ≤ (𝑖 − 1))
217211ltm1d 11662 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
218210, 214, 211, 216, 217lelttrd 10888 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑖)
219189adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑖 < 𝑀)
220210, 211, 212, 218, 219lttrd 10891 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑀)
221220adantlr 715 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑀)
222203adantl 485 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℤ)
223 0zd 12086 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ∈ ℤ)
224183ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑀 ∈ ℤ)
225222, 223, 224, 118syl3anc 1372 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤𝑤 < 𝑀)))
226209, 221, 225mpbir2and 713 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ (0..^𝑀))
227226adantlll 718 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ (0..^𝑀))
228200, 227, 138syl2anc 587 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑄𝑤) ≤ (𝑄‘(𝑤 + 1)))
229228adantlr 715 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑄𝑤) ≤ (𝑄‘(𝑤 + 1)))
230165, 199, 229monoord 13504 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → (𝑄𝑗) ≤ (𝑄𝑖))
2312303adantl3 1169 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → (𝑄𝑗) ≤ (𝑄𝑖))
232159, 231eqbrtrd 5062 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → 𝑋 ≤ (𝑄𝑖))
23325adantr 484 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
234 elfzofz 13156 . . . . . . . . . . . . . 14 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
235234adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
23617, 235ffvelrnd 6874 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
237233, 236lenltd 10876 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 ≤ (𝑄𝑖) ↔ ¬ (𝑄𝑖) < 𝑋))
238237adantr 484 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗𝑖) → (𝑋 ≤ (𝑄𝑖) ↔ ¬ (𝑄𝑖) < 𝑋))
2392383ad2antl1 1186 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → (𝑋 ≤ (𝑄𝑖) ↔ ¬ (𝑄𝑖) < 𝑋))
240232, 239mpbid 235 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → ¬ (𝑄𝑖) < 𝑋)
241155, 240syldan 594 . . . . . . 7 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → ¬ (𝑄𝑖) < 𝑋)
242241intnanrd 493 . . . . . 6 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → ¬ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1))))
243150, 242pm2.61dan 813 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → ¬ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1))))
244243intnand 492 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → ¬ (((𝑄𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ*𝑋 ∈ ℝ*) ∧ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1)))))
245 elioo3g 12862 . . . 4 (𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (((𝑄𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ*𝑋 ∈ ℝ*) ∧ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1)))))
246244, 245sylnibr 332 . . 3 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → ¬ 𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
247246rexlimdv3a 3197 . 2 ((𝜑𝑖 ∈ (0..^𝑀)) → (∃𝑗 ∈ (0...𝑀)(𝑄𝑗) = 𝑋 → ¬ 𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
24815, 247mpd 15 1 ((𝜑𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  wrex 3055  {crab 3058  wss 3853   class class class wbr 5040  cmpt 5120  ran crn 5536   Fn wfn 6344  wf 6345  cfv 6349  (class class class)co 7182  m cmap 8449  cr 10626  0cc0 10627  1c1 10628   + caddc 10630  *cxr 10764   < clt 10765  cle 10766  cmin 10960  cn 11728  cz 12074  cuz 12336  (,)cioo 12833  ...cfz 12993  ..^cfzo 13136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491  ax-cnex 10683  ax-resscn 10684  ax-1cn 10685  ax-icn 10686  ax-addcl 10687  ax-addrcl 10688  ax-mulcl 10689  ax-mulrcl 10690  ax-mulcom 10691  ax-addass 10692  ax-mulass 10693  ax-distr 10694  ax-i2m1 10695  ax-1ne0 10696  ax-1rid 10697  ax-rnegex 10698  ax-rrecex 10699  ax-cnre 10700  ax-pre-lttri 10701  ax-pre-lttrn 10702  ax-pre-ltadd 10703  ax-pre-mulgt0 10704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-om 7612  df-1st 7726  df-2nd 7727  df-wrecs 7988  df-recs 8049  df-rdg 8087  df-er 8332  df-map 8451  df-en 8568  df-dom 8569  df-sdom 8570  df-pnf 10767  df-mnf 10768  df-xr 10769  df-ltxr 10770  df-le 10771  df-sub 10962  df-neg 10963  df-nn 11729  df-n0 11989  df-z 12075  df-uz 12337  df-ioo 12837  df-fz 12994  df-fzo 13137
This theorem is referenced by:  fourierdlem38  43268  fourierdlem74  43303  fourierdlem75  43304  fourierdlem88  43317  fourierdlem103  43332  fourierdlem104  43333
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