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Theorem fourierdlem12 43660
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 43650 . . . . . . . . 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 231 . . . . . . 7 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
87simpld 495 . . . . . 6 (𝜑𝑄 ∈ (ℝ ↑m (0...𝑀)))
9 elmapi 8637 . . . . . 6 (𝑄 ∈ (ℝ ↑m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
10 ffn 6600 . . . . . 6 (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
118, 9, 103syl 18 . . . . 5 (𝜑𝑄 Fn (0...𝑀))
12 fvelrnb 6830 . . . . 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 231 . . 3 (𝜑 → ∃𝑗 ∈ (0...𝑀)(𝑄𝑗) = 𝑋)
1514adantr 481 . 2 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑗 ∈ (0...𝑀)(𝑄𝑗) = 𝑋)
168, 9syl 17 . . . . . . . . . . . 12 (𝜑𝑄:(0...𝑀)⟶ℝ)
1716adantr 481 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
18 fzofzp1 13484 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
1918adantl 482 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
2017, 19ffvelrnd 6962 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
2120adantr 481 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
22213ad2antl1 1184 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
23 frn 6607 . . . . . . . . . . . 12 (𝑄:(0...𝑀)⟶ℝ → ran 𝑄 ⊆ ℝ)
2416, 23syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑄 ⊆ ℝ)
2524, 1sseldd 3922 . . . . . . . . . 10 (𝜑𝑋 ∈ ℝ)
2625ad2antrr 723 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑖 < 𝑗) → 𝑋 ∈ ℝ)
27263ad2antl1 1184 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → 𝑋 ∈ ℝ)
2817ffvelrnda 6961 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑄𝑗) ∈ ℝ)
29283adant3 1131 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) ∈ ℝ)
3029adantr 481 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄𝑗) ∈ ℝ)
31 simpr 485 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
32 elfzoelz 13387 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
3332ad2antrr 723 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 ∈ ℤ)
34 elfzelz 13256 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
3534ad2antlr 724 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
36 zltp1le 12370 . . . . . . . . . . . . . . 15 ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
3733, 35, 36syl2anc 584 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
3831, 37mpbid 231 . . . . . . . . . . . . 13 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ≤ 𝑗)
3933peano2zd 12429 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ∈ ℤ)
40 eluz 12596 . . . . . . . . . . . . . 14 (((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ‘(𝑖 + 1)) ↔ (𝑖 + 1) ≤ 𝑗))
4139, 35, 40syl2anc 584 . . . . . . . . . . . . 13 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑗 ∈ (ℤ‘(𝑖 + 1)) ↔ (𝑖 + 1) ≤ 𝑗))
4238, 41mpbird 256 . . . . . . . . . . . 12 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (ℤ‘(𝑖 + 1)))
4342adantlll 715 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (ℤ‘(𝑖 + 1)))
4417ad2antrr 723 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑄:(0...𝑀)⟶ℝ)
45 0zd 12331 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ∈ ℤ)
46 elfzel2 13254 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
4746ad2antlr 724 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑀 ∈ ℤ)
48 elfzelz 13256 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ∈ ℤ)
4948adantl 482 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℤ)
50 0red 10978 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ∈ ℝ)
5148zred 12426 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ∈ ℝ)
5251adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℝ)
5332peano2zd 12429 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ ℤ)
5453zred 12426 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ ℝ)
5554adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑖 + 1) ∈ ℝ)
5632zred 12426 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
5756adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑖 ∈ ℝ)
58 elfzole1 13395 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 0 ≤ 𝑖)
5958adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑖)
6057ltp1d 11905 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑖 < (𝑖 + 1))
6150, 57, 55, 59, 60lelttrd 11133 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 < (𝑖 + 1))
62 elfzle1 13259 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ((𝑖 + 1)...𝑗) → (𝑖 + 1) ≤ 𝑤)
6362adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑖 + 1) ≤ 𝑤)
6450, 55, 52, 61, 63ltletrd 11135 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 < 𝑤)
6550, 52, 64ltled 11123 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑤)
6665adantlr 712 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑤)
6751adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℝ)
6834zred 12426 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
6968adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑗 ∈ ℝ)
7046zred 12426 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
7170adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑀 ∈ ℝ)
72 elfzle2 13260 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤𝑗)
7372adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤𝑗)
74 elfzle2 13260 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → 𝑗𝑀)
7574adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑗𝑀)
7667, 69, 71, 73, 75letrd 11132 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤𝑀)
7776adantll 711 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤𝑀)
7845, 47, 49, 66, 77elfzd 13247 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ (0...𝑀))
7978adantlll 715 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ (0...𝑀))
8044, 79ffvelrnd 6962 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑄𝑤) ∈ ℝ)
8180adantlr 712 . . . . . . . . . . 11 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑄𝑤) ∈ ℝ)
82 simp-4l 780 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝜑)
83 0red 10978 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℝ)
84 elfzelz 13256 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ∈ ℤ)
8584zred 12426 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ∈ ℝ)
8685adantl 482 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
87 0red 10978 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℝ)
8854adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑖 + 1) ∈ ℝ)
8985adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
90 0red 10978 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 0 ∈ ℝ)
9156ltp1d 11905 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 𝑖 < (𝑖 + 1))
9290, 56, 54, 58, 91lelttrd 11133 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑀) → 0 < (𝑖 + 1))
9392adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < (𝑖 + 1))
94 elfzle1 13259 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → (𝑖 + 1) ≤ 𝑤)
9594adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑖 + 1) ≤ 𝑤)
9687, 88, 89, 93, 95ltletrd 11135 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < 𝑤)
9796adantlr 712 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < 𝑤)
9883, 86, 97ltled 11123 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
9998adantlll 715 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
10099adantlr 712 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
10185adantl 482 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
102 peano2rem 11288 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℝ → (𝑗 − 1) ∈ ℝ)
10368, 102syl 17 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℝ)
104103adantr 481 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑗 − 1) ∈ ℝ)
10570adantr 481 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑀 ∈ ℝ)
106 elfzle2 13260 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ≤ (𝑗 − 1))
107106adantl 482 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ≤ (𝑗 − 1))
108 zlem1lt 12372 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗𝑀 ↔ (𝑗 − 1) < 𝑀))
10934, 46, 108syl2anc 584 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → (𝑗𝑀 ↔ (𝑗 − 1) < 𝑀))
11074, 109mpbid 231 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑀)
111110adantr 481 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑗 − 1) < 𝑀)
112101, 104, 105, 107, 111lelttrd 11133 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
113112adantlr 712 . . . . . . . . . . . . . 14 (((𝑗 ∈ (0...𝑀) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
114113adantlll 715 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
11584adantl 482 . . . . . . . . . . . . . 14 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℤ)
116 0zd 12331 . . . . . . . . . . . . . 14 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℤ)
11746ad3antlr 728 . . . . . . . . . . . . . 14 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑀 ∈ ℤ)
118 elfzo 13389 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤𝑤 < 𝑀)))
119115, 116, 117, 118syl3anc 1370 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤𝑤 < 𝑀)))
120100, 114, 119mpbir2and 710 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ (0..^𝑀))
12116adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
122 elfzofz 13403 . . . . . . . . . . . . . . 15 (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ (0...𝑀))
123122adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (0..^𝑀)) → 𝑤 ∈ (0...𝑀))
124121, 123ffvelrnd 6962 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄𝑤) ∈ ℝ)
125 fzofzp1 13484 . . . . . . . . . . . . . . 15 (𝑤 ∈ (0..^𝑀) → (𝑤 + 1) ∈ (0...𝑀))
126125adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑤 + 1) ∈ (0...𝑀))
127121, 126ffvelrnd 6962 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄‘(𝑤 + 1)) ∈ ℝ)
128 eleq1w 2821 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑤 → (𝑖 ∈ (0..^𝑀) ↔ 𝑤 ∈ (0..^𝑀)))
129128anbi2d 629 . . . . . . . . . . . . . . 15 (𝑖 = 𝑤 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑤 ∈ (0..^𝑀))))
130 fveq2 6774 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑤 → (𝑄𝑖) = (𝑄𝑤))
131 oveq1 7282 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑤 → (𝑖 + 1) = (𝑤 + 1))
132131fveq2d 6778 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑤 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑤 + 1)))
133130, 132breq12d 5087 . . . . . . . . . . . . . . 15 (𝑖 = 𝑤 → ((𝑄𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄𝑤) < (𝑄‘(𝑤 + 1))))
134129, 133imbi12d 345 . . . . . . . . . . . . . 14 (𝑖 = 𝑤 → (((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄𝑤) < (𝑄‘(𝑤 + 1)))))
1357simprrd 771 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
136135r19.21bi 3134 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
137134, 136chvarvv 2002 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄𝑤) < (𝑄‘(𝑤 + 1)))
138124, 127, 137ltled 11123 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄𝑤) ≤ (𝑄‘(𝑤 + 1)))
13982, 120, 138syl2anc 584 . . . . . . . . . . 11 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑄𝑤) ≤ (𝑄‘(𝑤 + 1)))
14043, 81, 139monoord 13753 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ (𝑄𝑗))
1411403adantl3 1167 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ (𝑄𝑗))
14216ffvelrnda 6961 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑄𝑗) ∈ ℝ)
1431423adant3 1131 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) ∈ ℝ)
144 simp3 1137 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) = 𝑋)
145143, 144eqled 11078 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) ≤ 𝑋)
1461453adant1r 1176 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) ≤ 𝑋)
147146adantr 481 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄𝑗) ≤ 𝑋)
14822, 30, 27, 141, 147letrd 11132 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ 𝑋)
14922, 27, 148lensymd 11126 . . . . . . 7 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → ¬ 𝑋 < (𝑄‘(𝑖 + 1)))
150149intnand 489 . . . . . 6 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → ¬ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1))))
15168ad2antlr 724 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ∈ ℝ)
15256ad3antlr 728 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑖 ∈ ℝ)
153 simpr 485 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → ¬ 𝑖 < 𝑗)
154151, 152, 153nltled 11125 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑗𝑖)
1551543adantl3 1167 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → 𝑗𝑖)
156 eqcom 2745 . . . . . . . . . . . . 13 ((𝑄𝑗) = 𝑋𝑋 = (𝑄𝑗))
157156biimpi 215 . . . . . . . . . . . 12 ((𝑄𝑗) = 𝑋𝑋 = (𝑄𝑗))
158157adantr 481 . . . . . . . . . . 11 (((𝑄𝑗) = 𝑋𝑗𝑖) → 𝑋 = (𝑄𝑗))
1591583ad2antl3 1186 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → 𝑋 = (𝑄𝑗))
16034ad2antlr 724 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑗 ∈ ℤ)
16132ad2antrr 723 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑖 ∈ ℤ)
162 simpr 485 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑗𝑖)
163 eluz2 12588 . . . . . . . . . . . . . 14 (𝑖 ∈ (ℤ𝑗) ↔ (𝑗 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗𝑖))
164160, 161, 162, 163syl3anbrc 1342 . . . . . . . . . . . . 13 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑖 ∈ (ℤ𝑗))
165164adantlll 715 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑖 ∈ (ℤ𝑗))
16617ad2antrr 723 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
167 0zd 12331 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ∈ ℤ)
16846ad2antlr 724 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑀 ∈ ℤ)
169 elfzelz 13256 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝑗...𝑖) → 𝑤 ∈ ℤ)
170169adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℤ)
171167, 168, 1703jca 1127 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ))
172 0red 10978 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ∈ ℝ)
17368adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑗 ∈ ℝ)
174169zred 12426 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝑗...𝑖) → 𝑤 ∈ ℝ)
175174adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℝ)
176 elfzle1 13259 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...𝑀) → 0 ≤ 𝑗)
177176adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑗)
178 elfzle1 13259 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝑗...𝑖) → 𝑗𝑤)
179178adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑗𝑤)
180172, 173, 175, 177, 179letrd 11132 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑤)
181180adantll 711 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑤)
182174adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℝ)
183 elfzoel2 13386 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
184183zred 12426 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 𝑀 ∈ ℝ)
185184adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑀 ∈ ℝ)
18656adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑖 ∈ ℝ)
187 elfzle2 13260 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ (𝑗...𝑖) → 𝑤𝑖)
188187adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤𝑖)
189 elfzolt2 13396 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑀) → 𝑖 < 𝑀)
190189adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑖 < 𝑀)
191182, 186, 185, 188, 190lelttrd 11133 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 < 𝑀)
192182, 185, 191ltled 11123 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤𝑀)
193192adantlr 712 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤𝑀)
194171, 181, 193jca32 516 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤𝑤𝑀)))
195194adantlll 715 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤𝑤𝑀)))
196 elfz2 13246 . . . . . . . . . . . . . . 15 (𝑤 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤𝑤𝑀)))
197195, 196sylibr 233 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ (0...𝑀))
198166, 197ffvelrnd 6962 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → (𝑄𝑤) ∈ ℝ)
199198adantlr 712 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) ∧ 𝑤 ∈ (𝑗...𝑖)) → (𝑄𝑤) ∈ ℝ)
200 simplll 772 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝜑)
201 0red 10978 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ∈ ℝ)
20268ad2antlr 724 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑗 ∈ ℝ)
203 elfzelz 13256 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ∈ ℤ)
204203zred 12426 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ∈ ℝ)
205204adantl 482 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℝ)
206176ad2antlr 724 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ≤ 𝑗)
207 elfzle1 13259 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑗𝑤)
208207adantl 482 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑗𝑤)
209201, 202, 205, 206, 208letrd 11132 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ≤ 𝑤)
210204adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℝ)
21156adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑖 ∈ ℝ)
212184adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑀 ∈ ℝ)
213 peano2rem 11288 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
214211, 213syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
215 elfzle2 13260 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ≤ (𝑖 − 1))
216215adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ≤ (𝑖 − 1))
217211ltm1d 11907 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
218210, 214, 211, 216, 217lelttrd 11133 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑖)
219189adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑖 < 𝑀)
220210, 211, 212, 218, 219lttrd 11136 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑀)
221220adantlr 712 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑀)
222203adantl 482 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℤ)
223 0zd 12331 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ∈ ℤ)
224183ad2antrr 723 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑀 ∈ ℤ)
225222, 223, 224, 118syl3anc 1370 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤𝑤 < 𝑀)))
226209, 221, 225mpbir2and 710 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ (0..^𝑀))
227226adantlll 715 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ (0..^𝑀))
228200, 227, 138syl2anc 584 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑄𝑤) ≤ (𝑄‘(𝑤 + 1)))
229228adantlr 712 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑄𝑤) ≤ (𝑄‘(𝑤 + 1)))
230165, 199, 229monoord 13753 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → (𝑄𝑗) ≤ (𝑄𝑖))
2312303adantl3 1167 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → (𝑄𝑗) ≤ (𝑄𝑖))
232159, 231eqbrtrd 5096 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → 𝑋 ≤ (𝑄𝑖))
23325adantr 481 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
234 elfzofz 13403 . . . . . . . . . . . . . 14 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
235234adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
23617, 235ffvelrnd 6962 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
237233, 236lenltd 11121 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 ≤ (𝑄𝑖) ↔ ¬ (𝑄𝑖) < 𝑋))
238237adantr 481 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗𝑖) → (𝑋 ≤ (𝑄𝑖) ↔ ¬ (𝑄𝑖) < 𝑋))
2392383ad2antl1 1184 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → (𝑋 ≤ (𝑄𝑖) ↔ ¬ (𝑄𝑖) < 𝑋))
240232, 239mpbid 231 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → ¬ (𝑄𝑖) < 𝑋)
241155, 240syldan 591 . . . . . . 7 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → ¬ (𝑄𝑖) < 𝑋)
242241intnanrd 490 . . . . . 6 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → ¬ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1))))
243150, 242pm2.61dan 810 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → ¬ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1))))
244243intnand 489 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → ¬ (((𝑄𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ*𝑋 ∈ ℝ*) ∧ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1)))))
245 elioo3g 13108 . . . 4 (𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (((𝑄𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ*𝑋 ∈ ℝ*) ∧ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1)))))
246244, 245sylnibr 329 . . 3 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → ¬ 𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
247246rexlimdv3a 3215 . 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 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065  {crab 3068  wss 3887   class class class wbr 5074  cmpt 5157  ran crn 5590   Fn wfn 6428  wf 6429  cfv 6433  (class class class)co 7275  m cmap 8615  cr 10870  0cc0 10871  1c1 10872   + caddc 10874  *cxr 11008   < clt 11009  cle 11010  cmin 11205  cn 11973  cz 12319  cuz 12582  (,)cioo 13079  ...cfz 13239  ..^cfzo 13382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-ioo 13083  df-fz 13240  df-fzo 13383
This theorem is referenced by:  fourierdlem38  43686  fourierdlem74  43721  fourierdlem75  43722  fourierdlem88  43735  fourierdlem103  43750  fourierdlem104  43751
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