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Theorem fourierdlem12 40973
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 40963 . . . . . . . . 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 223 . . . . . . 7 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
87simpld 488 . . . . . 6 (𝜑𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)))
9 elmapi 8082 . . . . . 6 (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
10 ffn 6223 . . . . . 6 (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
118, 9, 103syl 18 . . . . 5 (𝜑𝑄 Fn (0...𝑀))
12 fvelrnb 6432 . . . . 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 223 . . 3 (𝜑 → ∃𝑗 ∈ (0...𝑀)(𝑄𝑗) = 𝑋)
1514adantr 472 . 2 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑗 ∈ (0...𝑀)(𝑄𝑗) = 𝑋)
168, 9syl 17 . . . . . . . . . . . 12 (𝜑𝑄:(0...𝑀)⟶ℝ)
1716adantr 472 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
18 fzofzp1 12773 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
1918adantl 473 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
2017, 19ffvelrnd 6550 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
2120adantr 472 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
22213ad2antl1 1236 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
23 frn 6229 . . . . . . . . . . . 12 (𝑄:(0...𝑀)⟶ℝ → ran 𝑄 ⊆ ℝ)
2416, 23syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑄 ⊆ ℝ)
2524, 1sseldd 3762 . . . . . . . . . 10 (𝜑𝑋 ∈ ℝ)
2625ad2antrr 717 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑖 < 𝑗) → 𝑋 ∈ ℝ)
27263ad2antl1 1236 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → 𝑋 ∈ ℝ)
2817ffvelrnda 6549 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑄𝑗) ∈ ℝ)
29283adant3 1162 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) ∈ ℝ)
3029adantr 472 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄𝑗) ∈ ℝ)
31 simpr 477 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
32 elfzoelz 12678 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
3332ad2antrr 717 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 ∈ ℤ)
34 elfzelz 12549 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
3534ad2antlr 718 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
36 zltp1le 11674 . . . . . . . . . . . . . . 15 ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
3733, 35, 36syl2anc 579 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
3831, 37mpbid 223 . . . . . . . . . . . . 13 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ≤ 𝑗)
3933peano2zd 11732 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ∈ ℤ)
40 eluz 11900 . . . . . . . . . . . . . 14 (((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ‘(𝑖 + 1)) ↔ (𝑖 + 1) ≤ 𝑗))
4139, 35, 40syl2anc 579 . . . . . . . . . . . . 13 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑗 ∈ (ℤ‘(𝑖 + 1)) ↔ (𝑖 + 1) ≤ 𝑗))
4238, 41mpbird 248 . . . . . . . . . . . 12 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (ℤ‘(𝑖 + 1)))
4342adantlll 709 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (ℤ‘(𝑖 + 1)))
4417ad2antrr 717 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑄:(0...𝑀)⟶ℝ)
45 0red 10297 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ∈ ℝ)
46 elfzelz 12549 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ∈ ℤ)
4746zred 11729 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ∈ ℝ)
4847adantl 473 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℝ)
4932peano2zd 11732 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ ℤ)
5049zred 11729 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ ℝ)
5150adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑖 + 1) ∈ ℝ)
5232zred 11729 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
5352adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑖 ∈ ℝ)
54 elfzole1 12686 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 0 ≤ 𝑖)
5554adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑖)
5653ltp1d 11208 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑖 < (𝑖 + 1))
5745, 53, 51, 55, 56lelttrd 10449 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 < (𝑖 + 1))
58 elfzle1 12551 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ((𝑖 + 1)...𝑗) → (𝑖 + 1) ≤ 𝑤)
5958adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑖 + 1) ≤ 𝑤)
6045, 51, 48, 57, 59ltletrd 10451 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 < 𝑤)
6145, 48, 60ltled 10439 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑤)
6261adantlr 706 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑤)
6347adantl 473 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℝ)
6434zred 11729 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
6564adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑗 ∈ ℝ)
66 elfzel2 12547 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
6766zred 11729 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
6867adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑀 ∈ ℝ)
69 elfzle2 12552 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤𝑗)
7069adantl 473 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤𝑗)
71 elfzle2 12552 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → 𝑗𝑀)
7271adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑗𝑀)
7363, 65, 68, 70, 72letrd 10448 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤𝑀)
7473adantll 705 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤𝑀)
7546adantl 473 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℤ)
76 0zd 11636 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ∈ ℤ)
7766ad2antlr 718 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑀 ∈ ℤ)
78 elfz 12539 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑤 ∈ (0...𝑀) ↔ (0 ≤ 𝑤𝑤𝑀)))
7975, 76, 77, 78syl3anc 1490 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑤 ∈ (0...𝑀) ↔ (0 ≤ 𝑤𝑤𝑀)))
8062, 74, 79mpbir2and 704 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ (0...𝑀))
8180adantlll 709 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ (0...𝑀))
8244, 81ffvelrnd 6550 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑄𝑤) ∈ ℝ)
8382adantlr 706 . . . . . . . . . . 11 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑄𝑤) ∈ ℝ)
84 simp-4l 801 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝜑)
85 0red 10297 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℝ)
86 elfzelz 12549 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ∈ ℤ)
8786zred 11729 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ∈ ℝ)
8887adantl 473 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
89 0red 10297 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℝ)
9050adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑖 + 1) ∈ ℝ)
9187adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
92 0red 10297 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 0 ∈ ℝ)
9352ltp1d 11208 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 𝑖 < (𝑖 + 1))
9492, 52, 50, 54, 93lelttrd 10449 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑀) → 0 < (𝑖 + 1))
9594adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < (𝑖 + 1))
96 elfzle1 12551 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → (𝑖 + 1) ≤ 𝑤)
9796adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑖 + 1) ≤ 𝑤)
9889, 90, 91, 95, 97ltletrd 10451 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < 𝑤)
9998adantlr 706 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < 𝑤)
10085, 88, 99ltled 10439 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
101100adantlll 709 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
102101adantlr 706 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
10387adantl 473 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
104 peano2rem 10602 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℝ → (𝑗 − 1) ∈ ℝ)
10564, 104syl 17 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℝ)
106105adantr 472 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑗 − 1) ∈ ℝ)
10767adantr 472 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑀 ∈ ℝ)
108 elfzle2 12552 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ≤ (𝑗 − 1))
109108adantl 473 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ≤ (𝑗 − 1))
110 zlem1lt 11676 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗𝑀 ↔ (𝑗 − 1) < 𝑀))
11134, 66, 110syl2anc 579 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → (𝑗𝑀 ↔ (𝑗 − 1) < 𝑀))
11271, 111mpbid 223 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑀)
113112adantr 472 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑗 − 1) < 𝑀)
114103, 106, 107, 109, 113lelttrd 10449 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
115114adantlr 706 . . . . . . . . . . . . . 14 (((𝑗 ∈ (0...𝑀) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
116115adantlll 709 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
11786adantl 473 . . . . . . . . . . . . . 14 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℤ)
118 0zd 11636 . . . . . . . . . . . . . 14 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℤ)
11966ad3antlr 722 . . . . . . . . . . . . . 14 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑀 ∈ ℤ)
120 elfzo 12680 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤𝑤 < 𝑀)))
121117, 118, 119, 120syl3anc 1490 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤𝑤 < 𝑀)))
122102, 116, 121mpbir2and 704 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ (0..^𝑀))
12316adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
124 elfzofz 12693 . . . . . . . . . . . . . . 15 (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ (0...𝑀))
125124adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (0..^𝑀)) → 𝑤 ∈ (0...𝑀))
126123, 125ffvelrnd 6550 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄𝑤) ∈ ℝ)
127 fzofzp1 12773 . . . . . . . . . . . . . . 15 (𝑤 ∈ (0..^𝑀) → (𝑤 + 1) ∈ (0...𝑀))
128127adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑤 + 1) ∈ (0...𝑀))
129123, 128ffvelrnd 6550 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄‘(𝑤 + 1)) ∈ ℝ)
130 eleq1w 2827 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑤 → (𝑖 ∈ (0..^𝑀) ↔ 𝑤 ∈ (0..^𝑀)))
131130anbi2d 622 . . . . . . . . . . . . . . 15 (𝑖 = 𝑤 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑤 ∈ (0..^𝑀))))
132 fveq2 6375 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑤 → (𝑄𝑖) = (𝑄𝑤))
133 oveq1 6849 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑤 → (𝑖 + 1) = (𝑤 + 1))
134133fveq2d 6379 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑤 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑤 + 1)))
135132, 134breq12d 4822 . . . . . . . . . . . . . . 15 (𝑖 = 𝑤 → ((𝑄𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄𝑤) < (𝑄‘(𝑤 + 1))))
136131, 135imbi12d 335 . . . . . . . . . . . . . 14 (𝑖 = 𝑤 → (((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄𝑤) < (𝑄‘(𝑤 + 1)))))
1377simprrd 790 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
138137r19.21bi 3079 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
139136, 138chvarv 2369 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄𝑤) < (𝑄‘(𝑤 + 1)))
140126, 129, 139ltled 10439 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (0..^𝑀)) → (𝑄𝑤) ≤ (𝑄‘(𝑤 + 1)))
14184, 122, 140syl2anc 579 . . . . . . . . . . 11 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑄𝑤) ≤ (𝑄‘(𝑤 + 1)))
14243, 83, 141monoord 13038 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ (𝑄𝑗))
1431423adantl3 1209 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ (𝑄𝑗))
14416ffvelrnda 6549 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑄𝑗) ∈ ℝ)
1451443adant3 1162 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) ∈ ℝ)
146 simp3 1168 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) = 𝑋)
147145, 146eqled 10394 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) ≤ 𝑋)
1481473adant1r 1223 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → (𝑄𝑗) ≤ 𝑋)
149148adantr 472 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄𝑗) ≤ 𝑋)
15022, 30, 27, 143, 149letrd 10448 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ 𝑋)
15122, 27, 150lensymd 10442 . . . . . . 7 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → ¬ 𝑋 < (𝑄‘(𝑖 + 1)))
152151intnand 482 . . . . . 6 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → ¬ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1))))
15364ad2antlr 718 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ∈ ℝ)
15452ad3antlr 722 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑖 ∈ ℝ)
155 simpr 477 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → ¬ 𝑖 < 𝑗)
156153, 154, 155nltled 10441 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑗𝑖)
1571563adantl3 1209 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → 𝑗𝑖)
158 eqcom 2772 . . . . . . . . . . . . 13 ((𝑄𝑗) = 𝑋𝑋 = (𝑄𝑗))
159158biimpi 207 . . . . . . . . . . . 12 ((𝑄𝑗) = 𝑋𝑋 = (𝑄𝑗))
160159adantr 472 . . . . . . . . . . 11 (((𝑄𝑗) = 𝑋𝑗𝑖) → 𝑋 = (𝑄𝑗))
1611603ad2antl3 1238 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → 𝑋 = (𝑄𝑗))
16234ad2antlr 718 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑗 ∈ ℤ)
16332ad2antrr 717 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑖 ∈ ℤ)
164 simpr 477 . . . . . . . . . . . . . 14 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑗𝑖)
165 eluz2 11892 . . . . . . . . . . . . . 14 (𝑖 ∈ (ℤ𝑗) ↔ (𝑗 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗𝑖))
166162, 163, 164, 165syl3anbrc 1443 . . . . . . . . . . . . 13 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑖 ∈ (ℤ𝑗))
167166adantlll 709 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → 𝑖 ∈ (ℤ𝑗))
16817ad2antrr 717 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
169 0zd 11636 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ∈ ℤ)
17066ad2antlr 718 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑀 ∈ ℤ)
171 elfzelz 12549 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝑗...𝑖) → 𝑤 ∈ ℤ)
172171adantl 473 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℤ)
173169, 170, 1723jca 1158 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ))
174 0red 10297 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ∈ ℝ)
17564adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑗 ∈ ℝ)
176171zred 11729 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝑗...𝑖) → 𝑤 ∈ ℝ)
177176adantl 473 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℝ)
178 elfzle1 12551 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...𝑀) → 0 ≤ 𝑗)
179178adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑗)
180 elfzle1 12551 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝑗...𝑖) → 𝑗𝑤)
181180adantl 473 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑗𝑤)
182174, 175, 177, 179, 181letrd 10448 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑤)
183182adantll 705 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑤)
184176adantl 473 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℝ)
185 elfzoel2 12677 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
186185zred 11729 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑀) → 𝑀 ∈ ℝ)
187186adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑀 ∈ ℝ)
18852adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑖 ∈ ℝ)
189 elfzle2 12552 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ (𝑗...𝑖) → 𝑤𝑖)
190189adantl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤𝑖)
191 elfzolt2 12687 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑀) → 𝑖 < 𝑀)
192191adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑖 < 𝑀)
193184, 188, 187, 190, 192lelttrd 10449 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 < 𝑀)
194184, 187, 193ltled 10439 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤𝑀)
195194adantlr 706 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤𝑀)
196173, 183, 195jca32 511 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤𝑤𝑀)))
197196adantlll 709 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤𝑤𝑀)))
198 elfz2 12540 . . . . . . . . . . . . . . 15 (𝑤 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤𝑤𝑀)))
199197, 198sylibr 225 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ (0...𝑀))
200168, 199ffvelrnd 6550 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → (𝑄𝑤) ∈ ℝ)
201200adantlr 706 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) ∧ 𝑤 ∈ (𝑗...𝑖)) → (𝑄𝑤) ∈ ℝ)
202 simplll 791 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝜑)
203 0red 10297 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ∈ ℝ)
20464ad2antlr 718 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑗 ∈ ℝ)
205 elfzelz 12549 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ∈ ℤ)
206205zred 11729 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ∈ ℝ)
207206adantl 473 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℝ)
208178ad2antlr 718 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ≤ 𝑗)
209 elfzle1 12551 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑗𝑤)
210209adantl 473 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑗𝑤)
211203, 204, 207, 208, 210letrd 10448 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ≤ 𝑤)
212206adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℝ)
21352adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑖 ∈ ℝ)
214186adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑀 ∈ ℝ)
215 peano2rem 10602 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
216213, 215syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
217 elfzle2 12552 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ≤ (𝑖 − 1))
218217adantl 473 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ≤ (𝑖 − 1))
219213ltm1d 11210 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
220212, 216, 213, 218, 219lelttrd 10449 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑖)
221191adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑖 < 𝑀)
222212, 213, 214, 220, 221lttrd 10452 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑀)
223222adantlr 706 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑀)
224205adantl 473 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℤ)
225 0zd 11636 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ∈ ℤ)
226185ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑀 ∈ ℤ)
227224, 225, 226, 120syl3anc 1490 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤𝑤 < 𝑀)))
228211, 223, 227mpbir2and 704 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ (0..^𝑀))
229228adantlll 709 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ (0..^𝑀))
230202, 229, 140syl2anc 579 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑄𝑤) ≤ (𝑄‘(𝑤 + 1)))
231230adantlr 706 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑄𝑤) ≤ (𝑄‘(𝑤 + 1)))
232167, 201, 231monoord 13038 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗𝑖) → (𝑄𝑗) ≤ (𝑄𝑖))
2332323adantl3 1209 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → (𝑄𝑗) ≤ (𝑄𝑖))
234161, 233eqbrtrd 4831 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → 𝑋 ≤ (𝑄𝑖))
23525adantr 472 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
236 elfzofz 12693 . . . . . . . . . . . . . 14 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
237236adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
23817, 237ffvelrnd 6550 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
239235, 238lenltd 10437 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 ≤ (𝑄𝑖) ↔ ¬ (𝑄𝑖) < 𝑋))
240239adantr 472 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗𝑖) → (𝑋 ≤ (𝑄𝑖) ↔ ¬ (𝑄𝑖) < 𝑋))
2412403ad2antl1 1236 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → (𝑋 ≤ (𝑄𝑖) ↔ ¬ (𝑄𝑖) < 𝑋))
242234, 241mpbid 223 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ 𝑗𝑖) → ¬ (𝑄𝑖) < 𝑋)
243157, 242syldan 585 . . . . . . 7 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → ¬ (𝑄𝑖) < 𝑋)
244243intnanrd 483 . . . . . 6 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → ¬ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1))))
245152, 244pm2.61dan 847 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → ¬ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1))))
246245intnand 482 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → ¬ (((𝑄𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ*𝑋 ∈ ℝ*) ∧ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1)))))
247 elioo3g 12406 . . . 4 (𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (((𝑄𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ*𝑋 ∈ ℝ*) ∧ ((𝑄𝑖) < 𝑋𝑋 < (𝑄‘(𝑖 + 1)))))
248246, 247sylnibr 320 . . 3 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄𝑗) = 𝑋) → ¬ 𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
249248rexlimdv3a 3180 . 2 ((𝜑𝑖 ∈ (0..^𝑀)) → (∃𝑗 ∈ (0...𝑀)(𝑄𝑗) = 𝑋 → ¬ 𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
25015, 249mpd 15 1 ((𝜑𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  {crab 3059  wss 3732   class class class wbr 4809  cmpt 4888  ran crn 5278   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  cr 10188  0cc0 10189  1c1 10190   + caddc 10192  *cxr 10327   < clt 10328  cle 10329  cmin 10520  cn 11274  cz 11624  cuz 11886  (,)cioo 12377  ...cfz 12533  ..^cfzo 12673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-ioo 12381  df-fz 12534  df-fzo 12674
This theorem is referenced by:  fourierdlem38  40999  fourierdlem74  41034  fourierdlem75  41035  fourierdlem88  41048  fourierdlem103  41063  fourierdlem104  41064
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