Theorem fourierdlem12 42761

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 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem12.2	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem12.3	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem12.4	⊢ (𝜑 → 𝑋 ∈ ran 𝑄)

Assertion

Ref	Expression
fourierdlem12	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑄‘𝑖)(,)(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.

Step	Hyp	Ref	Expression
1		fourierdlem12.4	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝑄)
2		fourierdlem12.3	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
3		fourierdlem12.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4		fourierdlem12.1	. . . . . . . . . 10 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
5	4	fourierdlem2 42751	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
7	2, 6	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
8	7	simpld 498	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑m (0...𝑀)))
9		elmapi 8411	. . . . . 6 ⊢ (𝑄 ∈ (ℝ ↑m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
10		ffn 6487	. . . . . 6 ⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
11	8, 9, 10	3syl 18	. . . . 5 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
12		fvelrnb 6701	. . . . 5 ⊢ (𝑄 Fn (0...𝑀) → (𝑋 ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑋))
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑋))
14	1, 13	mpbid 235	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑋)
15	14	adantr 484	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑋)
16	8, 9	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
17	16	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
18		fzofzp1 13129	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
19	18	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
20	17, 19	ffvelrnd 6829	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
21	20	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
22	21	3ad2antl1 1182	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
23		frn 6493	. . . . . . . . . . . 12 ⊢ (𝑄:(0...𝑀)⟶ℝ → ran 𝑄 ⊆ ℝ)
24	16, 23	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑄 ⊆ ℝ)
25	24, 1	sseldd 3916	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ)
26	25	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑖 < 𝑗) → 𝑋 ∈ ℝ)
27	26	3ad2antl1 1182	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → 𝑋 ∈ ℝ)
28	17	ffvelrnda 6828	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
29	28	3adant3 1129	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) ∈ ℝ)
30	29	adantr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘𝑗) ∈ ℝ)
31		simpr 488	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
32		elfzoelz 13033	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
33	32	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 ∈ ℤ)
34		elfzelz 12902	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
35	34	ad2antlr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
36		zltp1le 12020	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
37	33, 35, 36	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
38	31, 37	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ≤ 𝑗)
39	33	peano2zd 12078	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ∈ ℤ)
40		eluz 12245	. . . . . . . . . . . . . 14 ⊢ (((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ≥‘(𝑖 + 1)) ↔ (𝑖 + 1) ≤ 𝑗))
41	39, 35, 40	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑗 ∈ (ℤ≥‘(𝑖 + 1)) ↔ (𝑖 + 1) ≤ 𝑗))
42	38, 41	mpbird 260	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (ℤ≥‘(𝑖 + 1)))
43	42	adantlll 717	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (ℤ≥‘(𝑖 + 1)))
44	17	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑄:(0...𝑀)⟶ℝ)
45		0red 10633	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ∈ ℝ)
46		elfzelz 12902	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ∈ ℤ)
47	46	zred 12075	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ∈ ℝ)
48	47	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℝ)
49	32	peano2zd 12078	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ ℤ)
50	49	zred 12075	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ ℝ)
51	50	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑖 + 1) ∈ ℝ)
52	32	zred 12075	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
53	52	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑖 ∈ ℝ)
54		elfzole1 13041	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 0 ≤ 𝑖)
55	54	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑖)
56	53	ltp1d 11559	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑖 < (𝑖 + 1))
57	45, 53, 51, 55, 56	lelttrd 10787	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 < (𝑖 + 1))
58		elfzle1 12905	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ((𝑖 + 1)...𝑗) → (𝑖 + 1) ≤ 𝑤)
59	58	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑖 + 1) ≤ 𝑤)
60	45, 51, 48, 57, 59	ltletrd 10789	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 < 𝑤)
61	45, 48, 60	ltled 10777	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑤)
62	61	adantlr 714	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑤)
63	47	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℝ)
64	34	zred 12075	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
65	64	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑗 ∈ ℝ)
66		elfzel2 12900	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
67	66	zred 12075	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
68	67	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑀 ∈ ℝ)
69		elfzle2 12906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ≤ 𝑗)
70	69	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ≤ 𝑗)
71		elfzle2 12906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ≤ 𝑀)
72	71	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑗 ≤ 𝑀)
73	63, 65, 68, 70, 72	letrd 10786	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ≤ 𝑀)
74	73	adantll 713	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ≤ 𝑀)
75	46	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℤ)
76		0zd 11981	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ∈ ℤ)
77	66	ad2antlr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑀 ∈ ℤ)
78		elfz 12891	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑤 ∈ (0...𝑀) ↔ (0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀)))
79	75, 76, 77, 78	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑤 ∈ (0...𝑀) ↔ (0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀)))
80	62, 74, 79	mpbir2and 712	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ (0...𝑀))
81	80	adantlll 717	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ (0...𝑀))
82	44, 81	ffvelrnd 6829	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑄‘𝑤) ∈ ℝ)
83	82	adantlr 714	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑄‘𝑤) ∈ ℝ)
84		simp-4l 782	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝜑)
85		0red 10633	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℝ)
86		elfzelz 12902	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ∈ ℤ)
87	86	zred 12075	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ∈ ℝ)
88	87	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
89		0red 10633	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℝ)
90	50	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑖 + 1) ∈ ℝ)
91	87	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
92		0red 10633	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 0 ∈ ℝ)
93	52	ltp1d 11559	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 < (𝑖 + 1))
94	92, 52, 50, 54, 93	lelttrd 10787	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑀) → 0 < (𝑖 + 1))
95	94	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < (𝑖 + 1))
96		elfzle1 12905	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → (𝑖 + 1) ≤ 𝑤)
97	96	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑖 + 1) ≤ 𝑤)
98	89, 90, 91, 95, 97	ltletrd 10789	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < 𝑤)
99	98	adantlr 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < 𝑤)
100	85, 88, 99	ltled 10777	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
101	100	adantlll 717	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
102	101	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
103	87	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
104		peano2rem 10942	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℝ → (𝑗 − 1) ∈ ℝ)
105	64, 104	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℝ)
106	105	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑗 − 1) ∈ ℝ)
107	67	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑀 ∈ ℝ)
108		elfzle2 12906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ≤ (𝑗 − 1))
109	108	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ≤ (𝑗 − 1))
110		zlem1lt 12022	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ≤ 𝑀 ↔ (𝑗 − 1) < 𝑀))
111	34, 66, 110	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 ≤ 𝑀 ↔ (𝑗 − 1) < 𝑀))
112	71, 111	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑀)
113	112	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑗 − 1) < 𝑀)
114	103, 106, 107, 109, 113	lelttrd 10787	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
115	114	adantlr 714	. . . . . . . . . . . . . 14 ⊢ (((𝑗 ∈ (0...𝑀) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
116	115	adantlll 717	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
117	86	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℤ)
118		0zd 11981	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℤ)
119	66	ad3antlr 730	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑀 ∈ ℤ)
120		elfzo 13035	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤 ∧ 𝑤 < 𝑀)))
121	117, 118, 119, 120	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤 ∧ 𝑤 < 𝑀)))
122	102, 116, 121	mpbir2and 712	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ (0..^𝑀))
123	16	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
124		elfzofz 13048	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ (0...𝑀))
125	124	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → 𝑤 ∈ (0...𝑀))
126	123, 125	ffvelrnd 6829	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘𝑤) ∈ ℝ)
127		fzofzp1 13129	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (0..^𝑀) → (𝑤 + 1) ∈ (0...𝑀))
128	127	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑤 + 1) ∈ (0...𝑀))
129	123, 128	ffvelrnd 6829	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘(𝑤 + 1)) ∈ ℝ)
130		eleq1w 2872	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑤 → (𝑖 ∈ (0..^𝑀) ↔ 𝑤 ∈ (0..^𝑀)))
131	130	anbi2d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑤 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑤 ∈ (0..^𝑀))))
132		fveq2 6645	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑤 → (𝑄‘𝑖) = (𝑄‘𝑤))
133		oveq1 7142	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑤 → (𝑖 + 1) = (𝑤 + 1))
134	133	fveq2d 6649	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑤 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑤 + 1)))
135	132, 134	breq12d 5043	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑤 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘𝑤) < (𝑄‘(𝑤 + 1))))
136	131, 135	imbi12d 348	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑤 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘𝑤) < (𝑄‘(𝑤 + 1)))))
137	7	simprrd 773	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
138	137	r19.21bi 3173	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
139	136, 138	chvarvv 2005	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘𝑤) < (𝑄‘(𝑤 + 1)))
140	126, 129, 139	ltled 10777	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘𝑤) ≤ (𝑄‘(𝑤 + 1)))
141	84, 122, 140	syl2anc 587	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑄‘𝑤) ≤ (𝑄‘(𝑤 + 1)))
142	43, 83, 141	monoord 13396	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ (𝑄‘𝑗))
143	142	3adantl3 1165	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ (𝑄‘𝑗))
144	16	ffvelrnda 6828	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
145	144	3adant3 1129	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) ∈ ℝ)
146		simp3 1135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) = 𝑋)
147	145, 146	eqled 10732	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) ≤ 𝑋)
148	147	3adant1r 1174	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) ≤ 𝑋)
149	148	adantr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘𝑗) ≤ 𝑋)
150	22, 30, 27, 143, 149	letrd 10786	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ 𝑋)
151	22, 27, 150	lensymd 10780	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → ¬ 𝑋 < (𝑄‘(𝑖 + 1)))
152	151	intnand 492	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → ¬ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1))))
153	64	ad2antlr 726	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ∈ ℝ)
154	52	ad3antlr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑖 ∈ ℝ)
155		simpr 488	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → ¬ 𝑖 < 𝑗)
156	153, 154, 155	nltled 10779	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ≤ 𝑖)
157	156	3adantl3 1165	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ≤ 𝑖)
158		eqcom 2805	. . . . . . . . . . . . 13 ⊢ ((𝑄‘𝑗) = 𝑋 ↔ 𝑋 = (𝑄‘𝑗))
159	158	biimpi 219	. . . . . . . . . . . 12 ⊢ ((𝑄‘𝑗) = 𝑋 → 𝑋 = (𝑄‘𝑗))
160	159	adantr 484	. . . . . . . . . . 11 ⊢ (((𝑄‘𝑗) = 𝑋 ∧ 𝑗 ≤ 𝑖) → 𝑋 = (𝑄‘𝑗))
161	160	3ad2antl3 1184	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → 𝑋 = (𝑄‘𝑗))
162	34	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑗 ∈ ℤ)
163	32	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑖 ∈ ℤ)
164		simpr 488	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑗 ≤ 𝑖)
165		eluz2 12237	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (ℤ≥‘𝑗) ↔ (𝑗 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ≤ 𝑖))
166	162, 163, 164, 165	syl3anbrc 1340	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑖 ∈ (ℤ≥‘𝑗))
167	166	adantlll 717	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑖 ∈ (ℤ≥‘𝑗))
168	17	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
169		0zd 11981	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ∈ ℤ)
170	66	ad2antlr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑀 ∈ ℤ)
171		elfzelz 12902	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (𝑗...𝑖) → 𝑤 ∈ ℤ)
172	171	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℤ)
173	169, 170, 172	3jca 1125	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ))
174		0red 10633	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ∈ ℝ)
175	64	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑗 ∈ ℝ)
176	171	zred 12075	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝑗...𝑖) → 𝑤 ∈ ℝ)
177	176	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℝ)
178		elfzle1 12905	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑀) → 0 ≤ 𝑗)
179	178	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑗)
180		elfzle1 12905	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝑗...𝑖) → 𝑗 ≤ 𝑤)
181	180	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑗 ≤ 𝑤)
182	174, 175, 177, 179, 181	letrd 10786	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑤)
183	182	adantll 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑤)
184	176	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℝ)
185		elfzoel2 13032	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
186	185	zred 12075	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑀 ∈ ℝ)
187	186	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑀 ∈ ℝ)
188	52	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑖 ∈ ℝ)
189		elfzle2 12906	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ (𝑗...𝑖) → 𝑤 ≤ 𝑖)
190	189	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ≤ 𝑖)
191		elfzolt2 13042	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 < 𝑀)
192	191	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑖 < 𝑀)
193	184, 188, 187, 190, 192	lelttrd 10787	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 < 𝑀)
194	184, 187, 193	ltled 10777	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ≤ 𝑀)
195	194	adantlr 714	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ≤ 𝑀)
196	173, 183, 195	jca32 519	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀)))
197	196	adantlll 717	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀)))
198		elfz2 12892	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀)))
199	197, 198	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ (0...𝑀))
200	168, 199	ffvelrnd 6829	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → (𝑄‘𝑤) ∈ ℝ)
201	200	adantlr 714	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) ∧ 𝑤 ∈ (𝑗...𝑖)) → (𝑄‘𝑤) ∈ ℝ)
202		simplll 774	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝜑)
203		0red 10633	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ∈ ℝ)
204	64	ad2antlr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑗 ∈ ℝ)
205		elfzelz 12902	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ∈ ℤ)
206	205	zred 12075	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ∈ ℝ)
207	206	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℝ)
208	178	ad2antlr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ≤ 𝑗)
209		elfzle1 12905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑗 ≤ 𝑤)
210	209	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑗 ≤ 𝑤)
211	203, 204, 207, 208, 210	letrd 10786	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ≤ 𝑤)
212	206	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℝ)
213	52	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑖 ∈ ℝ)
214	186	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑀 ∈ ℝ)
215		peano2rem 10942	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
216	213, 215	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
217		elfzle2 12906	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ≤ (𝑖 − 1))
218	217	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ≤ (𝑖 − 1))
219	213	ltm1d 11561	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
220	212, 216, 213, 218, 219	lelttrd 10787	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑖)
221	191	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑖 < 𝑀)
222	212, 213, 214, 220, 221	lttrd 10790	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑀)
223	222	adantlr 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑀)
224	205	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℤ)
225		0zd 11981	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ∈ ℤ)
226	185	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑀 ∈ ℤ)
227	224, 225, 226, 120	syl3anc 1368	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤 ∧ 𝑤 < 𝑀)))
228	211, 223, 227	mpbir2and 712	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ (0..^𝑀))
229	228	adantlll 717	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ (0..^𝑀))
230	202, 229, 140	syl2anc 587	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑄‘𝑤) ≤ (𝑄‘(𝑤 + 1)))
231	230	adantlr 714	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑄‘𝑤) ≤ (𝑄‘(𝑤 + 1)))
232	167, 201, 231	monoord 13396	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → (𝑄‘𝑗) ≤ (𝑄‘𝑖))
233	232	3adantl3 1165	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → (𝑄‘𝑗) ≤ (𝑄‘𝑖))
234	161, 233	eqbrtrd 5052	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → 𝑋 ≤ (𝑄‘𝑖))
235	25	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
236		elfzofz 13048	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
237	236	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
238	17, 237	ffvelrnd 6829	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
239	235, 238	lenltd 10775	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 ≤ (𝑄‘𝑖) ↔ ¬ (𝑄‘𝑖) < 𝑋))
240	239	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ≤ 𝑖) → (𝑋 ≤ (𝑄‘𝑖) ↔ ¬ (𝑄‘𝑖) < 𝑋))
241	240	3ad2antl1 1182	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → (𝑋 ≤ (𝑄‘𝑖) ↔ ¬ (𝑄‘𝑖) < 𝑋))
242	234, 241	mpbid 235	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → ¬ (𝑄‘𝑖) < 𝑋)
243	157, 242	syldan 594	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → ¬ (𝑄‘𝑖) < 𝑋)
244	243	intnanrd 493	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → ¬ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1))))
245	152, 244	pm2.61dan 812	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → ¬ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1))))
246	245	intnand 492	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → ¬ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*) ∧ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1)))))
247		elioo3g 12755	. . . 4 ⊢ (𝑋 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*) ∧ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1)))))
248	246, 247	sylnibr 332	. . 3 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → ¬ 𝑋 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
249	248	rexlimdv3a 3245	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑋 → ¬ 𝑋 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
250	15, 249	mpd 15	1 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110   ⊆ wss 3881   class class class wbr 5030   ↦ cmpt 5110  ran crn 5520   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529  ℝ^*cxr 10663   < clt 10664   ≤ cle 10665   − cmin 10859  ℕcn 11625  ℤcz 11969  ℤ_≥cuz 12231  (,)cioo 12726  ...cfz 12885  ..^cfzo 13028

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-ioo 12730  df-fz 12886  df-fzo 13029

This theorem is referenced by:  fourierdlem38  42787  fourierdlem74  42822  fourierdlem75  42823  fourierdlem88  42836  fourierdlem103  42851  fourierdlem104  42852