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 ∧ (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 43650	. . . . . . . . 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 231	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
8	7	simpld 495	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (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))
11	8, 9, 10	3syl 18	. . . . 5 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
12		fvelrnb 6830	. . . . 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 231	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑋)
15	14	adantr 481	. 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
18		fzofzp1 13484	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
19	18	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
20	17, 19	ffvelrnd 6962	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
21	20	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
22	21	3ad2antl1 1184	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
23		frn 6607	. . . . . . . . . . . 12 ⊢ (𝑄:(0...𝑀)⟶ℝ → ran 𝑄 ⊆ ℝ)
24	16, 23	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑄 ⊆ ℝ)
25	24, 1	sseldd 3922	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ)
26	25	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑖 < 𝑗) → 𝑋 ∈ ℝ)
27	26	3ad2antl1 1184	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → 𝑋 ∈ ℝ)
28	17	ffvelrnda 6961	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
29	28	3adant3 1131	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) ∈ ℝ)
30	29	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘𝑗) ∈ ℝ)
31		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
32		elfzoelz 13387	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
33	32	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 ∈ ℤ)
34		elfzelz 13256	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
35	34	ad2antlr 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))
37	33, 35, 36	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
38	31, 37	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ≤ 𝑗)
39	33	peano2zd 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))
41	39, 35, 40	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑗 ∈ (ℤ≥‘(𝑖 + 1)) ↔ (𝑖 + 1) ≤ 𝑗))
42	38, 41	mpbird 256	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (ℤ≥‘(𝑖 + 1)))
43	42	adantlll 715	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (ℤ≥‘(𝑖 + 1)))
44	17	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (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)
47	46	ad2antlr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑀 ∈ ℤ)
48		elfzelz 13256	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ∈ ℤ)
49	48	adantl 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)
51	48	zred 12426	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ∈ ℝ)
52	51	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℝ)
53	32	peano2zd 12429	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ ℤ)
54	53	zred 12426	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ ℝ)
55	54	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑖 + 1) ∈ ℝ)
56	32	zred 12426	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
57	56	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑖 ∈ ℝ)
58		elfzole1 13395	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 0 ≤ 𝑖)
59	58	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑖)
60	57	ltp1d 11905	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑖 < (𝑖 + 1))
61	50, 57, 55, 59, 60	lelttrd 11133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 < (𝑖 + 1))
62		elfzle1 13259	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ((𝑖 + 1)...𝑗) → (𝑖 + 1) ≤ 𝑤)
63	62	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑖 + 1) ≤ 𝑤)
64	50, 55, 52, 61, 63	ltletrd 11135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 < 𝑤)
65	50, 52, 64	ltled 11123	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑤)
66	65	adantlr 712	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑤)
67	51	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℝ)
68	34	zred 12426	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
69	68	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑗 ∈ ℝ)
70	46	zred 12426	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
71	70	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑀 ∈ ℝ)
72		elfzle2 13260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ≤ 𝑗)
73	72	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ≤ 𝑗)
74		elfzle2 13260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ≤ 𝑀)
75	74	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑗 ≤ 𝑀)
76	67, 69, 71, 73, 75	letrd 11132	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ≤ 𝑀)
77	76	adantll 711	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ≤ 𝑀)
78	45, 47, 49, 66, 77	elfzd 13247	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ (0...𝑀))
79	78	adantlll 715	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ (0...𝑀))
80	44, 79	ffvelrnd 6962	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑄‘𝑤) ∈ ℝ)
81	80	adantlr 712	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑄‘𝑤) ∈ ℝ)
82		simp-4l 780	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (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)
85	84	zred 12426	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ∈ ℝ)
86	85	adantl 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)
88	54	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑖 + 1) ∈ ℝ)
89	85	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
90		0red 10978	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 0 ∈ ℝ)
91	56	ltp1d 11905	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 < (𝑖 + 1))
92	90, 56, 54, 58, 91	lelttrd 11133	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑀) → 0 < (𝑖 + 1))
93	92	adantr 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)
95	94	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑖 + 1) ≤ 𝑤)
96	87, 88, 89, 93, 95	ltletrd 11135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < 𝑤)
97	96	adantlr 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < 𝑤)
98	83, 86, 97	ltled 11123	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
99	98	adantlll 715	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
100	99	adantlr 712	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
101	85	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
102		peano2rem 11288	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℝ → (𝑗 − 1) ∈ ℝ)
103	68, 102	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℝ)
104	103	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑗 − 1) ∈ ℝ)
105	70	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑀 ∈ ℝ)
106		elfzle2 13260	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ≤ (𝑗 − 1))
107	106	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ≤ (𝑗 − 1))
108		zlem1lt 12372	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ≤ 𝑀 ↔ (𝑗 − 1) < 𝑀))
109	34, 46, 108	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 ≤ 𝑀 ↔ (𝑗 − 1) < 𝑀))
110	74, 109	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑀)
111	110	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑗 − 1) < 𝑀)
112	101, 104, 105, 107, 111	lelttrd 11133	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
113	112	adantlr 712	. . . . . . . . . . . . . 14 ⊢ (((𝑗 ∈ (0...𝑀) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
114	113	adantlll 715	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
115	84	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℤ)
116		0zd 12331	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℤ)
117	46	ad3antlr 728	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (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 ≤ 𝑤 ∧ 𝑤 < 𝑀)))
119	115, 116, 117, 118	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤 ∧ 𝑤 < 𝑀)))
120	100, 114, 119	mpbir2and 710	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ (0..^𝑀))
121	16	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
122		elfzofz 13403	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ (0...𝑀))
123	122	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → 𝑤 ∈ (0...𝑀))
124	121, 123	ffvelrnd 6962	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘𝑤) ∈ ℝ)
125		fzofzp1 13484	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (0..^𝑀) → (𝑤 + 1) ∈ (0...𝑀))
126	125	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑤 + 1) ∈ (0...𝑀))
127	121, 126	ffvelrnd 6962	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘(𝑤 + 1)) ∈ ℝ)
128		eleq1w 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑤 → (𝑖 ∈ (0..^𝑀) ↔ 𝑤 ∈ (0..^𝑀)))
129	128	anbi2d 629	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑤 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑤 ∈ (0..^𝑀))))
130		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑤 → (𝑄‘𝑖) = (𝑄‘𝑤))
131		oveq1 7282	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑤 → (𝑖 + 1) = (𝑤 + 1))
132	131	fveq2d 6778	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑤 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑤 + 1)))
133	130, 132	breq12d 5087	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑤 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘𝑤) < (𝑄‘(𝑤 + 1))))
134	129, 133	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑤 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘𝑤) < (𝑄‘(𝑤 + 1)))))
135	7	simprrd 771	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
136	135	r19.21bi 3134	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
137	134, 136	chvarvv 2002	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘𝑤) < (𝑄‘(𝑤 + 1)))
138	124, 127, 137	ltled 11123	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘𝑤) ≤ (𝑄‘(𝑤 + 1)))
139	82, 120, 138	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑄‘𝑤) ≤ (𝑄‘(𝑤 + 1)))
140	43, 81, 139	monoord 13753	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ (𝑄‘𝑗))
141	140	3adantl3 1167	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ (𝑄‘𝑗))
142	16	ffvelrnda 6961	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
143	142	3adant3 1131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) ∈ ℝ)
144		simp3 1137	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) = 𝑋)
145	143, 144	eqled 11078	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) ≤ 𝑋)
146	145	3adant1r 1176	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) ≤ 𝑋)
147	146	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘𝑗) ≤ 𝑋)
148	22, 30, 27, 141, 147	letrd 11132	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ 𝑋)
149	22, 27, 148	lensymd 11126	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → ¬ 𝑋 < (𝑄‘(𝑖 + 1)))
150	149	intnand 489	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → ¬ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1))))
151	68	ad2antlr 724	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ∈ ℝ)
152	56	ad3antlr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑖 ∈ ℝ)
153		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → ¬ 𝑖 < 𝑗)
154	151, 152, 153	nltled 11125	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ≤ 𝑖)
155	154	3adantl3 1167	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ≤ 𝑖)
156		eqcom 2745	. . . . . . . . . . . . 13 ⊢ ((𝑄‘𝑗) = 𝑋 ↔ 𝑋 = (𝑄‘𝑗))
157	156	biimpi 215	. . . . . . . . . . . 12 ⊢ ((𝑄‘𝑗) = 𝑋 → 𝑋 = (𝑄‘𝑗))
158	157	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑄‘𝑗) = 𝑋 ∧ 𝑗 ≤ 𝑖) → 𝑋 = (𝑄‘𝑗))
159	158	3ad2antl3 1186	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → 𝑋 = (𝑄‘𝑗))
160	34	ad2antlr 724	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑗 ∈ ℤ)
161	32	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑖 ∈ ℤ)
162		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑗 ≤ 𝑖)
163		eluz2 12588	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (ℤ≥‘𝑗) ↔ (𝑗 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ≤ 𝑖))
164	160, 161, 162, 163	syl3anbrc 1342	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑖 ∈ (ℤ≥‘𝑗))
165	164	adantlll 715	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑖 ∈ (ℤ≥‘𝑗))
166	17	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (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)
168	46	ad2antlr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑀 ∈ ℤ)
169		elfzelz 13256	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (𝑗...𝑖) → 𝑤 ∈ ℤ)
170	169	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℤ)
171	167, 168, 170	3jca 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)
173	68	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑗 ∈ ℝ)
174	169	zred 12426	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝑗...𝑖) → 𝑤 ∈ ℝ)
175	174	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℝ)
176		elfzle1 13259	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑀) → 0 ≤ 𝑗)
177	176	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑗)
178		elfzle1 13259	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝑗...𝑖) → 𝑗 ≤ 𝑤)
179	178	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑗 ≤ 𝑤)
180	172, 173, 175, 177, 179	letrd 11132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑤)
181	180	adantll 711	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑤)
182	174	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℝ)
183		elfzoel2 13386	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
184	183	zred 12426	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑀 ∈ ℝ)
185	184	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑀 ∈ ℝ)
186	56	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑖 ∈ ℝ)
187		elfzle2 13260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ (𝑗...𝑖) → 𝑤 ≤ 𝑖)
188	187	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ≤ 𝑖)
189		elfzolt2 13396	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 < 𝑀)
190	189	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑖 < 𝑀)
191	182, 186, 185, 188, 190	lelttrd 11133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 < 𝑀)
192	182, 185, 191	ltled 11123	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ≤ 𝑀)
193	192	adantlr 712	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ≤ 𝑀)
194	171, 181, 193	jca32 516	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀)))
195	194	adantlll 715	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀)))
196		elfz2 13246	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀)))
197	195, 196	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ (0...𝑀))
198	166, 197	ffvelrnd 6962	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → (𝑄‘𝑤) ∈ ℝ)
199	198	adantlr 712	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) ∧ 𝑤 ∈ (𝑗...𝑖)) → (𝑄‘𝑤) ∈ ℝ)
200		simplll 772	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (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)
202	68	ad2antlr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑗 ∈ ℝ)
203		elfzelz 13256	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ∈ ℤ)
204	203	zred 12426	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ∈ ℝ)
205	204	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℝ)
206	176	ad2antlr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ≤ 𝑗)
207		elfzle1 13259	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑗 ≤ 𝑤)
208	207	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑗 ≤ 𝑤)
209	201, 202, 205, 206, 208	letrd 11132	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ≤ 𝑤)
210	204	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℝ)
211	56	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑖 ∈ ℝ)
212	184	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑀 ∈ ℝ)
213		peano2rem 11288	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
214	211, 213	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
215		elfzle2 13260	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ≤ (𝑖 − 1))
216	215	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ≤ (𝑖 − 1))
217	211	ltm1d 11907	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
218	210, 214, 211, 216, 217	lelttrd 11133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑖)
219	189	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑖 < 𝑀)
220	210, 211, 212, 218, 219	lttrd 11136	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑀)
221	220	adantlr 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑀)
222	203	adantl 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)
224	183	ad2antrr 723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑀 ∈ ℤ)
225	222, 223, 224, 118	syl3anc 1370	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤 ∧ 𝑤 < 𝑀)))
226	209, 221, 225	mpbir2and 710	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ (0..^𝑀))
227	226	adantlll 715	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ (0..^𝑀))
228	200, 227, 138	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑄‘𝑤) ≤ (𝑄‘(𝑤 + 1)))
229	228	adantlr 712	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑄‘𝑤) ≤ (𝑄‘(𝑤 + 1)))
230	165, 199, 229	monoord 13753	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → (𝑄‘𝑗) ≤ (𝑄‘𝑖))
231	230	3adantl3 1167	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → (𝑄‘𝑗) ≤ (𝑄‘𝑖))
232	159, 231	eqbrtrd 5096	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → 𝑋 ≤ (𝑄‘𝑖))
233	25	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
234		elfzofz 13403	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
235	234	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
236	17, 235	ffvelrnd 6962	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
237	233, 236	lenltd 11121	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 ≤ (𝑄‘𝑖) ↔ ¬ (𝑄‘𝑖) < 𝑋))
238	237	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ≤ 𝑖) → (𝑋 ≤ (𝑄‘𝑖) ↔ ¬ (𝑄‘𝑖) < 𝑋))
239	238	3ad2antl1 1184	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → (𝑋 ≤ (𝑄‘𝑖) ↔ ¬ (𝑄‘𝑖) < 𝑋))
240	232, 239	mpbid 231	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → ¬ (𝑄‘𝑖) < 𝑋)
241	155, 240	syldan 591	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → ¬ (𝑄‘𝑖) < 𝑋)
242	241	intnanrd 490	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → ¬ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1))))
243	150, 242	pm2.61dan 810	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → ¬ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1))))
244	243	intnand 489	. . . 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)))))
245		elioo3g 13108	. . . 4 ⊢ (𝑋 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*) ∧ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1)))))
246	244, 245	sylnibr 329	. . 3 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → ¬ 𝑋 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
247	246	rexlimdv3a 3215	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑋 → ¬ 𝑋 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
248	15, 247	mpd 15	1 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))

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  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010   − cmin 11205  ℕcn 11973  ℤcz 12319  ℤ_≥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