Theorem fourierdlem12 46718

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	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
3		fourierdlem12.2	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4		fourierdlem12.1	. . . . . . . . 9 ⊢ 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 46708	. . . . . . . 8 ⊢ (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 18	. . . . . . 7 ⊢ (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	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
8	7	simpld 499	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑m (0...𝑀)))
9		elmapi 8842	. . . . 5 ⊢ (𝑄 ∈ (ℝ ↑m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
10		ffn 6703	. . . . 5 ⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
11		fvelrnb 6939	. . . . 5 ⊢ (𝑄 Fn (0...𝑀) → (𝑋 ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑋))
12	8, 9, 10, 11	4syl 20	. . . 4 ⊢ (𝜑 → (𝑋 ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑋))
13	1, 12	mpbid 235	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑋)
14	13	adantr 485	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑋)
15	8, 9	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
16	15	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
17		fzofzp1 13789	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
18	17	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
19	16, 18	ffvelcdmd 7078	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
20	19	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
21	20	3ad2antl1 1202	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
22		frn 6711	. . . . . . . . . . . 12 ⊢ (𝑄:(0...𝑀)⟶ℝ → ran 𝑄 ⊆ ℝ)
23	15, 22	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑄 ⊆ ℝ)
24	23, 1	sseldd 3946	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ)
25	24	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑖 < 𝑗) → 𝑋 ∈ ℝ)
26	25	3ad2antl1 1202	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → 𝑋 ∈ ℝ)
27	16	ffvelcdmda 7077	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
28	27	3adant3 1148	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) ∈ ℝ)
29	28	adantr 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘𝑗) ∈ ℝ)
30		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
31		elfzoelz 13683	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
32	31	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 ∈ ℤ)
33		elfzelz 13548	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
34	33	ad2antlr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
35		zltp1le 12640	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
36	32, 34, 35	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
37	30, 36	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ≤ 𝑗)
38	32	peano2zd 12699	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ∈ ℤ)
39		eluz 12872	. . . . . . . . . . . . . 14 ⊢ (((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ≥‘(𝑖 + 1)) ↔ (𝑖 + 1) ≤ 𝑗))
40	38, 34, 39	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑗 ∈ (ℤ≥‘(𝑖 + 1)) ↔ (𝑖 + 1) ≤ 𝑗))
41	37, 40	mpbird 260	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (ℤ≥‘(𝑖 + 1)))
42	41	adantlll 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (ℤ≥‘(𝑖 + 1)))
43	16	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑄:(0...𝑀)⟶ℝ)
44		0zd 12599	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ∈ ℤ)
45		elfzel2 13546	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
46	45	ad2antlr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑀 ∈ ℤ)
47		elfzelz 13548	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ∈ ℤ)
48	47	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℤ)
49		0red 11207	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ∈ ℝ)
50	47	zred 12696	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ∈ ℝ)
51	50	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℝ)
52	31	peano2zd 12699	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ ℤ)
53	52	zred 12696	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ ℝ)
54	53	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑖 + 1) ∈ ℝ)
55	31	zred 12696	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
56	55	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑖 ∈ ℝ)
57		elfzole1 13692	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 0 ≤ 𝑖)
58	57	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑖)
59	56	ltp1d 12141	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑖 < (𝑖 + 1))
60	49, 56, 54, 58, 59	lelttrd 11364	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 < (𝑖 + 1))
61		elfzle1 13551	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ((𝑖 + 1)...𝑗) → (𝑖 + 1) ≤ 𝑤)
62	61	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑖 + 1) ≤ 𝑤)
63	49, 54, 51, 60, 62	ltletrd 11366	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 < 𝑤)
64	49, 51, 63	ltled 11354	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑤)
65	64	adantlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 0 ≤ 𝑤)
66	50	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ ℝ)
67	33	zred 12696	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
68	67	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑗 ∈ ℝ)
69	45	zred 12696	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
70	69	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑀 ∈ ℝ)
71		elfzle2 13552	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ((𝑖 + 1)...𝑗) → 𝑤 ≤ 𝑗)
72	71	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ≤ 𝑗)
73		elfzle2 13552	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ≤ 𝑀)
74	73	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑗 ≤ 𝑀)
75	66, 68, 70, 72, 74	letrd 11363	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ≤ 𝑀)
76	75	adantll 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ≤ 𝑀)
77	44, 46, 48, 65, 76	elfzd 13539	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ (0...𝑀))
78	77	adantlll 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → 𝑤 ∈ (0...𝑀))
79	43, 78	ffvelcdmd 7078	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑄‘𝑤) ∈ ℝ)
80	79	adantlr 727	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...𝑗)) → (𝑄‘𝑤) ∈ ℝ)
81		simp-4l 794	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝜑)
82		0red 11207	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℝ)
83		elfzelz 13548	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ∈ ℤ)
84	83	zred 12696	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ∈ ℝ)
85	84	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
86		0red 11207	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℝ)
87	53	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑖 + 1) ∈ ℝ)
88	84	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
89		0red 11207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 0 ∈ ℝ)
90	55	ltp1d 12141	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 < (𝑖 + 1))
91	89, 55, 53, 57, 90	lelttrd 11364	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑀) → 0 < (𝑖 + 1))
92	91	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < (𝑖 + 1))
93		elfzle1 13551	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → (𝑖 + 1) ≤ 𝑤)
94	93	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑖 + 1) ≤ 𝑤)
95	86, 87, 88, 92, 94	ltletrd 11366	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < 𝑤)
96	95	adantlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 < 𝑤)
97	82, 85, 96	ltled 11354	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
98	97	adantlll 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
99	98	adantlr 727	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ≤ 𝑤)
100	84	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℝ)
101		peano2rem 11521	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℝ → (𝑗 − 1) ∈ ℝ)
102	67, 101	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℝ)
103	102	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑗 − 1) ∈ ℝ)
104	69	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑀 ∈ ℝ)
105		elfzle2 13552	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1)) → 𝑤 ≤ (𝑗 − 1))
106	105	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ≤ (𝑗 − 1))
107		zlem1lt 12642	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ≤ 𝑀 ↔ (𝑗 − 1) < 𝑀))
108	33, 45, 107	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 ≤ 𝑀 ↔ (𝑗 − 1) < 𝑀))
109	73, 108	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑀)
110	109	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑗 − 1) < 𝑀)
111	100, 103, 104, 106, 110	lelttrd 11364	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
112	111	adantlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝑗 ∈ (0...𝑀) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
113	112	adantlll 730	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 < 𝑀)
114	83	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ ℤ)
115		0zd 12599	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 0 ∈ ℤ)
116	45	ad3antlr 743	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑀 ∈ ℤ)
117		elfzo 13685	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤 ∧ 𝑤 < 𝑀)))
118	114, 115, 116, 117	syl3anc 1396	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤 ∧ 𝑤 < 𝑀)))
119	99, 113, 118	mpbir2and 725	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → 𝑤 ∈ (0..^𝑀))
120	15	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
121		elfzofz 13700	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ (0...𝑀))
122	121	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → 𝑤 ∈ (0...𝑀))
123	120, 122	ffvelcdmd 7078	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘𝑤) ∈ ℝ)
124		fzofzp1 13789	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (0..^𝑀) → (𝑤 + 1) ∈ (0...𝑀))
125	124	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑤 + 1) ∈ (0...𝑀))
126	120, 125	ffvelcdmd 7078	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘(𝑤 + 1)) ∈ ℝ)
127		eleq1w 2852	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑤 → (𝑖 ∈ (0..^𝑀) ↔ 𝑤 ∈ (0..^𝑀)))
128	127	anbi2d 641	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑤 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑤 ∈ (0..^𝑀))))
129		fveq2 6879	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑤 → (𝑄‘𝑖) = (𝑄‘𝑤))
130		oveq1 7415	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑤 → (𝑖 + 1) = (𝑤 + 1))
131	130	fveq2d 6883	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑤 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑤 + 1)))
132	129, 131	breq12d 5123	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑤 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘𝑤) < (𝑄‘(𝑤 + 1))))
133	128, 132	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑤 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘𝑤) < (𝑄‘(𝑤 + 1)))))
134	7	simprrd 785	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
135	134	r19.21bi 3263	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
136	133, 135	chvarvv 2016	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘𝑤) < (𝑄‘(𝑤 + 1)))
137	123, 126, 136	ltled 11354	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (0..^𝑀)) → (𝑄‘𝑤) ≤ (𝑄‘(𝑤 + 1)))
138	81, 119, 137	syl2anc 595	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑤 ∈ ((𝑖 + 1)...(𝑗 − 1))) → (𝑄‘𝑤) ≤ (𝑄‘(𝑤 + 1)))
139	42, 80, 138	monoord 14064	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ (𝑄‘𝑗))
140	139	3adantl3 1185	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ (𝑄‘𝑗))
141	15	ffvelcdmda 7077	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
142	141	3adant3 1148	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) ∈ ℝ)
143		simp3 1154	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) = 𝑋)
144	142, 143	eqled 11309	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) ≤ 𝑋)
145	144	3adant1r 1194	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → (𝑄‘𝑗) ≤ 𝑋)
146	145	adantr 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘𝑗) ≤ 𝑋)
147	21, 29, 26, 140, 146	letrd 11363	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → (𝑄‘(𝑖 + 1)) ≤ 𝑋)
148	21, 26, 147	lensymd 11357	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → ¬ 𝑋 < (𝑄‘(𝑖 + 1)))
149	148	intnand 493	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑖 < 𝑗) → ¬ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1))))
150	67	ad2antlr 739	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ∈ ℝ)
151	55	ad3antlr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑖 ∈ ℝ)
152		simpr 489	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → ¬ 𝑖 < 𝑗)
153	150, 151, 152	nltled 11356	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ≤ 𝑖)
154	153	3adantl3 1185	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ≤ 𝑖)
155		eqcom 2776	. . . . . . . . . . . 12 ⊢ ((𝑄‘𝑗) = 𝑋 ↔ 𝑋 = (𝑄‘𝑗))
156	155	birani 508	. . . . . . . . . . 11 ⊢ (((𝑄‘𝑗) = 𝑋 ∧ 𝑗 ≤ 𝑖) → 𝑋 = (𝑄‘𝑗))
157	156	3ad2antl3 1204	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → 𝑋 = (𝑄‘𝑗))
158	33	ad2antlr 739	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑗 ∈ ℤ)
159	31	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑖 ∈ ℤ)
160		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑗 ≤ 𝑖)
161		eluz2 12864	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (ℤ≥‘𝑗) ↔ (𝑗 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ≤ 𝑖))
162	158, 159, 160, 161	syl3anbrc 1360	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑖 ∈ (ℤ≥‘𝑗))
163	162	adantlll 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → 𝑖 ∈ (ℤ≥‘𝑗))
164	16	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
165		0zd 12599	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ∈ ℤ)
166	45	ad2antlr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑀 ∈ ℤ)
167		elfzelz 13548	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (𝑗...𝑖) → 𝑤 ∈ ℤ)
168	167	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℤ)
169	165, 166, 168	3jca 1144	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ))
170		0red 11207	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ∈ ℝ)
171	67	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑗 ∈ ℝ)
172	167	zred 12696	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝑗...𝑖) → 𝑤 ∈ ℝ)
173	172	adantl 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℝ)
174		elfzle1 13551	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑀) → 0 ≤ 𝑗)
175	174	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑗)
176		elfzle1 13551	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝑗...𝑖) → 𝑗 ≤ 𝑤)
177	176	adantl 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑗 ≤ 𝑤)
178	170, 171, 173, 175, 177	letrd 11363	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ (0...𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑤)
179	178	adantll 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 0 ≤ 𝑤)
180	172	adantl 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ ℝ)
181		elfzoel2 13682	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
182	181	zred 12696	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑀 ∈ ℝ)
183	182	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑀 ∈ ℝ)
184	55	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑖 ∈ ℝ)
185		elfzle2 13552	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ (𝑗...𝑖) → 𝑤 ≤ 𝑖)
186	185	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ≤ 𝑖)
187		elfzolt2 13693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 < 𝑀)
188	187	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑖 < 𝑀)
189	180, 184, 183, 186, 188	lelttrd 11364	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 < 𝑀)
190	180, 183, 189	ltled 11354	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ≤ 𝑀)
191	190	adantlr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ≤ 𝑀)
192	169, 179, 191	jca32 524	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀)))
193	192	adantlll 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀)))
194		elfz2 13538	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ (0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀)))
195	193, 194	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → 𝑤 ∈ (0...𝑀))
196	164, 195	ffvelcdmd 7078	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...𝑖)) → (𝑄‘𝑤) ∈ ℝ)
197	196	adantlr 727	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) ∧ 𝑤 ∈ (𝑗...𝑖)) → (𝑄‘𝑤) ∈ ℝ)
198		simplll 786	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝜑)
199		0red 11207	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ∈ ℝ)
200	67	ad2antlr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑗 ∈ ℝ)
201		elfzelz 13548	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ∈ ℤ)
202	201	zred 12696	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ∈ ℝ)
203	202	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℝ)
204	174	ad2antlr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ≤ 𝑗)
205		elfzle1 13551	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑗 ≤ 𝑤)
206	205	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑗 ≤ 𝑤)
207	199, 200, 203, 204, 206	letrd 11363	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ≤ 𝑤)
208	202	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℝ)
209	55	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑖 ∈ ℝ)
210	182	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑀 ∈ ℝ)
211		peano2rem 11521	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
212	209, 211	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
213		elfzle2 13552	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝑗...(𝑖 − 1)) → 𝑤 ≤ (𝑖 − 1))
214	213	adantl 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ≤ (𝑖 − 1))
215	209	ltm1d 12143	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
216	208, 212, 209, 214, 215	lelttrd 11364	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑖)
217	187	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑖 < 𝑀)
218	208, 209, 210, 216, 217	lttrd 11367	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑀)
219	218	adantlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 < 𝑀)
220	201	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ ℤ)
221		0zd 12599	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 0 ∈ ℤ)
222	181	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑀 ∈ ℤ)
223	220, 221, 222, 117	syl3anc 1396	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑤 ∈ (0..^𝑀) ↔ (0 ≤ 𝑤 ∧ 𝑤 < 𝑀)))
224	207, 219, 223	mpbir2and 725	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ (0..^𝑀))
225	224	adantlll 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → 𝑤 ∈ (0..^𝑀))
226	198, 225, 137	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑄‘𝑤) ≤ (𝑄‘(𝑤 + 1)))
227	226	adantlr 727	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) ∧ 𝑤 ∈ (𝑗...(𝑖 − 1))) → (𝑄‘𝑤) ≤ (𝑄‘(𝑤 + 1)))
228	163, 197, 227	monoord 14064	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 ≤ 𝑖) → (𝑄‘𝑗) ≤ (𝑄‘𝑖))
229	228	3adantl3 1185	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → (𝑄‘𝑗) ≤ (𝑄‘𝑖))
230	157, 229	eqbrtrd 5134	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → 𝑋 ≤ (𝑄‘𝑖))
231	24	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
232		elfzofz 13700	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
233	232	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
234	16, 233	ffvelcdmd 7078	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
235	231, 234	lenltd 11352	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 ≤ (𝑄‘𝑖) ↔ ¬ (𝑄‘𝑖) < 𝑋))
236	235	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ≤ 𝑖) → (𝑋 ≤ (𝑄‘𝑖) ↔ ¬ (𝑄‘𝑖) < 𝑋))
237	236	3ad2antl1 1202	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → (𝑋 ≤ (𝑄‘𝑖) ↔ ¬ (𝑄‘𝑖) < 𝑋))
238	230, 237	mpbid 235	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ 𝑗 ≤ 𝑖) → ¬ (𝑄‘𝑖) < 𝑋)
239	154, 238	syldan 602	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → ¬ (𝑄‘𝑖) < 𝑋)
240	239	intnanrd 494	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) ∧ ¬ 𝑖 < 𝑗) → ¬ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1))))
241	149, 240	pm2.61dan 824	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → ¬ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1))))
242	241	intnand 493	. . . 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)))))
243		elioo3g 13397	. . . 4 ⊢ (𝑋 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*) ∧ ((𝑄‘𝑖) < 𝑋 ∧ 𝑋 < (𝑄‘(𝑖 + 1)))))
244	242, 243	sylnibr 332	. . 3 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = 𝑋) → ¬ 𝑋 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
245	244	rexlimdv3a 3176	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑋 → ¬ 𝑋 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
246	14, 245	mpd 16	1 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3423   ⊆ wss 3913   class class class wbr 5110   ↦ cmpt 5193  ran crn 5660   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7408   ↑_m cmap 8820  ℝcr 11095  0cc0 11096  1c1 11097   + caddc 11099  ℝ^*cxr 11238   < clt 11239   ≤ cle 11240   − cmin 11437  ℕcn 12229  ℤcz 12587  ℤ_≥cuz 12858  (,)cioo 13368  ...cfz 13531  ..^cfzo 13678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-n0 12501  df-z 12588  df-uz 12859  df-ioo 13372  df-fz 13532  df-fzo 13679

This theorem is referenced by:  fourierdlem38  46744  fourierdlem74  46779  fourierdlem75  46780  fourierdlem88  46793  fourierdlem103  46808  fourierdlem104  46809