fourierdlem65 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem65		Structured version Visualization version GIF version

Theorem fourierdlem65 45792

Description: The distance of two adjacent points in the moved partition is preserved in the original partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem65.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem65.t	⊢ 𝑇 = (𝐵 − 𝐴)
fourierdlem65.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem65.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem65.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem65.d	⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞))
fourierdlem65.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem65.n	⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
fourierdlem65.s	⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
fourierdlem65.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
fourierdlem65.l	⊢ 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
fourierdlem65.z	⊢ 𝑍 = ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))

**Assertion**
Ref	Expression
fourierdlem65	⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))

Distinct variable groups: 𝐴,𝑓,𝑘,𝑦 𝐴,𝑖,𝑥,𝑘,𝑦 𝐴,𝑚,𝑝,𝑖 𝐵,𝑓,𝑘,𝑦 𝐵,𝑖,𝑥 𝐵,𝑚,𝑝 𝐶,𝑓,𝑦 𝐶,𝑖,𝑚,𝑝 𝑥,𝐶 𝐷,𝑓,𝑦 𝐷,𝑖,𝑚,𝑝 𝑥,𝐷 𝑖,𝐸,𝑘,𝑥,𝑦 𝑖,𝑀,𝑚,𝑝 𝑓,𝑁,𝑦 𝑖,𝑁,𝑚,𝑝 𝑥,𝑁 𝑄,𝑓,𝑘,𝑦 𝑄,𝑖,𝑥 𝑄,𝑝 𝑆,𝑓,𝑘,𝑦 𝑆,𝑖,𝑥 𝑆,𝑝 𝑇,𝑖,𝑘,𝑥,𝑦 𝑖,𝑍,𝑘,𝑦 𝜑,𝑓,𝑘,𝑦 𝑖,𝑗,𝑘,𝑥,𝑦 𝜑,𝑖,𝑥 Allowed substitution hints: 𝜑(𝑗,𝑚,𝑝) 𝐴(𝑗) 𝐵(𝑗) 𝐶(𝑗,𝑘) 𝐷(𝑗,𝑘) 𝑃(𝑥,𝑦,𝑓,𝑖,𝑗,𝑘,𝑚,𝑝) 𝑄(𝑗,𝑚) 𝑆(𝑗,𝑚) 𝑇(𝑓,𝑗,𝑚,𝑝) 𝐸(𝑓,𝑗,𝑚,𝑝) 𝐿(𝑥,𝑦,𝑓,𝑖,𝑗,𝑘,𝑚,𝑝) 𝑀(𝑥,𝑦,𝑓,𝑗,𝑘) 𝑁(𝑗,𝑘) 𝑂(𝑥,𝑦,𝑓,𝑖,𝑗,𝑘,𝑚,𝑝) 𝑍(𝑥,𝑓,𝑗,𝑚,𝑝)

**Proof of Theorem fourierdlem65**
Step	Hyp	Ref	Expression
1		fourierdlem65.l	. . . . . 6 ⊢ 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
2	1	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)))
3		simpr 483	. . . . . . . 8 ⊢ (((𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → 𝑦 = (𝐸‘(𝑆‘𝑗)))
4		simpl 481	. . . . . . . 8 ⊢ (((𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → (𝐸‘(𝑆‘𝑗)) = 𝐵)
5	3, 4	eqtrd 2766	. . . . . . 7 ⊢ (((𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → 𝑦 = 𝐵)
6	5	iftrued 4541	. . . . . 6 ⊢ (((𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → if(𝑦 = 𝐵, 𝐴, 𝑦) = 𝐴)
7	6	adantll 712	. . . . 5 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → if(𝑦 = 𝐵, 𝐴, 𝑦) = 𝐴)
8		fourierdlem65.p	. . . . . . . . . . 11 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
9		fourierdlem65.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ)
10		fourierdlem65.q	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
11	8, 9, 10	fourierdlem11 45739	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
12	11	simp1d 1139	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
13	11	simp2d 1140	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
14	11	simp3d 1141	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 𝐵)
15		fourierdlem65.t	. . . . . . . . 9 ⊢ 𝑇 = (𝐵 − 𝐴)
16		fourierdlem65.e	. . . . . . . . 9 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
17	12, 13, 14, 15, 16	fourierdlem4 45732	. . . . . . . 8 ⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵))
18	17	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐸:ℝ⟶(𝐴(,]𝐵))
19		fourierdlem65.c	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ∈ ℝ)
20		ioossre 13439	. . . . . . . . . . . . . . . 16 ⊢ (𝐶(,)+∞) ⊆ ℝ
21		fourierdlem65.d	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞))
22	20, 21	sselid 3977	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ ℝ)
23	19	rexrd 11314	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
24		pnfxr 11318	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ^*
25	24	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → +∞ ∈ ℝ^*)
26		ioogtlb 45113	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐷 ∈ (𝐶(,)+∞)) → 𝐶 < 𝐷)
27	23, 25, 21, 26	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 < 𝐷)
28		fourierdlem65.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
29		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
30	15	eqcomi 2735	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐵 − 𝐴) = 𝑇
31	30	oveq2i 7435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 · (𝐵 − 𝐴)) = (𝑘 · 𝑇)
32	31	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (𝑘 · (𝐵 − 𝐴)) = (𝑘 · 𝑇))
33	29, 32	oveq12d 7442	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → (𝑦 + (𝑘 · (𝐵 − 𝐴))) = (𝑥 + (𝑘 · 𝑇)))
34	33	eleq1d 2811	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑥 → ((𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
35	34	rexbidv 3169	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
36	35	cbvrabv 3430	. . . . . . . . . . . . . . . 16 ⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} = {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}
37	36	uneq2i 4160	. . . . . . . . . . . . . . 15 ⊢ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
38		fourierdlem65.n	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
39		fourierdlem65.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
40	15, 8, 9, 10, 19, 22, 27, 28, 37, 38, 39	fourierdlem54 45781	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))))
41	40	simpld 493	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
42	41	simprd 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
43	41	simpld 493	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
44	28	fourierdlem2 45730	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))))
45	43, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))))
46	42, 45	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))
47	46	simpld 493	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ (ℝ ↑_m (0...𝑁)))
48		elmapi 8878	. . . . . . . . . 10 ⊢ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) → 𝑆:(0...𝑁)⟶ℝ)
49	47, 48	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶ℝ)
50	49	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑆:(0...𝑁)⟶ℝ)
51		elfzofz 13702	. . . . . . . . 9 ⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ (0...𝑁))
52	51	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0...𝑁))
53	50, 52	ffvelcdmd 7099	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ∈ ℝ)
54	18, 53	ffvelcdmd 7099	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐸‘(𝑆‘𝑗)) ∈ (𝐴(,]𝐵))
55	54	adantr 479	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) ∈ (𝐴(,]𝐵))
56	12	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐴 ∈ ℝ)
57	2, 7, 55, 56	fvmptd 7016	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐿‘(𝐸‘(𝑆‘𝑗))) = 𝐴)
58	57	oveq2d 7440	. . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘(𝑗 + 1))) − 𝐴))
59	13	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐵 ∈ ℝ)
60	14	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐴 < 𝐵)
61	53	adantr 479	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) ∈ ℝ)
62		simpr 483	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) = 𝐵)
63		fzofzp1 13784	. . . . . . . . 9 ⊢ (𝑗 ∈ (0..^𝑁) → (𝑗 + 1) ∈ (0...𝑁))
64	63	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 + 1) ∈ (0...𝑁))
65	50, 64	ffvelcdmd 7099	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ ℝ)
66	65	adantr 479	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘(𝑗 + 1)) ∈ ℝ)
67		elfzoelz 13686	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ ℤ)
68	67	zred 12718	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ ℝ)
69	68	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ ℝ)
70	69	ltp1d 12196	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 < (𝑗 + 1))
71	40	simprd 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
72	71	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
73		isorel 7338	. . . . . . . . 9 ⊢ ((𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) ∧ (𝑗 ∈ (0...𝑁) ∧ (𝑗 + 1) ∈ (0...𝑁))) → (𝑗 < (𝑗 + 1) ↔ (𝑆‘𝑗) < (𝑆‘(𝑗 + 1))))
74	72, 52, 64, 73	syl12anc 835	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 < (𝑗 + 1) ↔ (𝑆‘𝑗) < (𝑆‘(𝑗 + 1))))
75	70, 74	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)))
76	75	adantr 479	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)))
77		isof1o 7335	. . . . . . . . . . 11 ⊢ (𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) → 𝑆:(0...𝑁)–1-1-onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))
78		f1ofo 6850	. . . . . . . . . . 11 ⊢ (𝑆:(0...𝑁)–1-1-onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) → 𝑆:(0...𝑁)–onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))
79	71, 77, 78	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆:(0...𝑁)–onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))
80	79	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → 𝑆:(0...𝑁)–onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))
81	19	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → 𝐶 ∈ ℝ)
82	22	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → 𝐷 ∈ ℝ)
83	13, 12	resubcld 11692	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
84	15, 83	eqeltrid 2830	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ∈ ℝ)
85	84	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑇 ∈ ℝ)
86	53, 85	readdcld 11293	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + 𝑇) ∈ ℝ)
87	86	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) ∈ ℝ)
88	19	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ∈ ℝ)
89	28, 43, 42	fourierdlem15 45743	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
90	89	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
91	90, 52	ffvelcdmd 7099	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ∈ (𝐶[,]𝐷))
92	22	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐷 ∈ ℝ)
93		elicc2 13443	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆‘𝑗) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘𝑗) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘𝑗) ∧ (𝑆‘𝑗) ≤ 𝐷)))
94	88, 92, 93	syl2anc 582	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘𝑗) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘𝑗) ∧ (𝑆‘𝑗) ≤ 𝐷)))
95	91, 94	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘𝑗) ∧ (𝑆‘𝑗) ≤ 𝐷))
96	95	simp2d 1140	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ≤ (𝑆‘𝑗))
97	12, 13	posdifd 11851	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
98	14, 97	mpbid 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
99	98, 15	breqtrrdi 5195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 < 𝑇)
100	84, 99	elrpd 13067	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
101	100	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑇 ∈ ℝ⁺)
102	53, 101	ltaddrpd 13103	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) < ((𝑆‘𝑗) + 𝑇))
103	88, 53, 86, 96, 102	lelttrd 11422	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 < ((𝑆‘𝑗) + 𝑇))
104	88, 86, 103	ltled 11412	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ≤ ((𝑆‘𝑗) + 𝑇))
105	104	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → 𝐶 ≤ ((𝑆‘𝑗) + 𝑇))
106	65	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → (𝑆‘(𝑗 + 1)) ∈ ℝ)
107		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇))
108	87, 106	ltnled 11411	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → (((𝑆‘𝑗) + 𝑇) < (𝑆‘(𝑗 + 1)) ↔ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)))
109	107, 108	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) < (𝑆‘(𝑗 + 1)))
110	90, 64	ffvelcdmd 7099	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ (𝐶[,]𝐷))
111		elicc2 13443	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆‘(𝑗 + 1)) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘(𝑗 + 1)) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘(𝑗 + 1)) ∧ (𝑆‘(𝑗 + 1)) ≤ 𝐷)))
112	88, 92, 111	syl2anc 582	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘(𝑗 + 1)) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘(𝑗 + 1)) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘(𝑗 + 1)) ∧ (𝑆‘(𝑗 + 1)) ≤ 𝐷)))
113	110, 112	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘(𝑗 + 1)) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘(𝑗 + 1)) ∧ (𝑆‘(𝑗 + 1)) ≤ 𝐷))
114	113	simp3d 1141	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ≤ 𝐷)
115	114	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → (𝑆‘(𝑗 + 1)) ≤ 𝐷)
116	87, 106, 82, 109, 115	ltletrd 11424	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) < 𝐷)
117	87, 82, 116	ltled 11412	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) ≤ 𝐷)
118	81, 82, 87, 105, 117	eliccd 45122	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) ∈ (𝐶[,]𝐷))
119	118	adantlr 713	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) ∈ (𝐶[,]𝐷))
120	16	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
121		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑆‘𝑗) → 𝑥 = (𝑆‘𝑗))
122		oveq2 7432	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑆‘𝑗) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘𝑗)))
123	122	oveq1d 7439	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑆‘𝑗) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘𝑗)) / 𝑇))
124	123	fveq2d 6905	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑆‘𝑗) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)))
125	124	oveq1d 7439	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑆‘𝑗) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))
126	121, 125	oveq12d 7442	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑆‘𝑗) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)))
127	126	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑥 = (𝑆‘𝑗)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)))
128	13	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐵 ∈ ℝ)
129	128, 53	resubcld 11692	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝑆‘𝑗)) ∈ ℝ)
130	129, 101	rerpdivcld 13101	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∈ ℝ)
131	130	flcld 13818	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∈ ℤ)
132	131	zred 12718	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∈ ℝ)
133	132, 85	remulcld 11294	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇) ∈ ℝ)
134	53, 133	readdcld 11293	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) ∈ ℝ)
135	120, 127, 53, 134	fvmptd 7016	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐸‘(𝑆‘𝑗)) = ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)))
136	135	oveq1d 7439	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) = (((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) − (𝑆‘𝑗)))
137	136	oveq1d 7439	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) = ((((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) − (𝑆‘𝑗)) / 𝑇))
138	53	recnd 11292	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ∈ ℂ)
139	133	recnd 11292	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇) ∈ ℂ)
140	138, 139	pncan2d 11623	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) − (𝑆‘𝑗)) = ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))
141	140	oveq1d 7439	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) − (𝑆‘𝑗)) / 𝑇) = (((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇) / 𝑇))
142	132	recnd 11292	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∈ ℂ)
143	85	recnd 11292	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑇 ∈ ℂ)
144	101	rpne0d 13075	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑇 ≠ 0)
145	142, 143, 144	divcan4d 12047	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇) / 𝑇) = (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)))
146	137, 141, 145	3eqtrd 2770	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) = (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)))
147	146, 131	eqeltrd 2826	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) ∈ ℤ)
148		peano2zm 12657	. . . . . . . . . . . . . 14 ⊢ ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) ∈ ℤ → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) ∈ ℤ)
149	147, 148	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) ∈ ℤ)
150	149	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) ∈ ℤ)
151	30	oveq2i 7435	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴)) = (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇)
152	151	oveq2i 7435	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) = (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇))
153	152	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) = (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇)))
154	135	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) = ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)))
155		oveq1 7431	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐸‘(𝑆‘𝑗)) = 𝐵 → ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) = (𝐵 − (𝑆‘𝑗)))
156	155	eqcomd 2732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐸‘(𝑆‘𝑗)) = 𝐵 → (𝐵 − (𝑆‘𝑗)) = ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)))
157	156	oveq1d 7439	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸‘(𝑆‘𝑗)) = 𝐵 → ((𝐵 − (𝑆‘𝑗)) / 𝑇) = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇))
158	157	fveq2d 6905	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸‘(𝑆‘𝑗)) = 𝐵 → (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) = (⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)))
159	158	oveq1d 7439	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸‘(𝑆‘𝑗)) = 𝐵 → ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇) = ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇))
160	159	oveq2d 7440	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸‘(𝑆‘𝑗)) = 𝐵 → ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) = ((𝑆‘𝑗) + ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇)))
161	160	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) = ((𝑆‘𝑗) + ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇)))
162	146, 142	eqeltrd 2826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) ∈ ℂ)
163		1cnd 11259	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 1 ∈ ℂ)
164	162, 163, 143	subdird 11721	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇) = (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − (1 · 𝑇)))
165	84	recnd 11292	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑇 ∈ ℂ)
166	165	mullidd 11282	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (1 · 𝑇) = 𝑇)
167	166	oveq2d 7440	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − (1 · 𝑇)) = (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − 𝑇))
168	167	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − (1 · 𝑇)) = (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − 𝑇))
169	164, 168	eqtrd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇) = (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − 𝑇))
170	169	oveq2d 7440	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇)) = (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − 𝑇)))
171	162, 143	mulcld 11284	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) ∈ ℂ)
172	138, 143, 171	ppncand 11661	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − 𝑇)) = ((𝑆‘𝑗) + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇)))
173		flid 13828	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) ∈ ℤ → (⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇))
174	147, 173	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇))
175	174	eqcomd 2732	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) = (⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)))
176	175	oveq1d 7439	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) = ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇))
177	176	oveq2d 7440	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇)) = ((𝑆‘𝑗) + ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇)))
178	170, 172, 177	3eqtrrd 2771	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) = (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇)))
179	178	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝑆‘𝑗) + ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) = (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇)))
180	154, 161, 179	3eqtrrd 2771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇)) = (𝐸‘(𝑆‘𝑗)))
181	153, 180, 62	3eqtrd 2770	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) = 𝐵)
182	8	fourierdlem2 45730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
183	9, 182	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
184	10, 183	mpbid 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
185	184	simprd 494	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
186	185	simpld 493	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
187	186	simprd 494	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
188	187	eqcomd 2732	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 = (𝑄‘𝑀))
189	8, 9, 10	fourierdlem15 45743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
190		ffn 6728	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) → 𝑄 Fn (0...𝑀))
191	189, 190	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
192	9	nnnn0d 12584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
193		nn0uz 12916	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ₀ = (ℤ_≥‘0)
194	192, 193	eleqtrdi 2836	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
195		eluzfz2 13563	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
196	194, 195	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
197		fnfvelrn 7094	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 Fn (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)) → (𝑄‘𝑀) ∈ ran 𝑄)
198	191, 196, 197	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ran 𝑄)
199	188, 198	eqeltrd 2826	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ran 𝑄)
200	199	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐵 ∈ ran 𝑄)
201	181, 200	eqeltrd 2826	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) ∈ ran 𝑄)
202	201	adantr 479	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) ∈ ran 𝑄)
203		oveq1 7431	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) → (𝑘 · (𝐵 − 𝐴)) = (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴)))
204	203	oveq2d 7440	. . . . . . . . . . . . . 14 ⊢ (𝑘 = ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) → (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) = (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))))
205	204	eleq1d 2811	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) → ((((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) ∈ ran 𝑄))
206	205	rspcev 3608	. . . . . . . . . . . 12 ⊢ ((((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) ∈ ℤ ∧ (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) ∈ ran 𝑄) → ∃𝑘 ∈ ℤ (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
207	150, 202, 206	syl2anc 582	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ∃𝑘 ∈ ℤ (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
208		oveq1 7431	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((𝑆‘𝑗) + 𝑇) → (𝑦 + (𝑘 · (𝐵 − 𝐴))) = (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))))
209	208	eleq1d 2811	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝑆‘𝑗) + 𝑇) → ((𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄))
210	209	rexbidv 3169	. . . . . . . . . . . 12 ⊢ (𝑦 = ((𝑆‘𝑗) + 𝑇) → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄))
211	210	elrab 3681	. . . . . . . . . . 11 ⊢ (((𝑆‘𝑗) + 𝑇) ∈ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} ↔ (((𝑆‘𝑗) + 𝑇) ∈ (𝐶[,]𝐷) ∧ ∃𝑘 ∈ ℤ (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄))
212	119, 207, 211	sylanbrc 581	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) ∈ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})
213		elun2 4178	. . . . . . . . . 10 ⊢ (((𝑆‘𝑗) + 𝑇) ∈ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} → ((𝑆‘𝑗) + 𝑇) ∈ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))
214	212, 213	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) ∈ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))
215		foelrn 7121	. . . . . . . . 9 ⊢ ((𝑆:(0...𝑁)–onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) ∧ ((𝑆‘𝑗) + 𝑇) ∈ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) → ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) + 𝑇) = (𝑆‘𝑖))
216	80, 214, 215	syl2anc 582	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) + 𝑇) = (𝑆‘𝑖))
217		eqcom 2733	. . . . . . . . 9 ⊢ (((𝑆‘𝑗) + 𝑇) = (𝑆‘𝑖) ↔ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇))
218	217	rexbii 3084	. . . . . . . 8 ⊢ (∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) + 𝑇) = (𝑆‘𝑖) ↔ ∃𝑖 ∈ (0...𝑁)(𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇))
219	216, 218	sylib 217	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ∃𝑖 ∈ (0...𝑁)(𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇))
220	102	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → (𝑆‘𝑗) < ((𝑆‘𝑗) + 𝑇))
221	217	biimpri 227	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇) → ((𝑆‘𝑗) + 𝑇) = (𝑆‘𝑖))
222	221	adantl 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) = (𝑆‘𝑖))
223	220, 222	breqtrd 5179	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → (𝑆‘𝑗) < (𝑆‘𝑖))
224	109	adantr 479	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) < (𝑆‘(𝑗 + 1)))
225	222, 224	eqbrtrrd 5177	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))
226	223, 225	jca 510	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))
227	226	adantlr 713	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))
228		simplll 773	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → (𝜑 ∧ 𝑗 ∈ (0..^𝑁)))
229		simplr 767	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → 𝑖 ∈ (0...𝑁))
230		elfzelz 13555	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑁) → 𝑖 ∈ ℤ)
231	230	ad2antlr 725	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) → 𝑖 ∈ ℤ)
232	67	ad3antlr 729	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) → 𝑗 ∈ ℤ)
233		simpr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘𝑖)) → (𝑆‘𝑗) < (𝑆‘𝑖))
234	72	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘𝑖)) → 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
235	52	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘𝑖)) → 𝑗 ∈ (0...𝑁))
236		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘𝑖)) → 𝑖 ∈ (0...𝑁))
237		isorel 7338	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑖 ∈ (0...𝑁))) → (𝑗 < 𝑖 ↔ (𝑆‘𝑗) < (𝑆‘𝑖)))
238	234, 235, 236, 237	syl12anc 835	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘𝑖)) → (𝑗 < 𝑖 ↔ (𝑆‘𝑗) < (𝑆‘𝑖)))
239	233, 238	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘𝑖)) → 𝑗 < 𝑖)
240	239	adantrr 715	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) → 𝑗 < 𝑖)
241		simpr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) → (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))
242	72	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) → 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
243		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) → 𝑖 ∈ (0...𝑁))
244	64	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) → (𝑗 + 1) ∈ (0...𝑁))
245		isorel 7338	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) ∧ (𝑖 ∈ (0...𝑁) ∧ (𝑗 + 1) ∈ (0...𝑁))) → (𝑖 < (𝑗 + 1) ↔ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))
246	242, 243, 244, 245	syl12anc 835	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) → (𝑖 < (𝑗 + 1) ↔ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))
247	241, 246	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) → 𝑖 < (𝑗 + 1))
248	247	adantrl 714	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) → 𝑖 < (𝑗 + 1))
249		btwnnz 12690	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ℤ ∧ 𝑗 < 𝑖 ∧ 𝑖 < (𝑗 + 1)) → ¬ 𝑖 ∈ ℤ)
250	232, 240, 248, 249	syl3anc 1368	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) → ¬ 𝑖 ∈ ℤ)
251	231, 250	pm2.65da 815	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) → ¬ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))
252	228, 229, 251	syl2anc 582	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → ¬ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))
253	227, 252	pm2.65da 815	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ 𝑖 ∈ (0...𝑁)) → ¬ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇))
254	253	nrexdv 3139	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ¬ ∃𝑖 ∈ (0...𝑁)(𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇))
255	254	adantlr 713	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ¬ ∃𝑖 ∈ (0...𝑁)(𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇))
256	219, 255	condan 816	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇))
257	61	rexrd 11314	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) ∈ ℝ^*)
258	84	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝑇 ∈ ℝ)
259	61, 258	readdcld 11293	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝑆‘𝑗) + 𝑇) ∈ ℝ)
260		elioc2 13441	. . . . . . 7 ⊢ (((𝑆‘𝑗) ∈ ℝ^* ∧ ((𝑆‘𝑗) + 𝑇) ∈ ℝ) → ((𝑆‘(𝑗 + 1)) ∈ ((𝑆‘𝑗)(,]((𝑆‘𝑗) + 𝑇)) ↔ ((𝑆‘(𝑗 + 1)) ∈ ℝ ∧ (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)) ∧ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇))))
261	257, 259, 260	syl2anc 582	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝑆‘(𝑗 + 1)) ∈ ((𝑆‘𝑗)(,]((𝑆‘𝑗) + 𝑇)) ↔ ((𝑆‘(𝑗 + 1)) ∈ ℝ ∧ (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)) ∧ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇))))
262	66, 76, 256, 261	mpbir3and 1339	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘(𝑗 + 1)) ∈ ((𝑆‘𝑗)(,]((𝑆‘𝑗) + 𝑇)))
263	56, 59, 60, 15, 16, 61, 62, 262	fourierdlem26 45754	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘(𝑗 + 1))) = (𝐴 + ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))))
264	263	oveq1d 7439	. . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘(𝑗 + 1))) − 𝐴) = ((𝐴 + ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) − 𝐴))
265	56	recnd 11292	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐴 ∈ ℂ)
266	65	recnd 11292	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ ℂ)
267	266, 138	subcld 11621	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) ∈ ℂ)
268	267	adantr 479	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) ∈ ℂ)
269	265, 268	pncan2d 11623	. . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐴 + ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) − 𝐴) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))
270	58, 264, 269	3eqtrd 2770	. 2 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))
271	1	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)))
272		eqcom 2733	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐸‘(𝑆‘𝑗)) ↔ (𝐸‘(𝑆‘𝑗)) = 𝑦)
273	272	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐸‘(𝑆‘𝑗)) → (𝐸‘(𝑆‘𝑗)) = 𝑦)
274	273	adantl 480	. . . . . . . . . 10 ⊢ ((¬ (𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → (𝐸‘(𝑆‘𝑗)) = 𝑦)
275		neqne 2938	. . . . . . . . . . 11 ⊢ (¬ (𝐸‘(𝑆‘𝑗)) = 𝐵 → (𝐸‘(𝑆‘𝑗)) ≠ 𝐵)
276	275	adantr 479	. . . . . . . . . 10 ⊢ ((¬ (𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → (𝐸‘(𝑆‘𝑗)) ≠ 𝐵)
277	274, 276	eqnetrrd 2999	. . . . . . . . 9 ⊢ ((¬ (𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → 𝑦 ≠ 𝐵)
278	277	neneqd 2935	. . . . . . . 8 ⊢ ((¬ (𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → ¬ 𝑦 = 𝐵)
279	278	iffalsed 4544	. . . . . . 7 ⊢ ((¬ (𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → if(𝑦 = 𝐵, 𝐴, 𝑦) = 𝑦)
280		simpr 483	. . . . . . 7 ⊢ ((¬ (𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → 𝑦 = (𝐸‘(𝑆‘𝑗)))
281	279, 280	eqtrd 2766	. . . . . 6 ⊢ ((¬ (𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → if(𝑦 = 𝐵, 𝐴, 𝑦) = (𝐸‘(𝑆‘𝑗)))
282	281	adantll 712	. . . . 5 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → if(𝑦 = 𝐵, 𝐴, 𝑦) = (𝐸‘(𝑆‘𝑗)))
283	54	adantr 479	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) ∈ (𝐴(,]𝐵))
284	271, 282, 283, 283	fvmptd 7016	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐿‘(𝐸‘(𝑆‘𝑗))) = (𝐸‘(𝑆‘𝑗)))
285	284	oveq2d 7440	. . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐸‘(𝑆‘𝑗))))
286		id 22	. . . . . . . 8 ⊢ (𝑥 = (𝑆‘(𝑗 + 1)) → 𝑥 = (𝑆‘(𝑗 + 1)))
287		oveq2 7432	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑆‘(𝑗 + 1)) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘(𝑗 + 1))))
288	287	oveq1d 7439	. . . . . . . . . 10 ⊢ (𝑥 = (𝑆‘(𝑗 + 1)) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇))
289	288	fveq2d 6905	. . . . . . . . 9 ⊢ (𝑥 = (𝑆‘(𝑗 + 1)) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)))
290	289	oveq1d 7439	. . . . . . . 8 ⊢ (𝑥 = (𝑆‘(𝑗 + 1)) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇))
291	286, 290	oveq12d 7442	. . . . . . 7 ⊢ (𝑥 = (𝑆‘(𝑗 + 1)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)))
292	291	adantl 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑥 = (𝑆‘(𝑗 + 1))) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)))
293	128, 65	resubcld 11692	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝑆‘(𝑗 + 1))) ∈ ℝ)
294	293, 101	rerpdivcld 13101	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) ∈ ℝ)
295	294	flcld 13818	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℤ)
296	295	zred 12718	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℝ)
297	296, 85	remulcld 11294	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇) ∈ ℝ)
298	65, 297	readdcld 11293	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) ∈ ℝ)
299	120, 292, 65, 298	fvmptd 7016	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐸‘(𝑆‘(𝑗 + 1))) = ((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)))
300	299, 135	oveq12d 7442	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐸‘(𝑆‘𝑗))) = (((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) − ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))))
301	300	adantr 479	. . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐸‘(𝑆‘𝑗))) = (((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) − ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))))
302		flle 13819	. . . . . . . . . . . . 13 ⊢ (((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) ∈ ℝ → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇))
303	294, 302	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇))
304	53, 65, 75	ltled 11412	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ≤ (𝑆‘(𝑗 + 1)))
305	53, 65, 128, 304	lesub2dd 11881	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝑆‘(𝑗 + 1))) ≤ (𝐵 − (𝑆‘𝑗)))
306	293, 129, 101, 305	lediv1dd 13128	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇))
307	296, 294, 130, 303, 306	letrd 11421	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇))
308	307	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇))
309	296	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℝ)
310		1red 11265	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → 1 ∈ ℝ)
311	309, 310	readdcld 11293	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ∈ ℝ)
312	130	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∈ ℝ)
313		simpr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1))
314	311, 312, 313	nltled 11414	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇))
315	314	adantlr 713	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇))
316	79	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑆:(0...𝑁)–onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))
317	88	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝐶 ∈ ℝ)
318	92	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝐷 ∈ ℝ)
319		fourierdlem65.z	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑍 = ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))
320	135, 134	eqeltrd 2826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐸‘(𝑆‘𝑗)) ∈ ℝ)
321	128, 320	resubcld 11692	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝐸‘(𝑆‘𝑗))) ∈ ℝ)
322	53, 321	readdcld 11293	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) ∈ ℝ)
323	319, 322	eqeltrid 2830	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑍 ∈ ℝ)
324	323	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 ∈ ℝ)
325	12	rexrd 11314	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
326	325	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 ∈ ℝ^*)
327		elioc2 13441	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐸‘(𝑆‘𝑗)) ∈ (𝐴(,]𝐵) ↔ ((𝐸‘(𝑆‘𝑗)) ∈ ℝ ∧ 𝐴 < (𝐸‘(𝑆‘𝑗)) ∧ (𝐸‘(𝑆‘𝑗)) ≤ 𝐵)))
328	326, 128, 327	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘𝑗)) ∈ (𝐴(,]𝐵) ↔ ((𝐸‘(𝑆‘𝑗)) ∈ ℝ ∧ 𝐴 < (𝐸‘(𝑆‘𝑗)) ∧ (𝐸‘(𝑆‘𝑗)) ≤ 𝐵)))
329	54, 328	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘𝑗)) ∈ ℝ ∧ 𝐴 < (𝐸‘(𝑆‘𝑗)) ∧ (𝐸‘(𝑆‘𝑗)) ≤ 𝐵))
330	329	simp3d 1141	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐸‘(𝑆‘𝑗)) ≤ 𝐵)
331	128, 320	subge0d 11854	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (0 ≤ (𝐵 − (𝐸‘(𝑆‘𝑗))) ↔ (𝐸‘(𝑆‘𝑗)) ≤ 𝐵))
332	330, 331	mpbird 256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 0 ≤ (𝐵 − (𝐸‘(𝑆‘𝑗))))
333	53, 321	addge01d 11852	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (0 ≤ (𝐵 − (𝐸‘(𝑆‘𝑗))) ↔ (𝑆‘𝑗) ≤ ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))))
334	332, 333	mpbid 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ≤ ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))))
335	88, 53, 322, 96, 334	letrd 11421	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ≤ ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))))
336	335, 319	breqtrrdi 5195	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ≤ 𝑍)
337	336	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝐶 ≤ 𝑍)
338	65	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝑆‘(𝑗 + 1)) ∈ ℝ)
339	294	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) ∈ ℝ)
340		reflcl 13816	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) ∈ ℝ → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℝ)
341		peano2re 11437	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℝ → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ∈ ℝ)
342	339, 340, 341	3syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ∈ ℝ)
343	128	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝐵 ∈ ℝ)
344	343, 324	resubcld 11692	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝐵 − 𝑍) ∈ ℝ)
345	101	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑇 ∈ ℝ⁺)
346	344, 345	rerpdivcld 13101	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − 𝑍) / 𝑇) ∈ ℝ)
347		flltp1 13820	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) ∈ ℝ → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1))
348	294, 347	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1))
349	348	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1))
350	295	peano2zd 12721	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ∈ ℤ)
351	350	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ∈ ℤ)
352	130	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∈ ℝ)
353		simpr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇))
354	321, 101	rerpdivcld 13101	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇) ∈ ℝ)
355	354	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇) ∈ ℝ)
356	12	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 ∈ ℝ)
357	329	simp2d 1140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 < (𝐸‘(𝑆‘𝑗)))
358	356, 320, 128, 357	ltsub2dd 11877	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝐸‘(𝑆‘𝑗))) < (𝐵 − 𝐴))
359	358, 15	breqtrrdi 5195	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝐸‘(𝑆‘𝑗))) < 𝑇)
360	321, 85, 101, 359	ltdiv1dd 13127	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇) < (𝑇 / 𝑇))
361	143, 144	dividd 12039	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑇 / 𝑇) = 1)
362	360, 361	breqtrd 5179	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇) < 1)
363	362	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇) < 1)
364	129	recnd 11292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝑆‘𝑗)) ∈ ℂ)
365	321	recnd 11292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝐸‘(𝑆‘𝑗))) ∈ ℂ)
366	364, 365, 143, 144	divsubdird 12080	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝑆‘𝑗)) − (𝐵 − (𝐸‘(𝑆‘𝑗)))) / 𝑇) = (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)))
367	366	eqcomd 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)) = (((𝐵 − (𝑆‘𝑗)) − (𝐵 − (𝐸‘(𝑆‘𝑗)))) / 𝑇))
368	128	recnd 11292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐵 ∈ ℂ)
369	320	recnd 11292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐸‘(𝑆‘𝑗)) ∈ ℂ)
370	368, 138, 369	nnncan1d 11655	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝑆‘𝑗)) − (𝐵 − (𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)))
371	370	oveq1d 7439	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝑆‘𝑗)) − (𝐵 − (𝐸‘(𝑆‘𝑗)))) / 𝑇) = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇))
372	367, 371	eqtrd 2766	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)) = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇))
373	372, 147	eqeltrd 2826	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)) ∈ ℤ)
374	373	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)) ∈ ℤ)
375	351, 352, 353, 355, 363, 374	zltlesub 44900	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)))
376	319	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑍 = ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))))
377	376	oveq2d 7440	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − 𝑍) = (𝐵 − ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))))
378	138, 368, 369	addsub12d 11644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) = (𝐵 + ((𝑆‘𝑗) − (𝐸‘(𝑆‘𝑗)))))
379	368, 369, 138	subsub2d 11650	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) = (𝐵 + ((𝑆‘𝑗) − (𝐸‘(𝑆‘𝑗)))))
380	378, 379	eqtr4d 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) = (𝐵 − ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))))
381	380	oveq2d 7440	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))) = (𝐵 − (𝐵 − ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)))))
382	369, 138	subcld 11621	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) ∈ ℂ)
383	368, 382	nncand 11626	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝐵 − ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)))) = ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)))
384	377, 381, 383	3eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − 𝑍) = ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)))
385	384	oveq1d 7439	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − 𝑍) / 𝑇) = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇))
386	371, 367, 385	3eqtr4d 2776	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)) = ((𝐵 − 𝑍) / 𝑇))
387	386	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)) = ((𝐵 − 𝑍) / 𝑇))
388	375, 387	breqtrd 5179	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − 𝑍) / 𝑇))
389	339, 342, 346, 349, 388	ltletrd 11424	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) < ((𝐵 − 𝑍) / 𝑇))
390	293	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝐵 − (𝑆‘(𝑗 + 1))) ∈ ℝ)
391	390, 344, 345	ltdiv1d 13115	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝑆‘(𝑗 + 1))) < (𝐵 − 𝑍) ↔ ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) < ((𝐵 − 𝑍) / 𝑇)))
392	389, 391	mpbird 256	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝐵 − (𝑆‘(𝑗 + 1))) < (𝐵 − 𝑍))
393	324, 338, 343	ltsub2d 11874	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝑍 < (𝑆‘(𝑗 + 1)) ↔ (𝐵 − (𝑆‘(𝑗 + 1))) < (𝐵 − 𝑍)))
394	392, 393	mpbird 256	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 < (𝑆‘(𝑗 + 1)))
395	114	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝑆‘(𝑗 + 1)) ≤ 𝐷)
396	324, 338, 318, 394, 395	ltletrd 11424	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 < 𝐷)
397	324, 318, 396	ltled 11412	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 ≤ 𝐷)
398	317, 318, 324, 337, 397	eliccd 45122	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 ∈ (𝐶[,]𝐷))
399	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − 𝐴) = 𝑇)
400	399	oveq2d 7440	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴)) = ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇))
401	382, 143, 144	divcan1d 12042	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) = ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)))
402	400, 401	eqtrd 2766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴)) = ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)))
403	376, 402	oveq12d 7442	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑍 + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴))) = (((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) + ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))))
404	138, 365	addcomd 11466	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) = ((𝐵 − (𝐸‘(𝑆‘𝑗))) + (𝑆‘𝑗)))
405	404	oveq1d 7439	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) + ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) = (((𝐵 − (𝐸‘(𝑆‘𝑗))) + (𝑆‘𝑗)) + ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))))
406	365, 138, 369	ppncand 11661	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝐸‘(𝑆‘𝑗))) + (𝑆‘𝑗)) + ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) = ((𝐵 − (𝐸‘(𝑆‘𝑗))) + (𝐸‘(𝑆‘𝑗))))
407	368, 369	npcand 11625	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝐸‘(𝑆‘𝑗))) + (𝐸‘(𝑆‘𝑗))) = 𝐵)
408	406, 407	eqtrd 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝐸‘(𝑆‘𝑗))) + (𝑆‘𝑗)) + ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) = 𝐵)
409	403, 405, 408	3eqtrd 2770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑍 + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴))) = 𝐵)
410	199	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐵 ∈ ran 𝑄)
411	409, 410	eqeltrd 2826	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑍 + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴))) ∈ ran 𝑄)
412		oveq1 7431	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) → (𝑘 · (𝐵 − 𝐴)) = ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴)))
413	412	oveq2d 7440	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) → (𝑍 + (𝑘 · (𝐵 − 𝐴))) = (𝑍 + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴))))
414	413	eleq1d 2811	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) → ((𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (𝑍 + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴))) ∈ ran 𝑄))
415	414	rspcev 3608	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) ∈ ℤ ∧ (𝑍 + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴))) ∈ ran 𝑄) → ∃𝑘 ∈ ℤ (𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
416	147, 411, 415	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑘 ∈ ℤ (𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
417	416	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ∃𝑘 ∈ ℤ (𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
418		oveq1 7431	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑍 → (𝑦 + (𝑘 · (𝐵 − 𝐴))) = (𝑍 + (𝑘 · (𝐵 − 𝐴))))
419	418	eleq1d 2811	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑍 → ((𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄))
420	419	rexbidv 3169	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑍 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄))
421	420	elrab 3681	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} ↔ (𝑍 ∈ (𝐶[,]𝐷) ∧ ∃𝑘 ∈ ℤ (𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄))
422	398, 417, 421	sylanbrc 581	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 ∈ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})
423		elun2 4178	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} → 𝑍 ∈ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))
424	422, 423	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 ∈ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))
425		foelrn 7121	. . . . . . . . . . . . . 14 ⊢ ((𝑆:(0...𝑁)–onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) ∧ 𝑍 ∈ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) → ∃𝑖 ∈ (0...𝑁)𝑍 = (𝑆‘𝑖))
426	316, 424, 425	syl2anc 582	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ∃𝑖 ∈ (0...𝑁)𝑍 = (𝑆‘𝑖))
427	53	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) ∈ ℝ)
428	321	adantr 479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐵 − (𝐸‘(𝑆‘𝑗))) ∈ ℝ)
429	320	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) ∈ ℝ)
430	13	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐵 ∈ ℝ)
431	330	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) ≤ 𝐵)
432	275	necomd 2986	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ (𝐸‘(𝑆‘𝑗)) = 𝐵 → 𝐵 ≠ (𝐸‘(𝑆‘𝑗)))
433	432	adantl 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐵 ≠ (𝐸‘(𝑆‘𝑗)))
434	429, 430, 431, 433	leneltd 11418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) < 𝐵)
435	429, 430	posdifd 11851	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘𝑗)) < 𝐵 ↔ 0 < (𝐵 − (𝐸‘(𝑆‘𝑗)))))
436	434, 435	mpbid 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 0 < (𝐵 − (𝐸‘(𝑆‘𝑗))))
437	428, 436	elrpd 13067	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐵 − (𝐸‘(𝑆‘𝑗))) ∈ ℝ⁺)
438	427, 437	ltaddrpd 13103	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) < ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))))
439	438, 319	breqtrrdi 5195	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) < 𝑍)
440	439	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∧ 𝑍 = (𝑆‘𝑖)) → (𝑆‘𝑗) < 𝑍)
441		simpr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∧ 𝑍 = (𝑆‘𝑖)) → 𝑍 = (𝑆‘𝑖))
442	440, 441	breqtrd 5179	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∧ 𝑍 = (𝑆‘𝑖)) → (𝑆‘𝑗) < (𝑆‘𝑖))
443	394	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∧ 𝑍 = (𝑆‘𝑖)) → 𝑍 < (𝑆‘(𝑗 + 1)))
444	441, 443	eqbrtrrd 5177	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∧ 𝑍 = (𝑆‘𝑖)) → (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))
445	442, 444	jca 510	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∧ 𝑍 = (𝑆‘𝑖)) → ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))
446	445	ex 411	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝑍 = (𝑆‘𝑖) → ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))))
447	446	reximdv 3160	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (∃𝑖 ∈ (0...𝑁)𝑍 = (𝑆‘𝑖) → ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))))
448	426, 447	mpd 15	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))
449	315, 448	syldan 589	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))
450	251	nrexdv 3139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ¬ ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))
451	450	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ¬ ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))
452	449, 451	condan 816	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1))
453	308, 452	jca 510	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∧ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)))
454	130	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∈ ℝ)
455	295	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℤ)
456		flbi 13836	. . . . . . . . . 10 ⊢ ((((𝐵 − (𝑆‘𝑗)) / 𝑇) ∈ ℝ ∧ (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℤ) → ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ↔ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∧ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1))))
457	454, 455, 456	syl2anc 582	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ↔ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∧ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1))))
458	453, 457	mpbird 256	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)))
459	458	eqcomd 2732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)))
460	459	oveq1d 7439	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))
461	460	oveq2d 7440	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) = ((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)))
462	461	oveq1d 7439	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) − ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) = (((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) − ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))))
463	266	adantr 479	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘(𝑗 + 1)) ∈ ℂ)
464	138	adantr 479	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) ∈ ℂ)
465	139	adantr 479	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇) ∈ ℂ)
466	463, 464, 465	pnpcan2d 11659	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) − ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))
467	462, 466	eqtrd 2766	. . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) − ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))
468	285, 301, 467	3eqtrd 2770	. 2 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))
469	270, 468	pm2.61dan 811	1 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 {crab 3419 ∪ cun 3945 ifcif 4533 {cpr 4635 class class class wbr 5153 ↦ cmpt 5236 ran crn 5683 ℩cio 6504 Fn wfn 6549 ⟶wf 6550 –onto→wfo 6552 –1-1-onto→wf1o 6553 ‘cfv 6554 Isom wiso 6555 (class class class)co 7424 ↑_m cmap 8855 ℂcc 11156 ℝcr 11157 0cc0 11158 1c1 11159 + caddc 11161 · cmul 11163 +∞cpnf 11295 ℝ^*cxr 11297 < clt 11298 ≤ cle 11299 − cmin 11494 / cdiv 11921 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12610 ℤ_≥cuz 12874 ℝ⁺crp 13028 (,)cioo 13378 (,]cioc 13379 [,]cicc 13381 ...cfz 13538 ..^cfzo 13681 ⌊cfl 13810 ♯chash 14347

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12597 df-z 12611 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ioc 13383 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-rest 17437 df-topgen 17458 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-top 22887 df-topon 22904 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-cmp 23382

This theorem is referenced by: fourierdlem89 45816 fourierdlem90 45817 fourierdlem91 45818

W3C validator