fourierdlem41 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem41 41306

Description: Lemma used to prove that every real is a limit point for the domain of the derivative of the periodic function to be approximated. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem41.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem41.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem41.altb	⊢ (𝜑 → 𝐴 < 𝐵)
fourierdlem41.t	⊢ 𝑇 = (𝐵 − 𝐴)
fourierdlem41.dper	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷)
fourierdlem41.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem41.z	⊢ 𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))
fourierdlem41.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥)))
fourierdlem41.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem41.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem41.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem41.qssd	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷)

**Assertion**
Ref	Expression
fourierdlem41	⊢ (𝜑 → (∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ 𝐷) ∧ ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷)))

Distinct variable groups: 𝐴,𝑚,𝑝 𝑥,𝐴 𝐵,𝑖,𝑘 𝐵,𝑚,𝑝,𝑖 𝑥,𝐵,𝑦,𝑘 𝐷,𝑖,𝑘,𝑦 𝑥,𝐷 𝑖,𝐸,𝑘,𝑦 𝑖,𝑀,𝑘 𝑚,𝑀,𝑝 𝑦,𝑀 𝑄,𝑖,𝑘 𝑄,𝑝 𝑦,𝑄 𝑇,𝑘,𝑥,𝑦 𝑖,𝑋,𝑘 𝑥,𝑋,𝑦 𝑘,𝑍,𝑥,𝑦 𝜑,𝑖,𝑘 𝜑,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑚,𝑝) 𝐴(𝑦,𝑖,𝑘) 𝐷(𝑚,𝑝) 𝑃(𝑥,𝑦,𝑖,𝑘,𝑚,𝑝) 𝑄(𝑥,𝑚) 𝑇(𝑖,𝑚,𝑝) 𝐸(𝑥,𝑚,𝑝) 𝑀(𝑥) 𝑋(𝑚,𝑝) 𝑍(𝑖,𝑚,𝑝)

Proof of Theorem fourierdlem41 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ ran 𝑄) → (𝐸‘𝑋) ∈ ran 𝑄)
2		fourierdlem41.q	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
3		fourierdlem41.m	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4		fourierdlem41.p	. . . . . . . . . . . . 13 ⊢ 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 41267	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
6	3, 5	syl 17	. . . . . . . . . . 11 ⊢ (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 224	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
8	7	simpld 490	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)))
9		elmapi 8164	. . . . . . . . 9 ⊢ (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
10		ffn 6293	. . . . . . . . 9 ⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
11	8, 9, 10	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
12	11	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ ran 𝑄) → 𝑄 Fn (0...𝑀))
13		fvelrnb 6505	. . . . . . 7 ⊢ (𝑄 Fn (0...𝑀) → ((𝐸‘𝑋) ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋)))
14	12, 13	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ((𝐸‘𝑋) ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋)))
15	1, 14	mpbid 224	. . . . 5 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋))
16		0zd 11745	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 ∈ ℤ)
17		elfzelz 12664	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
18	17	3ad2ant2 1125	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ ℤ)
19		1zzd 11765	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 1 ∈ ℤ)
20	18, 19	zsubcld 11844	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ ℤ)
21		simpll 757	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 0 < 𝑗) → 𝜑)
22		elfzle1 12666	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑀) → 0 ≤ 𝑗)
23	22	anim1i 608	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ (0...𝑀) ∧ ¬ 0 < 𝑗) → (0 ≤ 𝑗 ∧ ¬ 0 < 𝑗))
24		0red 10382	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ ¬ 0 < 𝑗) → 0 ∈ ℝ)
25	17	zred 11839	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
26	25	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0...𝑀) ∧ ¬ 0 < 𝑗) → 𝑗 ∈ ℝ)
27	24, 26	eqleltd 10522	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ (0...𝑀) ∧ ¬ 0 < 𝑗) → (0 = 𝑗 ↔ (0 ≤ 𝑗 ∧ ¬ 0 < 𝑗)))
28	23, 27	mpbird 249	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ (0...𝑀) ∧ ¬ 0 < 𝑗) → 0 = 𝑗)
29	28	eqcomd 2784	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0...𝑀) ∧ ¬ 0 < 𝑗) → 𝑗 = 0)
30	29	adantll 704	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 0 < 𝑗) → 𝑗 = 0)
31		fveq2 6448	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 0 → (𝑄‘𝑗) = (𝑄‘0))
32	7	simprld 762	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
33	32	simpld 490	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
34	31, 33	sylan9eqr 2836	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 = 0) → (𝑄‘𝑗) = 𝐴)
35	21, 30, 34	syl2anc 579	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 0 < 𝑗) → (𝑄‘𝑗) = 𝐴)
36	35	3adantl3 1170	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ ¬ 0 < 𝑗) → (𝑄‘𝑗) = 𝐴)
37		simpr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) = (𝐸‘𝑋))
38		fourierdlem41.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ ℝ)
39	38	rexrd 10428	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
40		fourierdlem41.b	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐵 ∈ ℝ)
41	40	rexrd 10428	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
42		fourierdlem41.altb	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐴 < 𝐵)
43		fourierdlem41.t	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑇 = (𝐵 − 𝐴)
44		eqid 2778	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
45	38, 40, 42, 43, 44	fourierdlem4 41269	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))):ℝ⟶(𝐴(,]𝐵))
46		fourierdlem41.e	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥)))
47	46	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥))))
48		simpr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
49	40	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐵 ∈ ℝ)
50	49, 48	resubcld 10806	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐵 − 𝑥) ∈ ℝ)
51	40, 38	resubcld 10806	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
52	43, 51	syl5eqel 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 𝑇 ∈ ℝ)
53	52	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
54		0red 10382	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 0 ∈ ℝ)
55	38, 40	posdifd 10965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
56	42, 55	mpbid 224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
57	43	eqcomi 2787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝐵 − 𝐴) = 𝑇
58	57	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝐵 − 𝐴) = 𝑇)
59	56, 58	breqtrd 4914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 0 < 𝑇)
60	54, 59	gtned 10513	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 𝑇 ≠ 0)
61	60	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑇 ≠ 0)
62	50, 53, 61	redivcld 11206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐵 − 𝑥) / 𝑇) ∈ ℝ)
63	62	flcld 12923	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ)
64	63	zred 11839	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℝ)
65	64, 53	remulcld 10409	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) ∈ ℝ)
66		fourierdlem41.z	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))
67	66	fvmpt2 6554	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ ℝ ∧ ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) ∈ ℝ) → (𝑍‘𝑥) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))
68	48, 65, 67	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑍‘𝑥) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))
69	68	oveq2d 6940	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 + (𝑍‘𝑥)) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
70	69	mpteq2dva 4981	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥))) = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
71	47, 70	eqtrd 2814	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
72	71	feq1d 6278	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐸:ℝ⟶(𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))):ℝ⟶(𝐴(,]𝐵)))
73	45, 72	mpbird 249	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵))
74		fourierdlem41.x	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑋 ∈ ℝ)
75	73, 74	ffvelrnd 6626	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (𝐴(,]𝐵))
76		iocgtlb 40650	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐴 < (𝐸‘𝑋))
77	39, 41, 75, 76	syl3anc 1439	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 < (𝐸‘𝑋))
78	38, 77	gtned 10513	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸‘𝑋) ≠ 𝐴)
79	78	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≠ 𝐴)
80	37, 79	eqnetrd 3036	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) ≠ 𝐴)
81	80	adantr 474	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ ¬ 0 < 𝑗) → (𝑄‘𝑗) ≠ 𝐴)
82	81	3adantl2 1169	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ ¬ 0 < 𝑗) → (𝑄‘𝑗) ≠ 𝐴)
83	82	neneqd 2974	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ ¬ 0 < 𝑗) → ¬ (𝑄‘𝑗) = 𝐴)
84	36, 83	condan 808	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 < 𝑗)
85		zltlem1 11787	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (0 < 𝑗 ↔ 0 ≤ (𝑗 − 1)))
86	16, 18, 85	syl2anc 579	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (0 < 𝑗 ↔ 0 ≤ (𝑗 − 1)))
87	84, 86	mpbid 224	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 ≤ (𝑗 − 1))
88		eluz2 12003	. . . . . . . . . . 11 ⊢ ((𝑗 − 1) ∈ (ℤ_≥‘0) ↔ (0 ∈ ℤ ∧ (𝑗 − 1) ∈ ℤ ∧ 0 ≤ (𝑗 − 1)))
89	16, 20, 87, 88	syl3anbrc 1400	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ (ℤ_≥‘0))
90		elfzel2 12662	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
91	90	3ad2ant2 1125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑀 ∈ ℤ)
92		1red 10379	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑀) → 1 ∈ ℝ)
93	25, 92	resubcld 10806	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℝ)
94	90	zred 11839	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
95	25	ltm1d 11313	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑗)
96		elfzle2 12667	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ≤ 𝑀)
97	93, 25, 94, 95, 96	ltletrd 10538	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑀)
98	97	3ad2ant2 1125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) < 𝑀)
99		elfzo2 12797	. . . . . . . . . 10 ⊢ ((𝑗 − 1) ∈ (0..^𝑀) ↔ ((𝑗 − 1) ∈ (ℤ_≥‘0) ∧ 𝑀 ∈ ℤ ∧ (𝑗 − 1) < 𝑀))
100	89, 91, 98, 99	syl3anbrc 1400	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ (0..^𝑀))
101	8, 9	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
102	101	3ad2ant1 1124	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑄:(0...𝑀)⟶ℝ)
103	16, 91, 20	3jca 1119	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑗 − 1) ∈ ℤ))
104	93, 94, 97	ltled 10526	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ≤ 𝑀)
105	104	3ad2ant2 1125	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ≤ 𝑀)
106	103, 87, 105	jca32 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑗 − 1) ∈ ℤ) ∧ (0 ≤ (𝑗 − 1) ∧ (𝑗 − 1) ≤ 𝑀)))
107		elfz2 12655	. . . . . . . . . . . . 13 ⊢ ((𝑗 − 1) ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑗 − 1) ∈ ℤ) ∧ (0 ≤ (𝑗 − 1) ∧ (𝑗 − 1) ≤ 𝑀)))
108	106, 107	sylibr 226	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ (0...𝑀))
109	102, 108	ffvelrnd 6626	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) ∈ ℝ)
110	109	rexrd 10428	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) ∈ ℝ^*)
111	25	recnd 10407	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ)
112		1cnd 10373	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑀) → 1 ∈ ℂ)
113	111, 112	npcand 10740	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 − 1) + 1) = 𝑗)
114	113	fveq2d 6452	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑀) → (𝑄‘((𝑗 − 1) + 1)) = (𝑄‘𝑗))
115	114	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘((𝑗 − 1) + 1)) = (𝑄‘𝑗))
116	101	ffvelrnda 6625	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
117	116	rexrd 10428	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ^*)
118	115, 117	eqeltrd 2859	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘((𝑗 − 1) + 1)) ∈ ℝ^*)
119	118	3adant3 1123	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘((𝑗 − 1) + 1)) ∈ ℝ^*)
120		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
121		fveq2 6448	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑋 → (𝑍‘𝑥) = (𝑍‘𝑋))
122	120, 121	oveq12d 6942	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → (𝑥 + (𝑍‘𝑥)) = (𝑋 + (𝑍‘𝑋)))
123	122	adantl 475	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥 + (𝑍‘𝑥)) = (𝑋 + (𝑍‘𝑋)))
124	66	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
125		oveq2 6932	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑋 → (𝐵 − 𝑥) = (𝐵 − 𝑋))
126	125	oveq1d 6939	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑋 → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − 𝑋) / 𝑇))
127	126	fveq2d 6452	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑋 → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − 𝑋) / 𝑇)))
128	127	oveq1d 6939	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑋 → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
129	128	adantl 475	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
130	40, 74	resubcld 10806	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
131	130, 52, 60	redivcld 11206	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐵 − 𝑋) / 𝑇) ∈ ℝ)
132	131	flcld 12923	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
133	132	zred 11839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℝ)
134	133, 52	remulcld 10409	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) ∈ ℝ)
135	124, 129, 74, 134	fvmptd 6550	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑍‘𝑋) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
136	135, 134	eqeltrd 2859	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑍‘𝑋) ∈ ℝ)
137	74, 136	readdcld 10408	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + (𝑍‘𝑋)) ∈ ℝ)
138	47, 123, 74, 137	fvmptd 6550	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸‘𝑋) = (𝑋 + (𝑍‘𝑋)))
139	138, 137	eqeltrd 2859	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ℝ)
140	139	rexrd 10428	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ℝ^*)
141	140	3ad2ant1 1124	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ℝ^*)
142		simp1 1127	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝜑)
143		ovex 6956	. . . . . . . . . . . . . 14 ⊢ (𝑗 − 1) ∈ V
144		eleq1 2847	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝑗 − 1) → (𝑖 ∈ (0..^𝑀) ↔ (𝑗 − 1) ∈ (0..^𝑀)))
145	144	anbi2d 622	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = (𝑗 − 1) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ (𝑗 − 1) ∈ (0..^𝑀))))
146		fveq2 6448	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝑗 − 1) → (𝑄‘𝑖) = (𝑄‘(𝑗 − 1)))
147		oveq1 6931	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = (𝑗 − 1) → (𝑖 + 1) = ((𝑗 − 1) + 1))
148	147	fveq2d 6452	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝑗 − 1) → (𝑄‘(𝑖 + 1)) = (𝑄‘((𝑗 − 1) + 1)))
149	146, 148	breq12d 4901	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = (𝑗 − 1) → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1))))
150	145, 149	imbi12d 336	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑗 − 1) → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ (𝑗 − 1) ∈ (0..^𝑀)) → (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1)))))
151	7	simprrd 764	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
152	151	r19.21bi 3114	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
153	143, 150, 152	vtocl 3460	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 − 1) ∈ (0..^𝑀)) → (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1)))
154	142, 100, 153	syl2anc 579	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1)))
155	114	3ad2ant2 1125	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘((𝑗 − 1) + 1)) = (𝑄‘𝑗))
156	154, 155	breqtrd 4914	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) < (𝑄‘𝑗))
157		simp3 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) = (𝐸‘𝑋))
158	156, 157	breqtrd 4914	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) < (𝐸‘𝑋))
159	139	leidd 10944	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘𝑋) ≤ (𝐸‘𝑋))
160	159	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝐸‘𝑋))
161	37	eqcomd 2784	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) = (𝑄‘𝑗))
162	160, 161	breqtrd 4914	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝑄‘𝑗))
163	162	3adant2 1122	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝑄‘𝑗))
164	113	eqcomd 2784	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 = ((𝑗 − 1) + 1))
165	164	fveq2d 6452	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑀) → (𝑄‘𝑗) = (𝑄‘((𝑗 − 1) + 1)))
166	165	3ad2ant2 1125	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) = (𝑄‘((𝑗 − 1) + 1)))
167	163, 166	breqtrd 4914	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝑄‘((𝑗 − 1) + 1)))
168	110, 119, 141, 158, 167	eliocd 40656	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1))))
169	146, 148	oveq12d 6942	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 − 1) → ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) = ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1))))
170	169	eleq2d 2845	. . . . . . . . . 10 ⊢ (𝑖 = (𝑗 − 1) → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) ↔ (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1)))))
171	170	rspcev 3511	. . . . . . . . 9 ⊢ (((𝑗 − 1) ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1)))) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
172	100, 168, 171	syl2anc 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
173	172	3exp 1109	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ (0...𝑀) → ((𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))))
174	173	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ ran 𝑄) → (𝑗 ∈ (0...𝑀) → ((𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))))
175	174	rexlimdv 3212	. . . . 5 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ ran 𝑄) → (∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))))
176	15, 175	mpd 15	. . . 4 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
177	3	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → 𝑀 ∈ ℕ)
178	101	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
179		iocssicc 12579	. . . . . . . 8 ⊢ ((𝑄‘0)(,](𝑄‘𝑀)) ⊆ ((𝑄‘0)[,](𝑄‘𝑀))
180	32	simprd 491	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
181	33, 180	oveq12d 6942	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄‘0)(,](𝑄‘𝑀)) = (𝐴(,]𝐵))
182	75, 181	eleqtrrd 2862	. . . . . . . 8 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ((𝑄‘0)(,](𝑄‘𝑀)))
183	179, 182	sseldi 3819	. . . . . . 7 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
184	183	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → (𝐸‘𝑋) ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
185		simpr 479	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ¬ (𝐸‘𝑋) ∈ ran 𝑄)
186		fveq2 6448	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑄‘𝑘) = (𝑄‘𝑗))
187	186	breq1d 4898	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝑄‘𝑘) < (𝐸‘𝑋) ↔ (𝑄‘𝑗) < (𝐸‘𝑋)))
188	187	cbvrabv 3396	. . . . . . 7 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < (𝐸‘𝑋)} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < (𝐸‘𝑋)}
189	188	supeq1i 8643	. . . . . 6 ⊢ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < (𝐸‘𝑋)}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < (𝐸‘𝑋)}, ℝ, < )
190	177, 178, 184, 185, 189	fourierdlem25 41290	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
191		ioossioc 40639	. . . . . . . 8 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))
192	191	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
193	192	sseld 3820	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))))
194	193	reximdva 3198	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → (∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))))
195	190, 194	mpd 15	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
196	176, 195	pm2.61dan 803	. . 3 ⊢ (𝜑 → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
197	101	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
198		elfzofz 12809	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
199	198	adantl 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
200	197, 199	ffvelrnd 6626	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
201	200	3adant3 1123	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
202	136	3ad2ant1 1124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑍‘𝑋) ∈ ℝ)
203	201, 202	resubcld 10806	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ)
204	139	3ad2ant1 1124	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ℝ)
205	201	rexrd 10428	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
206		fzofzp1 12889	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
207	206	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
208	197, 207	ffvelrnd 6626	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
209	208	rexrd 10428	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
210	209	3adant3 1123	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
211		simp3 1129	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
212		iocgtlb 40650	. . . . . . . . 9 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝐸‘𝑋))
213	205, 210, 211, 212	syl3anc 1439	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝐸‘𝑋))
214	201, 204, 202, 213	ltsub1dd 10990	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < ((𝐸‘𝑋) − (𝑍‘𝑋)))
215	138	oveq1d 6939	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸‘𝑋) − (𝑍‘𝑋)) = ((𝑋 + (𝑍‘𝑋)) − (𝑍‘𝑋)))
216	74	recnd 10407	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℂ)
217	136	recnd 10407	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍‘𝑋) ∈ ℂ)
218	216, 217	pncand 10737	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = 𝑋)
219	215, 218	eqtrd 2814	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘𝑋) − (𝑍‘𝑋)) = 𝑋)
220	219	3ad2ant1 1124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐸‘𝑋) − (𝑍‘𝑋)) = 𝑋)
221	214, 220	breqtrd 4914	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋)
222		elioore 12522	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) → 𝑦 ∈ ℝ)
223	135	oveq2d 6940	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑦 + (𝑍‘𝑋)) = (𝑦 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
224	133	recnd 10407	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℂ)
225	52	recnd 10407	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 ∈ ℂ)
226	224, 225	mulneg1d 10831	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) = -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
227	223, 226	oveq12d 6942	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = ((𝑦 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) + -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
228	227	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = ((𝑦 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) + -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
229		simpr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
230	134	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) ∈ ℝ)
231	229, 230	readdcld 10408	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ ℝ)
232	231	recnd 10407	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ ℂ)
233	230	recnd 10407	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) ∈ ℂ)
234	232, 233	negsubd 10742	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) + -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = ((𝑦 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) − ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
235	229	recnd 10407	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
236	235, 233	pncand 10737	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) − ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = 𝑦)
237	228, 234, 236	3eqtrrd 2819	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
238	222, 237	sylan2 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
239	238	3ad2antl1 1193	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
240		simpl1 1199	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝜑)
241		fourierdlem41.qssd	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷)
242	241	3adant3 1123	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷)
243	242	adantr 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷)
244	205	adantr 474	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) ∈ ℝ^*)
245	210	adantr 474	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
246	222	adantl 475	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ ℝ)
247	136	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑍‘𝑋) ∈ ℝ)
248	246, 247	readdcld 10408	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ℝ)
249	248	3ad2antl1 1193	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ℝ)
250	136	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑍‘𝑋) ∈ ℝ)
251	200, 250	resubcld 10806	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ)
252	251	rexrd 10428	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^*)
253	252	adantr 474	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^*)
254	74	rexrd 10428	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
255	254	ad2antrr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑋 ∈ ℝ^*)
256		simpr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋))
257		ioogtlb 40643	. . . . . . . . . . . . . . 15 ⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^* ∧ 𝑋 ∈ ℝ^* ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑦)
258	253, 255, 256, 257	syl3anc 1439	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑦)
259	200	adantr 474	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) ∈ ℝ)
260	136	ad2antrr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑍‘𝑋) ∈ ℝ)
261	222	adantl 475	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ ℝ)
262	259, 260, 261	ltsubaddd 10974	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑦 ↔ (𝑄‘𝑖) < (𝑦 + (𝑍‘𝑋))))
263	258, 262	mpbid 224	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) < (𝑦 + (𝑍‘𝑋)))
264	263	3adantl3 1170	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) < (𝑦 + (𝑍‘𝑋)))
265	240, 139	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐸‘𝑋) ∈ ℝ)
266	208	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
267	266	3adantl3 1170	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
268	74	ad2antrr 716	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑋 ∈ ℝ)
269		iooltub 40659	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^* ∧ 𝑋 ∈ ℝ^* ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 < 𝑋)
270	253, 255, 256, 269	syl3anc 1439	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 < 𝑋)
271	261, 268, 260, 270	ltadd1dd 10989	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) < (𝑋 + (𝑍‘𝑋)))
272	138	eqcomd 2784	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑋 + (𝑍‘𝑋)) = (𝐸‘𝑋))
273	272	ad2antrr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑋 + (𝑍‘𝑋)) = (𝐸‘𝑋))
274	271, 273	breqtrd 4914	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) < (𝐸‘𝑋))
275	274	3adantl3 1170	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) < (𝐸‘𝑋))
276		iocleub 40651	. . . . . . . . . . . . . . 15 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1)))
277	205, 210, 211, 276	syl3anc 1439	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1)))
278	277	adantr 474	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1)))
279	249, 265, 267, 275, 278	ltletrd 10538	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) < (𝑄‘(𝑖 + 1)))
280	244, 245, 249, 264, 279	eliood 40646	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
281	243, 280	sseldd 3822	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ 𝐷)
282	240, 131	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝐵 − 𝑋) / 𝑇) ∈ ℝ)
283	282	flcld 12923	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
284	283	znegcld 11841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
285		negex 10622	. . . . . . . . . . 11 ⊢ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ V
286		eleq1 2847	. . . . . . . . . . . . 13 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 ∈ ℤ ↔ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ))
287	286	3anbi3d 1515	. . . . . . . . . . . 12 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)))
288		oveq1 6931	. . . . . . . . . . . . . 14 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 · 𝑇) = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
289	288	oveq2d 6940	. . . . . . . . . . . . 13 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
290	289	eleq1d 2844	. . . . . . . . . . . 12 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷 ↔ ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷))
291	287, 290	imbi12d 336	. . . . . . . . . . 11 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷) ↔ ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷)))
292		ovex 6956	. . . . . . . . . . . 12 ⊢ (𝑦 + (𝑍‘𝑋)) ∈ V
293		eleq1 2847	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → (𝑥 ∈ 𝐷 ↔ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷))
294	293	3anbi2d 1514	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ)))
295		oveq1 6931	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → (𝑥 + (𝑘 · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)))
296	295	eleq1d 2844	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → ((𝑥 + (𝑘 · 𝑇)) ∈ 𝐷 ↔ ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷))
297	294, 296	imbi12d 336	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → (((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷) ↔ ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷)))
298		fourierdlem41.dper	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷)
299	292, 297, 298	vtocl 3460	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷)
300	285, 291, 299	vtocl 3460	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷)
301	240, 281, 284, 300	syl3anc 1439	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷)
302	239, 301	eqeltrd 2859	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ 𝐷)
303	302	ralrimiva 3148	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ∀𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑦 ∈ 𝐷)
304		dfss3 3810	. . . . . . 7 ⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ 𝐷 ↔ ∀𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑦 ∈ 𝐷)
305	303, 304	sylibr 226	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ 𝐷)
306		breq1 4891	. . . . . . . 8 ⊢ (𝑦 = ((𝑄‘𝑖) − (𝑍‘𝑋)) → (𝑦 < 𝑋 ↔ ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋))
307		oveq1 6931	. . . . . . . . 9 ⊢ (𝑦 = ((𝑄‘𝑖) − (𝑍‘𝑋)) → (𝑦(,)𝑋) = (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋))
308	307	sseq1d 3851	. . . . . . . 8 ⊢ (𝑦 = ((𝑄‘𝑖) − (𝑍‘𝑋)) → ((𝑦(,)𝑋) ⊆ 𝐷 ↔ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ 𝐷))
309	306, 308	anbi12d 624	. . . . . . 7 ⊢ (𝑦 = ((𝑄‘𝑖) − (𝑍‘𝑋)) → ((𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ 𝐷) ↔ (((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋 ∧ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ 𝐷)))
310	309	rspcev 3511	. . . . . 6 ⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ ∧ (((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋 ∧ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ 𝐷)) → ∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ 𝐷))
311	203, 221, 305, 310	syl12anc 827	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ 𝐷))
312	311	3exp 1109	. . . 4 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) → ∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ 𝐷))))
313	312	rexlimdv 3212	. . 3 ⊢ (𝜑 → (∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) → ∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ 𝐷)))
314	196, 313	mpd 15	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ 𝐷))
315		0zd 11745	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
316	3	nnzd 11838	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
317		1zzd 11765	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
318	315, 316, 317	3jca 1119	. . . . . . . . 9 ⊢ (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 1 ∈ ℤ))
319		0le1 10901	. . . . . . . . . 10 ⊢ 0 ≤ 1
320	319	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 1)
321	3	nnge1d 11428	. . . . . . . . 9 ⊢ (𝜑 → 1 ≤ 𝑀)
322	318, 320, 321	jca32 511	. . . . . . . 8 ⊢ (𝜑 → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (0 ≤ 1 ∧ 1 ≤ 𝑀)))
323		elfz2 12655	. . . . . . . 8 ⊢ (1 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (0 ≤ 1 ∧ 1 ≤ 𝑀)))
324	322, 323	sylibr 226	. . . . . . 7 ⊢ (𝜑 → 1 ∈ (0...𝑀))
325	101, 324	ffvelrnd 6626	. . . . . 6 ⊢ (𝜑 → (𝑄‘1) ∈ ℝ)
326	136, 52	resubcld 10806	. . . . . 6 ⊢ (𝜑 → ((𝑍‘𝑋) − 𝑇) ∈ ℝ)
327	325, 326	resubcld 10806	. . . . 5 ⊢ (𝜑 → ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) ∈ ℝ)
328	327	adantr 474	. . . 4 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) ∈ ℝ)
329	38	recnd 10407	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℂ)
330	329, 225	pncand 10737	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝑇) − 𝑇) = 𝐴)
331	330	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → ((𝐴 + 𝑇) − 𝑇) = 𝐴)
332	43	oveq2i 6935	. . . . . . . . . . 11 ⊢ (𝐴 + 𝑇) = (𝐴 + (𝐵 − 𝐴))
333	332	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → (𝐴 + 𝑇) = (𝐴 + (𝐵 − 𝐴)))
334	40	recnd 10407	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℂ)
335	329, 334	pncan3d 10739	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
336	335	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
337		id 22	. . . . . . . . . . . 12 ⊢ ((𝐸‘𝑋) = 𝐵 → (𝐸‘𝑋) = 𝐵)
338	337	eqcomd 2784	. . . . . . . . . . 11 ⊢ ((𝐸‘𝑋) = 𝐵 → 𝐵 = (𝐸‘𝑋))
339	338	adantl 475	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → 𝐵 = (𝐸‘𝑋))
340	333, 336, 339	3eqtrrd 2819	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → (𝐸‘𝑋) = (𝐴 + 𝑇))
341	340	oveq1d 6939	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → ((𝐸‘𝑋) − 𝑇) = ((𝐴 + 𝑇) − 𝑇))
342	33	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → (𝑄‘0) = 𝐴)
343	331, 341, 342	3eqtr4rd 2825	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → (𝑄‘0) = ((𝐸‘𝑋) − 𝑇))
344	343	oveq1d 6939	. . . . . 6 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → ((𝑄‘0) − ((𝑍‘𝑋) − 𝑇)) = (((𝐸‘𝑋) − 𝑇) − ((𝑍‘𝑋) − 𝑇)))
345	139	recnd 10407	. . . . . . . 8 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ℂ)
346	345, 217, 225	nnncan2d 10771	. . . . . . 7 ⊢ (𝜑 → (((𝐸‘𝑋) − 𝑇) − ((𝑍‘𝑋) − 𝑇)) = ((𝐸‘𝑋) − (𝑍‘𝑋)))
347	346	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → (((𝐸‘𝑋) − 𝑇) − ((𝑍‘𝑋) − 𝑇)) = ((𝐸‘𝑋) − (𝑍‘𝑋)))
348	219	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → ((𝐸‘𝑋) − (𝑍‘𝑋)) = 𝑋)
349	344, 347, 348	3eqtrrd 2819	. . . . 5 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → 𝑋 = ((𝑄‘0) − ((𝑍‘𝑋) − 𝑇)))
350	33, 38	eqeltrd 2859	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
351	3	nngt0d 11429	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝑀)
352		fzolb 12800	. . . . . . . . . 10 ⊢ (0 ∈ (0..^𝑀) ↔ (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
353	315, 316, 351, 352	syl3anbrc 1400	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0..^𝑀))
354		0re 10380	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
355		eleq1 2847	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑖 ∈ (0..^𝑀) ↔ 0 ∈ (0..^𝑀)))
356	355	anbi2d 622	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 0 ∈ (0..^𝑀))))
357		fveq2 6448	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑄‘𝑖) = (𝑄‘0))
358		oveq1 6931	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → (𝑖 + 1) = (0 + 1))
359	358	fveq2d 6452	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑄‘(𝑖 + 1)) = (𝑄‘(0 + 1)))
360	357, 359	breq12d 4901	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘0) < (𝑄‘(0 + 1))))
361	356, 360	imbi12d 336	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 0 ∈ (0..^𝑀)) → (𝑄‘0) < (𝑄‘(0 + 1)))))
362	361, 152	vtoclg 3467	. . . . . . . . . 10 ⊢ (0 ∈ ℝ → ((𝜑 ∧ 0 ∈ (0..^𝑀)) → (𝑄‘0) < (𝑄‘(0 + 1))))
363	354, 362	ax-mp 5	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ∈ (0..^𝑀)) → (𝑄‘0) < (𝑄‘(0 + 1)))
364	353, 363	mpdan 677	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘0) < (𝑄‘(0 + 1)))
365		0p1e1 11509	. . . . . . . . . 10 ⊢ (0 + 1) = 1
366	365	fveq2i 6451	. . . . . . . . 9 ⊢ (𝑄‘(0 + 1)) = (𝑄‘1)
367	366	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘(0 + 1)) = (𝑄‘1))
368	364, 367	breqtrd 4914	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) < (𝑄‘1))
369	350, 325, 326, 368	ltsub1dd 10990	. . . . . 6 ⊢ (𝜑 → ((𝑄‘0) − ((𝑍‘𝑋) − 𝑇)) < ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))
370	369	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → ((𝑄‘0) − ((𝑍‘𝑋) − 𝑇)) < ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))
371	349, 370	eqbrtrd 4910	. . . 4 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → 𝑋 < ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))
372		elioore 12522	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇))) → 𝑦 ∈ ℝ)
373	135	eqcomd 2784	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) = (𝑍‘𝑋))
374	373	negeqd 10618	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) = -(𝑍‘𝑋))
375	226, 374	eqtrd 2814	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) = -(𝑍‘𝑋))
376	225	mulid2d 10397	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 · 𝑇) = 𝑇)
377	375, 376	oveq12d 6942	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) + (1 · 𝑇)) = (-(𝑍‘𝑋) + 𝑇))
378	224	negcld 10723	. . . . . . . . . . . . . 14 ⊢ (𝜑 → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℂ)
379		1cnd 10373	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℂ)
380	378, 379, 225	adddird 10404	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇) = ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) + (1 · 𝑇)))
381	217, 225	negsubdid 10751	. . . . . . . . . . . . 13 ⊢ (𝜑 → -((𝑍‘𝑋) − 𝑇) = (-(𝑍‘𝑋) + 𝑇))
382	377, 380, 381	3eqtr4d 2824	. . . . . . . . . . . 12 ⊢ (𝜑 → ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇) = -((𝑍‘𝑋) − 𝑇))
383	382	oveq2d 6940	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇)) = ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + -((𝑍‘𝑋) − 𝑇)))
384	383	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇)) = ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + -((𝑍‘𝑋) − 𝑇)))
385	326	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑍‘𝑋) − 𝑇) ∈ ℝ)
386	229, 385	readdcld 10408	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ ℝ)
387	386	recnd 10407	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ ℂ)
388	385	recnd 10407	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑍‘𝑋) − 𝑇) ∈ ℂ)
389	387, 388	negsubd 10742	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + -((𝑍‘𝑋) − 𝑇)) = ((𝑦 + ((𝑍‘𝑋) − 𝑇)) − ((𝑍‘𝑋) − 𝑇)))
390	235, 388	pncand 10737	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 + ((𝑍‘𝑋) − 𝑇)) − ((𝑍‘𝑋) − 𝑇)) = 𝑦)
391	384, 389, 390	3eqtrrd 2819	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 = ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇)))
392	372, 391	sylan2 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → 𝑦 = ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇)))
393	392	adantlr 705	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → 𝑦 = ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇)))
394		simpll 757	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → 𝜑)
395	367	eqcomd 2784	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘1) = (𝑄‘(0 + 1)))
396	395	oveq2d 6940	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄‘0)(,)(𝑄‘1)) = ((𝑄‘0)(,)(𝑄‘(0 + 1))))
397	357, 359	oveq12d 6942	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 0 → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄‘0)(,)(𝑄‘(0 + 1))))
398	397	sseq1d 3851	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 0 → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷 ↔ ((𝑄‘0)(,)(𝑄‘(0 + 1))) ⊆ 𝐷))
399	356, 398	imbi12d 336	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷) ↔ ((𝜑 ∧ 0 ∈ (0..^𝑀)) → ((𝑄‘0)(,)(𝑄‘(0 + 1))) ⊆ 𝐷)))
400	399, 241	vtoclg 3467	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → ((𝜑 ∧ 0 ∈ (0..^𝑀)) → ((𝑄‘0)(,)(𝑄‘(0 + 1))) ⊆ 𝐷))
401	354, 400	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 ∈ (0..^𝑀)) → ((𝑄‘0)(,)(𝑄‘(0 + 1))) ⊆ 𝐷)
402	353, 401	mpdan 677	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄‘0)(,)(𝑄‘(0 + 1))) ⊆ 𝐷)
403	396, 402	eqsstrd 3858	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄‘0)(,)(𝑄‘1)) ⊆ 𝐷)
404	403	ad2antrr 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → ((𝑄‘0)(,)(𝑄‘1)) ⊆ 𝐷)
405	33, 39	eqeltrd 2859	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ^*)
406	405	ad2antrr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑄‘0) ∈ ℝ^*)
407	325	rexrd 10428	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄‘1) ∈ ℝ^*)
408	407	ad2antrr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑄‘1) ∈ ℝ^*)
409	372, 386	sylan2 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ ℝ)
410	409	adantlr 705	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ ℝ)
411	345, 216, 217	subaddd 10754	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝐸‘𝑋) − 𝑋) = (𝑍‘𝑋) ↔ (𝑋 + (𝑍‘𝑋)) = (𝐸‘𝑋)))
412	272, 411	mpbird 249	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸‘𝑋) − 𝑋) = (𝑍‘𝑋))
413		oveq1 6931	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸‘𝑋) = 𝐵 → ((𝐸‘𝑋) − 𝑋) = (𝐵 − 𝑋))
414	412, 413	sylan9req 2835	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → (𝑍‘𝑋) = (𝐵 − 𝑋))
415	414	oveq1d 6939	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → ((𝑍‘𝑋) − 𝑇) = ((𝐵 − 𝑋) − 𝑇))
416	415	oveq2d 6940	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → (𝑋 + ((𝑍‘𝑋) − 𝑇)) = (𝑋 + ((𝐵 − 𝑋) − 𝑇)))
417	130	recnd 10407	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℂ)
418	216, 417, 225	addsubassd 10756	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑋 + (𝐵 − 𝑋)) − 𝑇) = (𝑋 + ((𝐵 − 𝑋) − 𝑇)))
419	418	eqcomd 2784	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + ((𝐵 − 𝑋) − 𝑇)) = ((𝑋 + (𝐵 − 𝑋)) − 𝑇))
420	419	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → (𝑋 + ((𝐵 − 𝑋) − 𝑇)) = ((𝑋 + (𝐵 − 𝑋)) − 𝑇))
421	334, 225, 329	subsub23d 40423	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐵 − 𝑇) = 𝐴 ↔ (𝐵 − 𝐴) = 𝑇))
422	58, 421	mpbird 249	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 𝑇) = 𝐴)
423	216, 334	pncan3d 10739	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑋 + (𝐵 − 𝑋)) = 𝐵)
424	423	oveq1d 6939	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑋 + (𝐵 − 𝑋)) − 𝑇) = (𝐵 − 𝑇))
425	422, 424, 33	3eqtr4d 2824	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑋 + (𝐵 − 𝑋)) − 𝑇) = (𝑄‘0))
426	425	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → ((𝑋 + (𝐵 − 𝑋)) − 𝑇) = (𝑄‘0))
427	416, 420, 426	3eqtrrd 2819	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → (𝑄‘0) = (𝑋 + ((𝑍‘𝑋) − 𝑇)))
428	427	adantr 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑄‘0) = (𝑋 + ((𝑍‘𝑋) − 𝑇)))
429	74	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → 𝑋 ∈ ℝ)
430	372	adantl 475	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → 𝑦 ∈ ℝ)
431	326	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → ((𝑍‘𝑋) − 𝑇) ∈ ℝ)
432	254	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → 𝑋 ∈ ℝ^*)
433	327	rexrd 10428	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) ∈ ℝ^*)
434	433	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) ∈ ℝ^*)
435		simpr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇))))
436		ioogtlb 40643	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℝ^* ∧ ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) ∈ ℝ^* ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → 𝑋 < 𝑦)
437	432, 434, 435, 436	syl3anc 1439	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → 𝑋 < 𝑦)
438	429, 430, 431, 437	ltadd1dd 10989	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑋 + ((𝑍‘𝑋) − 𝑇)) < (𝑦 + ((𝑍‘𝑋) − 𝑇)))
439	438	adantlr 705	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑋 + ((𝑍‘𝑋) − 𝑇)) < (𝑦 + ((𝑍‘𝑋) − 𝑇)))
440	428, 439	eqbrtrd 4910	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑄‘0) < (𝑦 + ((𝑍‘𝑋) − 𝑇)))
441		iooltub 40659	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ ℝ^* ∧ ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) ∈ ℝ^* ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → 𝑦 < ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))
442	432, 434, 435, 441	syl3anc 1439	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → 𝑦 < ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))
443	325	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑄‘1) ∈ ℝ)
444	430, 431, 443	ltaddsubd 10978	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → ((𝑦 + ((𝑍‘𝑋) − 𝑇)) < (𝑄‘1) ↔ 𝑦 < ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇))))
445	442, 444	mpbird 249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑦 + ((𝑍‘𝑋) − 𝑇)) < (𝑄‘1))
446	445	adantlr 705	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑦 + ((𝑍‘𝑋) − 𝑇)) < (𝑄‘1))
447	406, 408, 410, 440, 446	eliood 40646	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ ((𝑄‘0)(,)(𝑄‘1)))
448	404, 447	sseldd 3822	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ 𝐷)
449	132	znegcld 11841	. . . . . . . . . 10 ⊢ (𝜑 → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
450	449	peano2zd 11842	. . . . . . . . 9 ⊢ (𝜑 → (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) ∈ ℤ)
451	450	ad2antrr 716	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) ∈ ℤ)
452		ovex 6956	. . . . . . . . 9 ⊢ (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) ∈ V
453		eleq1 2847	. . . . . . . . . . 11 ⊢ (𝑘 = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) → (𝑘 ∈ ℤ ↔ (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) ∈ ℤ))
454	453	3anbi3d 1515	. . . . . . . . . 10 ⊢ (𝑘 = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) → ((𝜑 ∧ (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ 𝐷 ∧ (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) ∈ ℤ)))
455		oveq1 6931	. . . . . . . . . . . 12 ⊢ (𝑘 = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) → (𝑘 · 𝑇) = ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇))
456	455	oveq2d 6940	. . . . . . . . . . 11 ⊢ (𝑘 = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) → ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + (𝑘 · 𝑇)) = ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇)))
457	456	eleq1d 2844	. . . . . . . . . 10 ⊢ (𝑘 = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) → (((𝑦 + ((𝑍‘𝑋) − 𝑇)) + (𝑘 · 𝑇)) ∈ 𝐷 ↔ ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇)) ∈ 𝐷))
458	454, 457	imbi12d 336	. . . . . . . . 9 ⊢ (𝑘 = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) → (((𝜑 ∧ (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + (𝑘 · 𝑇)) ∈ 𝐷) ↔ ((𝜑 ∧ (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ 𝐷 ∧ (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) ∈ ℤ) → ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇)) ∈ 𝐷)))
459		ovex 6956	. . . . . . . . . 10 ⊢ (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ V
460		eleq1 2847	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 + ((𝑍‘𝑋) − 𝑇)) → (𝑥 ∈ 𝐷 ↔ (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ 𝐷))
461	460	3anbi2d 1514	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 + ((𝑍‘𝑋) − 𝑇)) → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ)))
462		oveq1 6931	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 + ((𝑍‘𝑋) − 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + (𝑘 · 𝑇)))
463	462	eleq1d 2844	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 + ((𝑍‘𝑋) − 𝑇)) → ((𝑥 + (𝑘 · 𝑇)) ∈ 𝐷 ↔ ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + (𝑘 · 𝑇)) ∈ 𝐷))
464	461, 463	imbi12d 336	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 + ((𝑍‘𝑋) − 𝑇)) → (((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷) ↔ ((𝜑 ∧ (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + (𝑘 · 𝑇)) ∈ 𝐷)))
465	459, 464, 298	vtocl 3460	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + (𝑘 · 𝑇)) ∈ 𝐷)
466	452, 458, 465	vtocl 3460	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 + ((𝑍‘𝑋) − 𝑇)) ∈ 𝐷 ∧ (-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) ∈ ℤ) → ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇)) ∈ 𝐷)
467	394, 448, 451, 466	syl3anc 1439	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → ((𝑦 + ((𝑍‘𝑋) − 𝑇)) + ((-(⌊‘((𝐵 − 𝑋) / 𝑇)) + 1) · 𝑇)) ∈ 𝐷)
468	393, 467	eqeltrd 2859	. . . . . 6 ⊢ (((𝜑 ∧ (𝐸‘𝑋) = 𝐵) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))) → 𝑦 ∈ 𝐷)
469	468	ralrimiva 3148	. . . . 5 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → ∀𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))𝑦 ∈ 𝐷)
470		dfss3 3810	. . . . 5 ⊢ ((𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇))) ⊆ 𝐷 ↔ ∀𝑦 ∈ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)))𝑦 ∈ 𝐷)
471	469, 470	sylibr 226	. . . 4 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇))) ⊆ 𝐷)
472		breq2 4892	. . . . . 6 ⊢ (𝑦 = ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) → (𝑋 < 𝑦 ↔ 𝑋 < ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇))))
473		oveq2 6932	. . . . . . 7 ⊢ (𝑦 = ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) → (𝑋(,)𝑦) = (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇))))
474	473	sseq1d 3851	. . . . . 6 ⊢ (𝑦 = ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) → ((𝑋(,)𝑦) ⊆ 𝐷 ↔ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇))) ⊆ 𝐷))
475	472, 474	anbi12d 624	. . . . 5 ⊢ (𝑦 = ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) → ((𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷) ↔ (𝑋 < ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) ∧ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇))) ⊆ 𝐷)))
476	475	rspcev 3511	. . . 4 ⊢ ((((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) ∈ ℝ ∧ (𝑋 < ((𝑄‘1) − ((𝑍‘𝑋) − 𝑇)) ∧ (𝑋(,)((𝑄‘1) − ((𝑍‘𝑋) − 𝑇))) ⊆ 𝐷)) → ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷))
477	328, 371, 471, 476	syl12anc 827	. . 3 ⊢ ((𝜑 ∧ (𝐸‘𝑋) = 𝐵) → ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷))
478	15	adantlr 705	. . . . . 6 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋))
479		simp2 1128	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ (0...𝑀))
480	25	3ad2ant2 1125	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ ℝ)
481	94	3ad2ant2 1125	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑀 ∈ ℝ)
482	96	3ad2ant2 1125	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ≤ 𝑀)
483		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄‘𝑗) = (𝐸‘𝑋) → (𝑄‘𝑗) = (𝐸‘𝑋))
484	483	eqcomd 2784	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄‘𝑗) = (𝐸‘𝑋) → (𝐸‘𝑋) = (𝑄‘𝑗))
485	484	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ (((𝑄‘𝑗) = (𝐸‘𝑋) ∧ 𝑀 = 𝑗) → (𝐸‘𝑋) = (𝑄‘𝑗))
486	485	3ad2antl3 1195	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑀 = 𝑗) → (𝐸‘𝑋) = (𝑄‘𝑗))
487		fveq2 6448	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 = 𝑗 → (𝑄‘𝑀) = (𝑄‘𝑗))
488	487	eqcomd 2784	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 = 𝑗 → (𝑄‘𝑗) = (𝑄‘𝑀))
489	488	adantl 475	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑀 = 𝑗) → (𝑄‘𝑗) = (𝑄‘𝑀))
490	180	ad2antrr 716	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑀 = 𝑗) → (𝑄‘𝑀) = 𝐵)
491	490	3ad2antl1 1193	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑀 = 𝑗) → (𝑄‘𝑀) = 𝐵)
492	486, 489, 491	3eqtrd 2818	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑀 = 𝑗) → (𝐸‘𝑋) = 𝐵)
493		neneq 2975	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸‘𝑋) ≠ 𝐵 → ¬ (𝐸‘𝑋) = 𝐵)
494	493	ad2antlr 717	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑀 = 𝑗) → ¬ (𝐸‘𝑋) = 𝐵)
495	494	3ad2antl1 1193	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑀 = 𝑗) → ¬ (𝐸‘𝑋) = 𝐵)
496	492, 495	pm2.65da 807	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ¬ 𝑀 = 𝑗)
497	496	neqned 2976	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑀 ≠ 𝑗)
498	480, 481, 482, 497	leneltd 10532	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 < 𝑀)
499		elfzfzo 40412	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) ↔ (𝑗 ∈ (0...𝑀) ∧ 𝑗 < 𝑀))
500	479, 498, 499	sylanbrc 578	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ (0..^𝑀))
501	117	adantlr 705	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ^*)
502	501	3adant3 1123	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) ∈ ℝ^*)
503		simp1l 1211	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝜑)
504	101	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
505		fzofzp1 12889	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
506	505	adantl 475	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
507	504, 506	ffvelrnd 6626	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
508	507	rexrd 10428	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘(𝑗 + 1)) ∈ ℝ^*)
509	503, 500, 508	syl2anc 579	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 + 1)) ∈ ℝ^*)
510	141	3adant1r 1180	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ℝ^*)
511	37, 160	eqbrtrd 4910	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) ≤ (𝐸‘𝑋))
512	511	adantlr 705	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) ≤ (𝐸‘𝑋))
513	512	3adant2 1122	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) ≤ (𝐸‘𝑋))
514	484	3ad2ant3 1126	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) = (𝑄‘𝑗))
515		eleq1 2847	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
516	515	anbi2d 622	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (0..^𝑀))))
517		fveq2 6448	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → (𝑄‘𝑖) = (𝑄‘𝑗))
518		oveq1 6931	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
519	518	fveq2d 6452	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
520	517, 519	breq12d 4901	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘𝑗) < (𝑄‘(𝑗 + 1))))
521	516, 520	imbi12d 336	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))))
522	521, 152	chvarv 2361	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))
523	503, 500, 522	syl2anc 579	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))
524	514, 523	eqbrtrd 4910	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) < (𝑄‘(𝑗 + 1)))
525	502, 509, 510, 513, 524	elicod 12541	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ((𝑄‘𝑗)[,)(𝑄‘(𝑗 + 1))))
526	517, 519	oveq12d 6942	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))) = ((𝑄‘𝑗)[,)(𝑄‘(𝑗 + 1))))
527	526	eleq2d 2845	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))) ↔ (𝐸‘𝑋) ∈ ((𝑄‘𝑗)[,)(𝑄‘(𝑗 + 1)))))
528	527	rspcev 3511	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑗)[,)(𝑄‘(𝑗 + 1)))) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))))
529	500, 525, 528	syl2anc 579	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ 𝑗 ∈ (0...𝑀) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))))
530	529	3exp 1109	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) → (𝑗 ∈ (0...𝑀) → ((𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))))))
531	530	adantr 474	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → (𝑗 ∈ (0...𝑀) → ((𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))))))
532	531	rexlimdv 3212	. . . . . 6 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → (∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))))
533	478, 532	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))))
534		ioossico 12580	. . . . . . . . . . 11 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))
535		simpr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
536	534, 535	sseldi 3819	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))))
537	536	ex 403	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))))
538	537	adantlr 705	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))))
539	538	reximdva 3198	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → (∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))))
540	190, 539	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))))
541	540	adantlr 705	. . . . 5 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))))
542	533, 541	pm2.61dan 803	. . . 4 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))))
543	208, 250	resubcld 10806	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) ∈ ℝ)
544	543	3adant3 1123	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) ∈ ℝ)
545	219	eqcomd 2784	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ((𝐸‘𝑋) − (𝑍‘𝑋)))
546	545	3ad2ant1 1124	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → 𝑋 = ((𝐸‘𝑋) − (𝑍‘𝑋)))
547	139	3ad2ant1 1124	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ℝ)
548	208	3adant3 1123	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
549	136	3ad2ant1 1124	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → (𝑍‘𝑋) ∈ ℝ)
550	200	rexrd 10428	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
551	550	3adant3 1123	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
552	209	3adant3 1123	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
553		simp3 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))))
554		icoltub 40657	. . . . . . . . . . 11 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) < (𝑄‘(𝑖 + 1)))
555	551, 552, 553, 554	syl3anc 1439	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) < (𝑄‘(𝑖 + 1)))
556	547, 548, 549, 555	ltsub1dd 10990	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → ((𝐸‘𝑋) − (𝑍‘𝑋)) < ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))
557	546, 556	eqbrtrd 4910	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → 𝑋 < ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))
558		elioore 12522	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋))) → 𝑦 ∈ ℝ)
559	558, 237	sylan2 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝑦 = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
560	559	3ad2antl1 1193	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝑦 = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
561		simpl1 1199	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝜑)
562	241	3adant3 1123	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷)
563	562	adantr 474	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷)
564	551	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑄‘𝑖) ∈ ℝ^*)
565	552	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
566	558	adantl 475	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝑦 ∈ ℝ)
567	136	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑍‘𝑋) ∈ ℝ)
568	566, 567	readdcld 10408	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑦 + (𝑍‘𝑋)) ∈ ℝ)
569	568	3ad2antl1 1193	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑦 + (𝑍‘𝑋)) ∈ ℝ)
570	200	3adant3 1123	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
571	570	adantr 474	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑄‘𝑖) ∈ ℝ)
572	561, 139	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝐸‘𝑋) ∈ ℝ)
573		icogelb 12542	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ≤ (𝐸‘𝑋))
574	551, 552, 553, 573	syl3anc 1439	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ≤ (𝐸‘𝑋))
575	574	adantr 474	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑄‘𝑖) ≤ (𝐸‘𝑋))
576	138	ad2antrr 716	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝐸‘𝑋) = (𝑋 + (𝑍‘𝑋)))
577	74	ad2antrr 716	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝑋 ∈ ℝ)
578	558	adantl 475	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝑦 ∈ ℝ)
579	136	ad2antrr 716	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑍‘𝑋) ∈ ℝ)
580	254	ad2antrr 716	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝑋 ∈ ℝ^*)
581	543	rexrd 10428	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) ∈ ℝ^*)
582	581	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) ∈ ℝ^*)
583		simpr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋))))
584		ioogtlb 40643	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 ∈ ℝ^* ∧ ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) ∈ ℝ^* ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝑋 < 𝑦)
585	580, 582, 583, 584	syl3anc 1439	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝑋 < 𝑦)
586	577, 578, 579, 585	ltadd1dd 10989	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑋 + (𝑍‘𝑋)) < (𝑦 + (𝑍‘𝑋)))
587	576, 586	eqbrtrd 4910	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝐸‘𝑋) < (𝑦 + (𝑍‘𝑋)))
588	587	3adantl3 1170	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝐸‘𝑋) < (𝑦 + (𝑍‘𝑋)))
589	571, 572, 569, 575, 588	lelttrd 10536	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑄‘𝑖) < (𝑦 + (𝑍‘𝑋)))
590	543	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) ∈ ℝ)
591		iooltub 40659	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ ℝ^* ∧ ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) ∈ ℝ^* ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝑦 < ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))
592	580, 582, 583, 591	syl3anc 1439	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝑦 < ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))
593	578, 590, 579, 592	ltadd1dd 10989	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑦 + (𝑍‘𝑋)) < (((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) + (𝑍‘𝑋)))
594	208	recnd 10407	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
595	217	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑍‘𝑋) ∈ ℂ)
596	594, 595	npcand 10740	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) + (𝑍‘𝑋)) = (𝑄‘(𝑖 + 1)))
597	596	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) + (𝑍‘𝑋)) = (𝑄‘(𝑖 + 1)))
598	593, 597	breqtrd 4914	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑦 + (𝑍‘𝑋)) < (𝑄‘(𝑖 + 1)))
599	598	3adantl3 1170	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑦 + (𝑍‘𝑋)) < (𝑄‘(𝑖 + 1)))
600	564, 565, 569, 589, 599	eliood 40646	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑦 + (𝑍‘𝑋)) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
601	563, 600	sseldd 3822	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → (𝑦 + (𝑍‘𝑋)) ∈ 𝐷)
602	561, 449	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
603	561, 601, 602, 300	syl3anc 1439	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷)
604	560, 603	eqeltrd 2859	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))) → 𝑦 ∈ 𝐷)
605	604	ralrimiva 3148	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → ∀𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))𝑦 ∈ 𝐷)
606		dfss3 3810	. . . . . . . . 9 ⊢ ((𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋))) ⊆ 𝐷 ↔ ∀𝑦 ∈ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)))𝑦 ∈ 𝐷)
607	605, 606	sylibr 226	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋))) ⊆ 𝐷)
608		breq2 4892	. . . . . . . . . 10 ⊢ (𝑦 = ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) → (𝑋 < 𝑦 ↔ 𝑋 < ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋))))
609		oveq2 6932	. . . . . . . . . . 11 ⊢ (𝑦 = ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) → (𝑋(,)𝑦) = (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋))))
610	609	sseq1d 3851	. . . . . . . . . 10 ⊢ (𝑦 = ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) → ((𝑋(,)𝑦) ⊆ 𝐷 ↔ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋))) ⊆ 𝐷))
611	608, 610	anbi12d 624	. . . . . . . . 9 ⊢ (𝑦 = ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) → ((𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷) ↔ (𝑋 < ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) ∧ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋))) ⊆ 𝐷)))
612	611	rspcev 3511	. . . . . . . 8 ⊢ ((((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) ∈ ℝ ∧ (𝑋 < ((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋)) ∧ (𝑋(,)((𝑄‘(𝑖 + 1)) − (𝑍‘𝑋))) ⊆ 𝐷)) → ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷))
613	544, 557, 607, 612	syl12anc 827	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) → ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷))
614	613	3exp 1109	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))) → ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷))))
615	614	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) → (𝑖 ∈ (0..^𝑀) → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))) → ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷))))
616	615	rexlimdv 3212	. . . 4 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) → (∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1))) → ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷)))
617	542, 616	mpd 15	. . 3 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ≠ 𝐵) → ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷))
618	477, 617	pm2.61dane 3057	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷))
619	314, 618	jca 507	1 ⊢ (𝜑 → (∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ 𝐷) ∧ ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∀wral 3090 ∃wrex 3091 {crab 3094 ⊆ wss 3792 class class class wbr 4888 ↦ cmpt 4967 ran crn 5358 Fn wfn 6132 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ↑_𝑚 cmap 8142 supcsup 8636 ℂcc 10272 ℝcr 10273 0cc0 10274 1c1 10275 + caddc 10277 · cmul 10279 ℝ^*cxr 10412 < clt 10413 ≤ cle 10414 − cmin 10608 -cneg 10609 / cdiv 11035 ℕcn 11379 ℤcz 11733 ℤ_≥cuz 11997 (,)cioo 12492 (,]cioc 12493 [,)cico 12494 [,]cicc 12495 ...cfz 12648 ..^cfzo 12789 ⌊cfl 12915

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-n0 11648 df-z 11734 df-uz 11998 df-rp 12143 df-ioo 12496 df-ioc 12497 df-ico 12498 df-icc 12499 df-fz 12649 df-fzo 12790 df-fl 12917

This theorem is referenced by: fourierdlem113 41377

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