| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | prfi 9363 | . . . 4
⊢ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼} ∈
Fin | 
| 2 | 1 | a1i 11 | . . 3
⊢ (𝜑 → {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} ∈ Fin) | 
| 3 |  | fourierdlem71.f | . . . . . . 7
⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) | 
| 4 | 3 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼}) → 𝐹:dom 𝐹⟶ℝ) | 
| 5 |  | simpl 482 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼}) → 𝜑) | 
| 6 |  | simpr 484 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼}) → 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) | 
| 7 |  | fourierdlem71.q | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ) | 
| 8 |  | ovex 7464 | . . . . . . . . . . . . 13
⊢
(0...𝑀) ∈
V | 
| 9 | 8 | a1i 11 | . . . . . . . . . . . 12
⊢ (𝜑 → (0...𝑀) ∈ V) | 
| 10 | 7, 9 | fexd 7247 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑄 ∈ V) | 
| 11 |  | rnexg 7924 | . . . . . . . . . . 11
⊢ (𝑄 ∈ V → ran 𝑄 ∈ V) | 
| 12 |  | inex1g 5319 | . . . . . . . . . . 11
⊢ (ran
𝑄 ∈ V → (ran
𝑄 ∩ dom 𝐹) ∈ V) | 
| 13 | 10, 11, 12 | 3syl 18 | . . . . . . . . . 10
⊢ (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ V) | 
| 14 | 13 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼}) → (ran 𝑄 ∩ dom 𝐹) ∈ V) | 
| 15 |  | fourierdlem71.i | . . . . . . . . . . . . . 14
⊢ 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) | 
| 16 |  | ovex 7464 | . . . . . . . . . . . . . . 15
⊢
(0..^𝑀) ∈
V | 
| 17 | 16 | mptex 7243 | . . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ V | 
| 18 | 15, 17 | eqeltri 2837 | . . . . . . . . . . . . 13
⊢ 𝐼 ∈ V | 
| 19 | 18 | rnex 7932 | . . . . . . . . . . . 12
⊢ ran 𝐼 ∈ V | 
| 20 | 19 | a1i 11 | . . . . . . . . . . 11
⊢ (𝜑 → ran 𝐼 ∈ V) | 
| 21 | 20 | uniexd 7762 | . . . . . . . . . 10
⊢ (𝜑 → ∪ ran 𝐼 ∈ V) | 
| 22 | 21 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼}) → ∪ ran 𝐼 ∈ V) | 
| 23 |  | uniprg 4923 | . . . . . . . . 9
⊢ (((ran
𝑄 ∩ dom 𝐹) ∈ V ∧ ∪ ran 𝐼 ∈ V) → ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) | 
| 24 | 14, 22, 23 | syl2anc 584 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼}) → ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) | 
| 25 | 6, 24 | eleqtrd 2843 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼}) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) | 
| 26 |  | elinel2 4202 | . . . . . . . . 9
⊢ (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹) | 
| 27 | 26 | adantl 481 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹) | 
| 28 |  | simpll 767 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝜑) | 
| 29 |  | elunnel1 4154 | . . . . . . . . . 10
⊢ ((𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ ∪ ran
𝐼) | 
| 30 | 29 | adantll 714 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ ∪ ran
𝐼) | 
| 31 | 15 | funmpt2 6605 | . . . . . . . . . . . . 13
⊢ Fun 𝐼 | 
| 32 |  | elunirn 7271 | . . . . . . . . . . . . 13
⊢ (Fun
𝐼 → (𝑥 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖))) | 
| 33 | 31, 32 | ax-mp 5 | . . . . . . . . . . . 12
⊢ (𝑥 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖)) | 
| 34 | 33 | biimpi 216 | . . . . . . . . . . 11
⊢ (𝑥 ∈ ∪ ran 𝐼 → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖)) | 
| 35 | 34 | adantl 481 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran
𝐼) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖)) | 
| 36 |  | id 22 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ dom 𝐼) | 
| 37 |  | ovex 7464 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V | 
| 38 | 37, 15 | dmmpti 6712 | . . . . . . . . . . . . . . . . . . 19
⊢ dom 𝐼 = (0..^𝑀) | 
| 39 | 36, 38 | eleqtrdi 2851 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ (0..^𝑀)) | 
| 40 | 39 | adantl 481 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀)) | 
| 41 | 37 | a1i 11 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) | 
| 42 | 15 | fvmpt2 7027 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) | 
| 43 | 40, 41, 42 | syl2anc 584 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) | 
| 44 |  | fourierdlem71.fcn | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) | 
| 45 |  | cncff 24919 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ) | 
| 46 |  | fdm 6745 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) | 
| 47 | 44, 45, 46 | 3syl 18 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) | 
| 48 | 39, 47 | sylan2 593 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) | 
| 49 |  | ssdmres 6031 | . . . . . . . . . . . . . . . . 17
⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) | 
| 50 | 48, 49 | sylibr 234 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹) | 
| 51 | 43, 50 | eqsstrd 4018 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) ⊆ dom 𝐹) | 
| 52 | 51 | 3adant3 1133 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) → (𝐼‘𝑖) ⊆ dom 𝐹) | 
| 53 |  | simp3 1139 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) → 𝑥 ∈ (𝐼‘𝑖)) | 
| 54 | 52, 53 | sseldd 3984 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) → 𝑥 ∈ dom 𝐹) | 
| 55 | 54 | 3exp 1120 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼‘𝑖) → 𝑥 ∈ dom 𝐹))) | 
| 56 | 55 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran
𝐼) → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼‘𝑖) → 𝑥 ∈ dom 𝐹))) | 
| 57 | 56 | rexlimdv 3153 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran
𝐼) → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖) → 𝑥 ∈ dom 𝐹)) | 
| 58 | 35, 57 | mpd 15 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran
𝐼) → 𝑥 ∈ dom 𝐹) | 
| 59 | 28, 30, 58 | syl2anc 584 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹) | 
| 60 | 27, 59 | pm2.61dan 813 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) → 𝑥 ∈ dom 𝐹) | 
| 61 | 5, 25, 60 | syl2anc 584 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼}) → 𝑥 ∈ dom 𝐹) | 
| 62 | 4, 61 | ffvelcdmd 7105 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼}) → (𝐹‘𝑥) ∈ ℝ) | 
| 63 | 62 | recnd 11289 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼}) → (𝐹‘𝑥) ∈ ℂ) | 
| 64 | 63 | abscld 15475 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼}) →
(abs‘(𝐹‘𝑥)) ∈
ℝ) | 
| 65 |  | simpr 484 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = (ran 𝑄 ∩ dom 𝐹)) | 
| 66 |  | fzfid 14014 | . . . . . . . . . 10
⊢ (𝜑 → (0...𝑀) ∈ Fin) | 
| 67 |  | rnffi 45180 | . . . . . . . . . 10
⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin) | 
| 68 | 7, 66, 67 | syl2anc 584 | . . . . . . . . 9
⊢ (𝜑 → ran 𝑄 ∈ Fin) | 
| 69 |  | infi 9302 | . . . . . . . . 9
⊢ (ran
𝑄 ∈ Fin → (ran
𝑄 ∩ dom 𝐹) ∈ Fin) | 
| 70 | 68, 69 | syl 17 | . . . . . . . 8
⊢ (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ Fin) | 
| 71 | 70 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → (ran 𝑄 ∩ dom 𝐹) ∈ Fin) | 
| 72 | 65, 71 | eqeltrd 2841 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 ∈ Fin) | 
| 73 |  | simpll 767 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥 ∈ 𝑤) → 𝜑) | 
| 74 |  | simpr 484 | . . . . . . . . . 10
⊢ ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ 𝑤) | 
| 75 |  | simpl 482 | . . . . . . . . . 10
⊢ ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥 ∈ 𝑤) → 𝑤 = (ran 𝑄 ∩ dom 𝐹)) | 
| 76 | 74, 75 | eleqtrd 2843 | . . . . . . . . 9
⊢ ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) | 
| 77 | 76 | adantll 714 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) | 
| 78 | 3 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝐹:dom 𝐹⟶ℝ) | 
| 79 | 26 | adantl 481 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹) | 
| 80 | 78, 79 | ffvelcdmd 7105 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹‘𝑥) ∈ ℝ) | 
| 81 | 80 | recnd 11289 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹‘𝑥) ∈ ℂ) | 
| 82 | 81 | abscld 15475 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (abs‘(𝐹‘𝑥)) ∈ ℝ) | 
| 83 | 73, 77, 82 | syl2anc 584 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥 ∈ 𝑤) → (abs‘(𝐹‘𝑥)) ∈ ℝ) | 
| 84 | 83 | ralrimiva 3146 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ∈ ℝ) | 
| 85 |  | fimaxre3 12214 | . . . . . 6
⊢ ((𝑤 ∈ Fin ∧ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦) | 
| 86 | 72, 84, 85 | syl2anc 584 | . . . . 5
⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦) | 
| 87 | 86 | adantlr 715 | . . . 4
⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦) | 
| 88 |  | simpll 767 | . . . . 5
⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝜑) | 
| 89 |  | neqne 2948 | . . . . . . 7
⊢ (¬
𝑤 = (ran 𝑄 ∩ dom 𝐹) → 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹)) | 
| 90 |  | elprn1 45648 | . . . . . . 7
⊢ ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} ∧ 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ∪ ran 𝐼) | 
| 91 | 89, 90 | sylan2 593 | . . . . . 6
⊢ ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ∪ ran 𝐼) | 
| 92 | 91 | adantll 714 | . . . . 5
⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ∪ ran 𝐼) | 
| 93 |  | fzofi 14015 | . . . . . . . 8
⊢
(0..^𝑀) ∈
Fin | 
| 94 | 15 | rnmptfi 45176 | . . . . . . . 8
⊢
((0..^𝑀) ∈ Fin
→ ran 𝐼 ∈
Fin) | 
| 95 | 93, 94 | ax-mp 5 | . . . . . . 7
⊢ ran 𝐼 ∈ Fin | 
| 96 | 95 | a1i 11 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ran 𝐼 ∈ Fin) | 
| 97 | 3 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran
𝐼) → 𝐹:dom 𝐹⟶ℝ) | 
| 98 | 97, 58 | ffvelcdmd 7105 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran
𝐼) → (𝐹‘𝑥) ∈ ℝ) | 
| 99 | 98 | recnd 11289 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran
𝐼) → (𝐹‘𝑥) ∈ ℂ) | 
| 100 | 99 | adantlr 715 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑥 ∈ ∪ ran
𝐼) → (𝐹‘𝑥) ∈ ℂ) | 
| 101 | 100 | abscld 15475 | . . . . . 6
⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑥 ∈ ∪ ran
𝐼) → (abs‘(𝐹‘𝑥)) ∈ ℝ) | 
| 102 | 37, 15 | fnmpti 6711 | . . . . . . . . . . 11
⊢ 𝐼 Fn (0..^𝑀) | 
| 103 |  | fvelrnb 6969 | . . . . . . . . . . 11
⊢ (𝐼 Fn (0..^𝑀) → (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)) | 
| 104 | 102, 103 | ax-mp 5 | . . . . . . . . . 10
⊢ (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡) | 
| 105 | 104 | biimpi 216 | . . . . . . . . 9
⊢ (𝑡 ∈ ran 𝐼 → ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡) | 
| 106 | 105 | adantl 481 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡) | 
| 107 | 7 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ) | 
| 108 |  | elfzofz 13715 | . . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀)) | 
| 109 | 108 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀)) | 
| 110 | 107, 109 | ffvelcdmd 7105 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ) | 
| 111 |  | fzofzp1 13803 | . . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀)) | 
| 112 | 111 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀)) | 
| 113 | 107, 112 | ffvelcdmd 7105 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ) | 
| 114 |  | fourierdlem71.l | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) | 
| 115 |  | fourierdlem71.r | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) | 
| 116 | 110, 113,
44, 114, 115 | cncfioobd 45912 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏) | 
| 117 | 116 | 3adant3 1133 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏) | 
| 118 |  | fvres 6925 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥)) | 
| 119 | 118 | fveq2d 6910 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = (abs‘(𝐹‘𝑥))) | 
| 120 | 119 | breq1d 5153 | . . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑏)) | 
| 121 | 120 | adantl 481 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑏)) | 
| 122 | 121 | ralbidva 3176 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑥)) ≤ 𝑏)) | 
| 123 | 122 | rexbidv 3179 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑥)) ≤ 𝑏)) | 
| 124 | 123 | 3adant3 1133 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑥)) ≤ 𝑏)) | 
| 125 | 37, 42 | mpan2 691 | . . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (0..^𝑀) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) | 
| 126 |  | id 22 | . . . . . . . . . . . . . . . . 17
⊢ ((𝐼‘𝑖) = 𝑡 → (𝐼‘𝑖) = 𝑡) | 
| 127 | 125, 126 | sylan9req 2798 | . . . . . . . . . . . . . . . 16
⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡) | 
| 128 | 127 | 3adant1 1131 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡) | 
| 129 | 128 | raleqdv 3326 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)) | 
| 130 | 129 | rexbidv 3179 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)) | 
| 131 | 124, 130 | bitrd 279 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)) | 
| 132 | 117, 131 | mpbid 232 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏) | 
| 133 | 132 | 3exp 1120 | . . . . . . . . . 10
⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏))) | 
| 134 | 133 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏))) | 
| 135 | 134 | rexlimdv 3153 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → (∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)) | 
| 136 | 106, 135 | mpd 15 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏) | 
| 137 | 136 | adantlr 715 | . . . . . 6
⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏) | 
| 138 |  | eqimss 4042 | . . . . . . 7
⊢ (𝑤 = ∪
ran 𝐼 → 𝑤 ⊆ ∪ ran 𝐼) | 
| 139 | 138 | adantl 481 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → 𝑤 ⊆ ∪ ran
𝐼) | 
| 140 | 96, 101, 137, 139 | ssfiunibd 45321 | . . . . 5
⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦) | 
| 141 | 88, 92, 140 | syl2anc 584 | . . . 4
⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦) | 
| 142 | 87, 141 | pm2.61dan 813 | . . 3
⊢ ((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦) | 
| 143 |  | simpr 484 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ran 𝑄) | 
| 144 |  | elinel2 4202 | . . . . . . . . . 10
⊢ (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹) | 
| 145 | 144 | ad2antlr 727 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ dom 𝐹) | 
| 146 | 143, 145 | elind 4200 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) | 
| 147 |  | elun1 4182 | . . . . . . . 8
⊢ (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) | 
| 148 | 146, 147 | syl 17 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) | 
| 149 |  | fourierdlem71.7 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℕ) | 
| 150 | 149 | ad2antrr 726 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑀 ∈ ℕ) | 
| 151 | 7 | ad2antrr 726 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ) | 
| 152 |  | elinel1 4201 | . . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ (𝐴[,]𝐵)) | 
| 153 | 152 | adantl 481 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ (𝐴[,]𝐵)) | 
| 154 |  | fourierdlem71.q0 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑄‘0) = 𝐴) | 
| 155 | 154 | eqcomd 2743 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐴 = (𝑄‘0)) | 
| 156 | 155 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐴 = (𝑄‘0)) | 
| 157 |  | fourierdlem71.10 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑄‘𝑀) = 𝐵) | 
| 158 | 157 | eqcomd 2743 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐵 = (𝑄‘𝑀)) | 
| 159 | 158 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐵 = (𝑄‘𝑀)) | 
| 160 | 156, 159 | oveq12d 7449 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀))) | 
| 161 | 153, 160 | eleqtrd 2843 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) | 
| 162 | 161 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) | 
| 163 |  | simpr 484 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ¬ 𝑥 ∈ ran 𝑄) | 
| 164 |  | fveq2 6906 | . . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑗 → (𝑄‘𝑘) = (𝑄‘𝑗)) | 
| 165 | 164 | breq1d 5153 | . . . . . . . . . . . . 13
⊢ (𝑘 = 𝑗 → ((𝑄‘𝑘) < 𝑥 ↔ (𝑄‘𝑗) < 𝑥)) | 
| 166 | 165 | cbvrabv 3447 | . . . . . . . . . . . 12
⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑥} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑥} | 
| 167 | 166 | supeq1i 9487 | . . . . . . . . . . 11
⊢
sup({𝑘 ∈
(0..^𝑀) ∣ (𝑄‘𝑘) < 𝑥}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑥}, ℝ, < ) | 
| 168 | 150, 151,
162, 163, 167 | fourierdlem25 46147 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) | 
| 169 | 39 | ad2antrl 728 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → 𝑖 ∈ (0..^𝑀)) | 
| 170 |  | simprr 773 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → 𝑥 ∈ (𝐼‘𝑖)) | 
| 171 | 169, 125 | syl 17 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) | 
| 172 | 170, 171 | eleqtrd 2843 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) | 
| 173 | 169, 172 | jca 511 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) | 
| 174 |  | id 22 | . . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0..^𝑀)) | 
| 175 | 174, 38 | eleqtrrdi 2852 | . . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ dom 𝐼) | 
| 176 | 175 | ad2antrl 728 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑖 ∈ dom 𝐼) | 
| 177 |  | simprr 773 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) | 
| 178 | 125 | eqcomd 2743 | . . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (0..^𝑀) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖)) | 
| 179 | 178 | ad2antrl 728 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖)) | 
| 180 | 177, 179 | eleqtrd 2843 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ (𝐼‘𝑖)) | 
| 181 | 176, 180 | jca 511 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) | 
| 182 | 173, 181 | impbida 801 | . . . . . . . . . . . 12
⊢ (𝜑 → ((𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) ↔ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))) | 
| 183 | 182 | rexbidv2 3175 | . . . . . . . . . . 11
⊢ (𝜑 → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) | 
| 184 | 183 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) | 
| 185 | 168, 184 | mpbird 257 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖)) | 
| 186 | 185, 33 | sylibr 234 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ∪ ran
𝐼) | 
| 187 |  | elun2 4183 | . . . . . . . 8
⊢ (𝑥 ∈ ∪ ran 𝐼 → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) | 
| 188 | 186, 187 | syl 17 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) | 
| 189 | 148, 188 | pm2.61dan 813 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) | 
| 190 | 189 | ralrimiva 3146 | . . . . 5
⊢ (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) | 
| 191 |  | dfss3 3972 | . . . . 5
⊢ (((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼) ↔ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) | 
| 192 | 190, 191 | sylibr 234 | . . . 4
⊢ (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) | 
| 193 | 13, 21, 23 | syl2anc 584 | . . . 4
⊢ (𝜑 → ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran
𝐼)) | 
| 194 | 192, 193 | sseqtrrd 4021 | . . 3
⊢ (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ∪ {(ran
𝑄 ∩ dom 𝐹), ∪
ran 𝐼}) | 
| 195 | 2, 64, 142, 194 | ssfiunibd 45321 | . 2
⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) | 
| 196 |  | nfv 1914 | . . . . . 6
⊢
Ⅎ𝑥𝜑 | 
| 197 |  | nfra1 3284 | . . . . . 6
⊢
Ⅎ𝑥∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 | 
| 198 | 196, 197 | nfan 1899 | . . . . 5
⊢
Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) | 
| 199 |  | fourierdlem71.dmf | . . . . . . . . . . . . 13
⊢ (𝜑 → dom 𝐹 ⊆ ℝ) | 
| 200 | 199 | sselda 3983 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ℝ) | 
| 201 |  | fourierdlem71.b | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐵 ∈ ℝ) | 
| 202 | 201 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐵 ∈ ℝ) | 
| 203 | 202, 200 | resubcld 11691 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐵 − 𝑥) ∈ ℝ) | 
| 204 |  | fourierdlem71.t | . . . . . . . . . . . . . . . . . . 19
⊢ 𝑇 = (𝐵 − 𝐴) | 
| 205 |  | fourierdlem71.a | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| 206 | 201, 205 | resubcld 11691 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) | 
| 207 | 204, 206 | eqeltrid 2845 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑇 ∈ ℝ) | 
| 208 | 207 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝑇 ∈ ℝ) | 
| 209 |  | fourierdlem71.altb | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝐴 < 𝐵) | 
| 210 | 205, 201 | posdifd 11850 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) | 
| 211 | 209, 210 | mpbid 232 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 0 < (𝐵 − 𝐴)) | 
| 212 | 211, 204 | breqtrrdi 5185 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 0 < 𝑇) | 
| 213 | 212 | gt0ne0d 11827 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑇 ≠ 0) | 
| 214 | 213 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝑇 ≠ 0) | 
| 215 | 203, 208,
214 | redivcld 12095 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((𝐵 − 𝑥) / 𝑇) ∈ ℝ) | 
| 216 | 215 | flcld 13838 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) | 
| 217 | 216 | zred 12722 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℝ) | 
| 218 | 217, 208 | remulcld 11291 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) ∈ ℝ) | 
| 219 | 200, 218 | readdcld 11290 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ ℝ) | 
| 220 |  | fourierdlem71.e | . . . . . . . . . . . . 13
⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) | 
| 221 | 220 | fvmpt2 7027 | . . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ ℝ) → (𝐸‘𝑥) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) | 
| 222 | 200, 219,
221 | syl2anc 584 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) | 
| 223 | 222 | fveq2d 6910 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘(𝐸‘𝑥)) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))) | 
| 224 |  | fvex 6919 | . . . . . . . . . . . 12
⊢
(⌊‘((𝐵
− 𝑥) / 𝑇)) ∈ V | 
| 225 |  | eleq1 2829 | . . . . . . . . . . . . . 14
⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝑘 ∈ ℤ ↔ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ)) | 
| 226 | 225 | anbi2d 630 | . . . . . . . . . . . . 13
⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) ↔ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ))) | 
| 227 |  | oveq1 7438 | . . . . . . . . . . . . . . . 16
⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝑘 · 𝑇) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) | 
| 228 | 227 | oveq2d 7447 | . . . . . . . . . . . . . . 15
⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) | 
| 229 | 228 | fveq2d 6910 | . . . . . . . . . . . . . 14
⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))) | 
| 230 | 229 | eqeq1d 2739 | . . . . . . . . . . . . 13
⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥) ↔ (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))) | 
| 231 | 226, 230 | imbi12d 344 | . . . . . . . . . . . 12
⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → ((((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥)) ↔ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)))) | 
| 232 |  | fourierdlem71.fxpt | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥)) | 
| 233 | 224, 231,
232 | vtocl 3558 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)) | 
| 234 | 216, 233 | mpdan 687 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)) | 
| 235 | 223, 234 | eqtr2d 2778 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = (𝐹‘(𝐸‘𝑥))) | 
| 236 | 235 | fveq2d 6910 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘(𝐸‘𝑥)))) | 
| 237 | 236 | adantlr 715 | . . . . . . 7
⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘(𝐸‘𝑥)))) | 
| 238 |  | fveq2 6906 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤)) | 
| 239 | 238 | fveq2d 6910 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝑤 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘𝑤))) | 
| 240 | 239 | breq1d 5153 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑤 → ((abs‘(𝐹‘𝑥)) ≤ 𝑦 ↔ (abs‘(𝐹‘𝑤)) ≤ 𝑦)) | 
| 241 | 240 | cbvralvw 3237 | . . . . . . . . . 10
⊢
(∀𝑥 ∈
((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 ↔ ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑤)) ≤ 𝑦) | 
| 242 | 241 | biimpi 216 | . . . . . . . . 9
⊢
(∀𝑥 ∈
((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑤)) ≤ 𝑦) | 
| 243 | 242 | ad2antlr 727 | . . . . . . . 8
⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑤)) ≤ 𝑦) | 
| 244 |  | iocssicc 13477 | . . . . . . . . . . 11
⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵) | 
| 245 | 205 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐴 ∈ ℝ) | 
| 246 | 209 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐴 < 𝐵) | 
| 247 |  | id 22 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | 
| 248 |  | oveq2 7439 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑦 → (𝐵 − 𝑥) = (𝐵 − 𝑦)) | 
| 249 | 248 | oveq1d 7446 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − 𝑦) / 𝑇)) | 
| 250 | 249 | fveq2d 6910 | . . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − 𝑦) / 𝑇))) | 
| 251 | 250 | oveq1d 7446 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇)) | 
| 252 | 247, 251 | oveq12d 7449 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = (𝑦 + ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇))) | 
| 253 | 252 | cbvmptv 5255 | . . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇))) | 
| 254 | 220, 253 | eqtri 2765 | . . . . . . . . . . . . 13
⊢ 𝐸 = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇))) | 
| 255 | 245, 202,
246, 204, 254 | fourierdlem4 46126 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐸:ℝ⟶(𝐴(,]𝐵)) | 
| 256 | 255, 200 | ffvelcdmd 7105 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ (𝐴(,]𝐵)) | 
| 257 | 244, 256 | sselid 3981 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ (𝐴[,]𝐵)) | 
| 258 | 228 | eleq1d 2826 | . . . . . . . . . . . . . 14
⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → ((𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹 ↔ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)) | 
| 259 | 226, 258 | imbi12d 344 | . . . . . . . . . . . . 13
⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → ((((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹) ↔ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹))) | 
| 260 |  | fourierdlem71.xpt | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹) | 
| 261 | 224, 259,
260 | vtocl 3558 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹) | 
| 262 | 216, 261 | mpdan 687 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹) | 
| 263 | 222, 262 | eqeltrd 2841 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ dom 𝐹) | 
| 264 | 257, 263 | elind 4200 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) | 
| 265 | 264 | adantlr 715 | . . . . . . . 8
⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) | 
| 266 |  | fveq2 6906 | . . . . . . . . . . 11
⊢ (𝑤 = (𝐸‘𝑥) → (𝐹‘𝑤) = (𝐹‘(𝐸‘𝑥))) | 
| 267 | 266 | fveq2d 6910 | . . . . . . . . . 10
⊢ (𝑤 = (𝐸‘𝑥) → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘(𝐸‘𝑥)))) | 
| 268 | 267 | breq1d 5153 | . . . . . . . . 9
⊢ (𝑤 = (𝐸‘𝑥) → ((abs‘(𝐹‘𝑤)) ≤ 𝑦 ↔ (abs‘(𝐹‘(𝐸‘𝑥))) ≤ 𝑦)) | 
| 269 | 268 | rspccva 3621 | . . . . . . . 8
⊢
((∀𝑤 ∈
((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑤)) ≤ 𝑦 ∧ (𝐸‘𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (abs‘(𝐹‘(𝐸‘𝑥))) ≤ 𝑦) | 
| 270 | 243, 265,
269 | syl2anc 584 | . . . . . . 7
⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘(𝐸‘𝑥))) ≤ 𝑦) | 
| 271 | 237, 270 | eqbrtrd 5165 | . . . . . 6
⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘𝑥)) ≤ 𝑦) | 
| 272 | 271 | ex 412 | . . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) → (𝑥 ∈ dom 𝐹 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)) | 
| 273 | 198, 272 | ralrimi 3257 | . . . 4
⊢ ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦) | 
| 274 | 273 | ex 412 | . . 3
⊢ (𝜑 → (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦)) | 
| 275 | 274 | reximdv 3170 | . 2
⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦)) | 
| 276 | 195, 275 | mpd 15 | 1
⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦) |