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Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem2 | Structured version Visualization version GIF version |
Description: Membership in a partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fourierdlem2.1 | ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
Ref | Expression |
---|---|
fourierdlem2 | ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7345 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀)) | |
2 | 1 | oveq2d 7353 | . . . . 5 ⊢ (𝑚 = 𝑀 → (ℝ ↑m (0...𝑚)) = (ℝ ↑m (0...𝑀))) |
3 | fveqeq2 6834 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑝‘𝑚) = 𝐵 ↔ (𝑝‘𝑀) = 𝐵)) | |
4 | 3 | anbi2d 629 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵))) |
5 | oveq2 7345 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀)) | |
6 | 5 | raleqdv 3309 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))) |
7 | 4, 6 | anbi12d 631 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))))) |
8 | 2, 7 | rabeqbidv 3420 | . . . 4 ⊢ (𝑚 = 𝑀 → {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} = {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
9 | fourierdlem2.1 | . . . 4 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) | |
10 | ovex 7370 | . . . . 5 ⊢ (ℝ ↑m (0...𝑀)) ∈ V | |
11 | 10 | rabex 5276 | . . . 4 ⊢ {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} ∈ V |
12 | 8, 9, 11 | fvmpt 6931 | . . 3 ⊢ (𝑀 ∈ ℕ → (𝑃‘𝑀) = {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
13 | 12 | eleq2d 2822 | . 2 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ 𝑄 ∈ {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})) |
14 | fveq1 6824 | . . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0)) | |
15 | 14 | eqeq1d 2738 | . . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘0) = 𝐴 ↔ (𝑄‘0) = 𝐴)) |
16 | fveq1 6824 | . . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑀) = (𝑄‘𝑀)) | |
17 | 16 | eqeq1d 2738 | . . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑀) = 𝐵 ↔ (𝑄‘𝑀) = 𝐵)) |
18 | 15, 17 | anbi12d 631 | . . . 4 ⊢ (𝑝 = 𝑄 → (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ↔ ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))) |
19 | fveq1 6824 | . . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑖) = (𝑄‘𝑖)) | |
20 | fveq1 6824 | . . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1))) | |
21 | 19, 20 | breq12d 5105 | . . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))) |
22 | 21 | ralbidv 3170 | . . . 4 ⊢ (𝑝 = 𝑄 → (∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))) |
23 | 18, 22 | anbi12d 631 | . . 3 ⊢ (𝑝 = 𝑄 → ((((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))) |
24 | 23 | elrab 3634 | . 2 ⊢ (𝑄 ∈ {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))) |
25 | 13, 24 | bitrdi 286 | 1 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 {crab 3403 class class class wbr 5092 ↦ cmpt 5175 ‘cfv 6479 (class class class)co 7337 ↑m cmap 8686 ℝcr 10971 0cc0 10972 1c1 10973 + caddc 10975 < clt 11110 ℕcn 12074 ...cfz 13340 ..^cfzo 13483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-iota 6431 df-fun 6481 df-fv 6487 df-ov 7340 |
This theorem is referenced by: fourierdlem11 43995 fourierdlem12 43996 fourierdlem13 43997 fourierdlem14 43998 fourierdlem15 43999 fourierdlem34 44018 fourierdlem37 44021 fourierdlem41 44025 fourierdlem48 44031 fourierdlem49 44032 fourierdlem50 44033 fourierdlem54 44037 fourierdlem63 44046 fourierdlem64 44047 fourierdlem65 44048 fourierdlem69 44052 fourierdlem70 44053 fourierdlem72 44055 fourierdlem74 44057 fourierdlem75 44058 fourierdlem76 44059 fourierdlem79 44062 fourierdlem81 44064 fourierdlem85 44068 fourierdlem88 44071 fourierdlem89 44072 fourierdlem90 44073 fourierdlem91 44074 fourierdlem92 44075 fourierdlem93 44076 fourierdlem94 44077 fourierdlem97 44080 fourierdlem102 44085 fourierdlem103 44086 fourierdlem104 44087 fourierdlem111 44094 fourierdlem113 44096 fourierdlem114 44097 |
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