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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpart | Structured version Visualization version GIF version |
Description: A special partition. Corresponds to fourierdlem2 42484 in GS's mathbox. (Contributed by AV, 9-Jul-2020.) |
Ref | Expression |
---|---|
iccpart | ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccpval 43665 | . . 3 ⊢ (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))}) | |
2 | 1 | eleq2d 2898 | . 2 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ 𝑃 ∈ {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})) |
3 | fveq1 6655 | . . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑖) = (𝑃‘𝑖)) | |
4 | fveq1 6655 | . . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1))) | |
5 | 3, 4 | breq12d 5065 | . . . 4 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) |
6 | 5 | ralbidv 3197 | . . 3 ⊢ (𝑝 = 𝑃 → (∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) |
7 | 6 | elrab 3671 | . 2 ⊢ (𝑃 ∈ {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))} ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) |
8 | 2, 7 | syl6bb 289 | 1 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 class class class wbr 5052 ‘cfv 6341 (class class class)co 7142 ↑m cmap 8392 0cc0 10523 1c1 10524 + caddc 10526 ℝ*cxr 10660 < clt 10661 ℕcn 11624 ...cfz 12882 ..^cfzo 13023 RePartciccp 43663 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-iota 6300 df-fun 6343 df-fv 6349 df-ov 7145 df-iccp 43664 |
This theorem is referenced by: iccpartimp 43667 iccpartres 43668 iccpartxr 43669 iccpartrn 43680 iccpartf 43681 iccpartnel 43688 |
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