![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > vdwapfval | Structured version Visualization version GIF version |
Description: Define the arithmetic progression function, which takes as input a length 𝑘, a start point 𝑎, and a step 𝑑 and outputs the set of points in this progression. (Contributed by Mario Carneiro, 18-Aug-2014.) |
Ref | Expression |
---|---|
vdwapfval | ⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾) = (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . . . . . 7 ⊢ ((𝑘 = 𝐾 ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → 𝑘 = 𝐾) | |
2 | 1 | oveq1d 7150 | . . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑘 − 1) = (𝐾 − 1)) |
3 | 2 | oveq2d 7151 | . . . . 5 ⊢ ((𝑘 = 𝐾 ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (0...(𝑘 − 1)) = (0...(𝐾 − 1))) |
4 | 3 | mpteq1d 5119 | . . . 4 ⊢ ((𝑘 = 𝐾 ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))) = (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))) |
5 | 4 | rneqd 5772 | . . 3 ⊢ ((𝑘 = 𝐾 ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → ran (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))) = ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))) |
6 | 5 | mpoeq3dva 7210 | . 2 ⊢ (𝑘 = 𝐾 → (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))) = (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))))) |
7 | df-vdwap 16294 | . 2 ⊢ AP = (𝑘 ∈ ℕ0 ↦ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))))) | |
8 | nnex 11631 | . . 3 ⊢ ℕ ∈ V | |
9 | 8, 8 | mpoex 7760 | . 2 ⊢ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))) ∈ V |
10 | 6, 7, 9 | fvmpt 6745 | 1 ⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾) = (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ↦ cmpt 5110 ran crn 5520 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 − cmin 10859 ℕcn 11625 ℕ0cn0 11885 ...cfz 12885 APcvdwa 16291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 df-vdwap 16294 |
This theorem is referenced by: vdwapf 16298 vdwapval 16299 |
Copyright terms: Public domain | W3C validator |