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Mirrors > Home > MPE Home > Th. List > vdwapf | Structured version Visualization version GIF version |
Description: The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.) |
Ref | Expression |
---|---|
vdwapf | ⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 767 | . . . . . . . 8 ⊢ (((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℕ) | |
2 | elfznn0 13103 | . . . . . . . . . 10 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ0) | |
3 | 2 | adantl 485 | . . . . . . . . 9 ⊢ (((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℕ0) |
4 | nnnn0 11995 | . . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℕ0) | |
5 | 4 | ad2antlr 727 | . . . . . . . . 9 ⊢ (((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℕ0) |
6 | 3, 5 | nn0mulcld 12053 | . . . . . . . 8 ⊢ (((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝑑) ∈ ℕ0) |
7 | nnnn0addcl 12018 | . . . . . . . 8 ⊢ ((𝑎 ∈ ℕ ∧ (𝑚 · 𝑑) ∈ ℕ0) → (𝑎 + (𝑚 · 𝑑)) ∈ ℕ) | |
8 | 1, 6, 7 | syl2anc 587 | . . . . . . 7 ⊢ (((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ ℕ) |
9 | 8 | fmpttd 6901 | . . . . . 6 ⊢ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))):(0...(𝐾 − 1))⟶ℕ) |
10 | 9 | frnd 6522 | . . . . 5 ⊢ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))) ⊆ ℕ) |
11 | nnex 11734 | . . . . . 6 ⊢ ℕ ∈ V | |
12 | 11 | elpw2 5223 | . . . . 5 ⊢ (ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))) ∈ 𝒫 ℕ ↔ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))) ⊆ ℕ) |
13 | 10, 12 | sylibr 237 | . . . 4 ⊢ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))) ∈ 𝒫 ℕ) |
14 | 13 | rgen2 3116 | . . 3 ⊢ ∀𝑎 ∈ ℕ ∀𝑑 ∈ ℕ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))) ∈ 𝒫 ℕ |
15 | eqid 2739 | . . . 4 ⊢ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))) = (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))) | |
16 | 15 | fmpo 7803 | . . 3 ⊢ (∀𝑎 ∈ ℕ ∀𝑑 ∈ ℕ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))) ∈ 𝒫 ℕ ↔ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))):(ℕ × ℕ)⟶𝒫 ℕ) |
17 | 14, 16 | mpbi 233 | . 2 ⊢ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))):(ℕ × ℕ)⟶𝒫 ℕ |
18 | vdwapfval 16419 | . . 3 ⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾) = (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))))) | |
19 | 18 | feq1d 6499 | . 2 ⊢ (𝐾 ∈ ℕ0 → ((AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ ↔ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))):(ℕ × ℕ)⟶𝒫 ℕ)) |
20 | 17, 19 | mpbiri 261 | 1 ⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2114 ∀wral 3054 ⊆ wss 3853 𝒫 cpw 4498 ↦ cmpt 5120 × cxp 5533 ran crn 5536 ⟶wf 6345 ‘cfv 6349 (class class class)co 7182 ∈ cmpo 7184 0cc0 10627 1c1 10628 + caddc 10630 · cmul 10632 − cmin 10960 ℕcn 11728 ℕ0cn0 11988 ...cfz 12993 APcvdwa 16413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-n0 11989 df-z 12075 df-uz 12337 df-fz 12994 df-vdwap 16416 |
This theorem is referenced by: vdwmc 16426 |
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