Theorem vdwapfval 16302

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.)

Assertion

Ref	Expression
vdwapfval	⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾) = (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))

Distinct variable group:   𝑎,𝑑,𝑚,𝐾

Proof of Theorem vdwapfval

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1131	. . . . . . 7 ⊢ ((𝑘 = 𝐾 ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → 𝑘 = 𝐾)
2	1	oveq1d 7164	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑘 − 1) = (𝐾 − 1))
3	2	oveq2d 7165	. . . . 5 ⊢ ((𝑘 = 𝐾 ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (0...(𝑘 − 1)) = (0...(𝐾 − 1)))
4	3	mpteq1d 5148	. . . 4 ⊢ ((𝑘 = 𝐾 ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))) = (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))))
5	4	rneqd 5801	. . 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 7224	. 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 16299	. 2 ⊢ AP = (𝑘 ∈ ℕ0 ↦ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))
8		nnex 11637	. . 3 ⊢ ℕ ∈ V
9	8, 8	mpoex 7770	. 2 ⊢ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))) ∈ V
10	6, 7, 9	fvmpt 6761	1 ⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾) = (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))

Syntax hints:   → wi 4   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113   ↦ cmpt 5139  ran crn 5549  ‘cfv 6348  (class class class)co 7149   ∈ cmpo 7151  0cc0 10530  1c1 10531   + caddc 10533   · cmul 10535   − cmin 10863  ℕcn 11631  ℕ₀cn0 11891  ...cfz 12889  APcvdwa 16296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-cnex 10586  ax-1cn 10588  ax-addcl 10590

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-1st 7682  df-2nd 7683  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-nn 11632  df-vdwap 16299

This theorem is referenced by:  vdwapf  16303  vdwapval  16304