Theorem vdwapval 16309

Description: Value of the arithmetic progression function. (Contributed by Mario Carneiro, 18-Aug-2014.)

Assertion

Ref	Expression
vdwapval	⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑋 ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑋 = (𝐴 + (𝑚 · 𝐷))))

Distinct variable groups:   𝐴,𝑚   𝐷,𝑚   𝑚,𝐾   𝑚,𝑋

Proof of Theorem vdwapval

Dummy variables 𝑎 𝑑 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdwapfval 16307	. . . . . . 7 ⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾) = (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))
2	1	3ad2ant1 1129	. . . . . 6 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (AP‘𝐾) = (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))
3	2	oveqd 7173	. . . . 5 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘𝐾)𝐷) = (𝐴(𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))))𝐷))
4		oveq2 7164	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (𝑚 · 𝑑) = (𝑚 · 𝐷))
5		oveq12 7165	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ (𝑚 · 𝑑) = (𝑚 · 𝐷)) → (𝑎 + (𝑚 · 𝑑)) = (𝐴 + (𝑚 · 𝐷)))
6	4, 5	sylan2 594	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑑 = 𝐷) → (𝑎 + (𝑚 · 𝑑)) = (𝐴 + (𝑚 · 𝐷)))
7	6	mpteq2dv 5162	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑑 = 𝐷) → (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))) = (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝐴 + (𝑚 · 𝐷))))
8	7	rneqd 5808	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑑 = 𝐷) → ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))) = ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝐴 + (𝑚 · 𝐷))))
9		eqid 2821	. . . . . . 7 ⊢ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))) = (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))))
10		ovex 7189	. . . . . . . . 9 ⊢ (0...(𝐾 − 1)) ∈ V
11	10	mptex 6986	. . . . . . . 8 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝐴 + (𝑚 · 𝐷))) ∈ V
12	11	rnex 7617	. . . . . . 7 ⊢ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝐴 + (𝑚 · 𝐷))) ∈ V
13	8, 9, 12	ovmpoa 7305	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))))𝐷) = ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝐴 + (𝑚 · 𝐷))))
14	13	3adant1 1126	. . . . 5 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))))𝐷) = ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝐴 + (𝑚 · 𝐷))))
15	3, 14	eqtrd 2856	. . . 4 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘𝐾)𝐷) = ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝐴 + (𝑚 · 𝐷))))
16		eqid 2821	. . . . 5 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝐴 + (𝑚 · 𝐷))) = (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝐴 + (𝑚 · 𝐷)))
17	16	rnmpt 5827	. . . 4 ⊢ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝐴 + (𝑚 · 𝐷))) = {𝑥 ∣ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝐴 + (𝑚 · 𝐷))}
18	15, 17	syl6eq 2872	. . 3 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘𝐾)𝐷) = {𝑥 ∣ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝐴 + (𝑚 · 𝐷))})
19	18	eleq2d 2898	. 2 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑋 ∈ (𝐴(AP‘𝐾)𝐷) ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝐴 + (𝑚 · 𝐷))}))
20		id 22	. . . . 5 ⊢ (𝑋 = (𝐴 + (𝑚 · 𝐷)) → 𝑋 = (𝐴 + (𝑚 · 𝐷)))
21		ovex 7189	. . . . 5 ⊢ (𝐴 + (𝑚 · 𝐷)) ∈ V
22	20, 21	eqeltrdi 2921	. . . 4 ⊢ (𝑋 = (𝐴 + (𝑚 · 𝐷)) → 𝑋 ∈ V)
23	22	rexlimivw 3282	. . 3 ⊢ (∃𝑚 ∈ (0...(𝐾 − 1))𝑋 = (𝐴 + (𝑚 · 𝐷)) → 𝑋 ∈ V)
24		eqeq1 2825	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = (𝐴 + (𝑚 · 𝐷)) ↔ 𝑋 = (𝐴 + (𝑚 · 𝐷))))
25	24	rexbidv 3297	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝐴 + (𝑚 · 𝐷)) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑋 = (𝐴 + (𝑚 · 𝐷))))
26	23, 25	elab3 3674	. 2 ⊢ (𝑋 ∈ {𝑥 ∣ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝐴 + (𝑚 · 𝐷))} ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑋 = (𝐴 + (𝑚 · 𝐷)))
27	19, 26	syl6bb 289	1 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑋 ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑋 = (𝐴 + (𝑚 · 𝐷))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {cab 2799  ∃wrex 3139  Vcvv 3494   ↦ cmpt 5146  ran crn 5556  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  0cc0 10537  1c1 10538   + caddc 10540   · cmul 10542   − cmin 10870  ℕcn 11638  ℕ₀cn0 11898  ...cfz 12893  APcvdwa 16301

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-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-1cn 10595  ax-addcl 10597

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  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-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-nn 11639  df-vdwap 16304

This theorem is referenced by:  vdwapun  16310  vdwap0  16312  vdwmc2  16315  vdwlem1  16317  vdwlem2  16318  vdwlem6  16322  vdwlem8  16324