Theorem vdwpc 16889

Description: The predicate " The coloring 𝐹 contains a polychromatic 𝑀-tuple of AP's of length 𝐾". A polychromatic 𝑀-tuple of AP's is a set of AP's with the same base point but different step lengths, such that each individual AP is monochromatic, but the AP's all have mutually distinct colors. (The common basepoint is not required to have the same color as any of the AP's.) (Contributed by Mario Carneiro, 18-Aug-2014.)

Hypotheses

Ref	Expression
vdwmc.1	⊢ 𝑋 ∈ V
vdwmc.2	⊢ (𝜑 → 𝐾 ∈ ℕ0)
vdwmc.3	⊢ (𝜑 → 𝐹:𝑋⟶𝑅)
vdwpc.4	⊢ (𝜑 → 𝑀 ∈ ℕ)
vdwpc.5	⊢ 𝐽 = (1...𝑀)

Assertion

Ref	Expression
vdwpc	⊢ (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m 𝐽)(∀𝑖 ∈ 𝐽 ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)))

Distinct variable groups:   𝑎,𝑑,𝑖,𝐹   𝐾,𝑎,𝑑,𝑖   𝐽,𝑑,𝑖   𝑀,𝑎,𝑑,𝑖

Allowed substitution hints:   𝜑(𝑖,𝑎,𝑑)   𝑅(𝑖,𝑎,𝑑)   𝐽(𝑎)   𝑋(𝑖,𝑎,𝑑)

Proof of Theorem vdwpc

Dummy variables 𝑓 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdwpc.4	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2		vdwmc.2	. 2 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
3		vdwmc.3	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑅)
4		vdwmc.1	. . 3 ⊢ 𝑋 ∈ V
5		fex 7160	. . 3 ⊢ ((𝐹:𝑋⟶𝑅 ∧ 𝑋 ∈ V) → 𝐹 ∈ V)
6	3, 4, 5	sylancl 586	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
7		df-br 5092	. . . 4 ⊢ (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ PolyAP )
8		df-vdwpc 16879	. . . . 5 ⊢ PolyAP = {⟨⟨𝑚, 𝑘⟩, 𝑓⟩ ∣ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = 𝑚)}
9	8	eleq2i 2823	. . . 4 ⊢ (⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ PolyAP ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ {⟨⟨𝑚, 𝑘⟩, 𝑓⟩ ∣ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = 𝑚)})
10	7, 9	bitri 275	. . 3 ⊢ (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ {⟨⟨𝑚, 𝑘⟩, 𝑓⟩ ∣ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = 𝑚)})
11		simp1 1136	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → 𝑚 = 𝑀)
12	11	oveq2d 7362	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (1...𝑚) = (1...𝑀))
13		vdwpc.5	. . . . . . . 8 ⊢ 𝐽 = (1...𝑀)
14	12, 13	eqtr4di 2784	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (1...𝑚) = 𝐽)
15	14	oveq2d 7362	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (ℕ ↑m (1...𝑚)) = (ℕ ↑m 𝐽))
16		simp2 1137	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → 𝑘 = 𝐾)
17	16	fveq2d 6826	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (AP‘𝑘) = (AP‘𝐾))
18	17	oveqd 7363	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) = ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)))
19		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
20	19	cnveqd 5815	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ◡𝑓 = ◡𝐹)
21	19	fveq1d 6824	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (𝑓‘(𝑎 + (𝑑‘𝑖))) = (𝐹‘(𝑎 + (𝑑‘𝑖))))
22	21	sneqd 4588	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → {(𝑓‘(𝑎 + (𝑑‘𝑖)))} = {(𝐹‘(𝑎 + (𝑑‘𝑖)))})
23	20, 22	imaeq12d 6010	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) = (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}))
24	18, 23	sseq12d 3968	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ↔ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))})))
25	14, 24	raleqbidv 3312	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ↔ ∀𝑖 ∈ 𝐽 ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))})))
26	14, 21	mpteq12dv 5178	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) = (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖)))))
27	26	rneqd 5878	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) = ran (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖)))))
28	27	fveq2d 6826	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = (♯‘ran (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))))
29	28, 11	eqeq12d 2747	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ((♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = 𝑚 ↔ (♯‘ran (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 𝑀))
30	25, 29	anbi12d 632	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ((∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = 𝑚) ↔ (∀𝑖 ∈ 𝐽 ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)))
31	15, 30	rexeqbidv 3313	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (∃𝑑 ∈ (ℕ ↑m (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = 𝑚) ↔ ∃𝑑 ∈ (ℕ ↑m 𝐽)(∀𝑖 ∈ 𝐽 ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)))
32	31	rexbidv 3156	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = 𝑚) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m 𝐽)(∀𝑖 ∈ 𝐽 ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)))
33	32	eloprabga 7455	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝐾 ∈ ℕ0 ∧ 𝐹 ∈ V) → (⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ {⟨⟨𝑚, 𝑘⟩, 𝑓⟩ ∣ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = 𝑚)} ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m 𝐽)(∀𝑖 ∈ 𝐽 ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)))
34	10, 33	bitrid 283	. 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝐾 ∈ ℕ0 ∧ 𝐹 ∈ V) → (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m 𝐽)(∀𝑖 ∈ 𝐽 ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)))
35	1, 2, 6, 34	syl3anc 1373	1 ⊢ (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m 𝐽)(∀𝑖 ∈ 𝐽 ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ⊆ wss 3902  {csn 4576  ⟨cop 4582   class class class wbr 5091   ↦ cmpt 5172  ^◡ccnv 5615  ran crn 5617   “ cima 5619  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  {coprab 7347   ↑_m cmap 8750  1c1 11004   + caddc 11006  ℕcn 12122  ℕ₀cn0 12378  ...cfz 13404  ♯chash 14234  APcvdwa 16874   PolyAP cvdwp 16876

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-vdwpc 16879

This theorem is referenced by:  vdwlem6  16895  vdwlem7  16896  vdwlem8  16897  vdwlem11  16900