Theorem vdwpc 16949

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 7177	. . 3 ⊢ ((𝐹:𝑋⟶𝑅 ∧ 𝑋 ∈ V) → 𝐹 ∈ V)
6	3, 4, 5	sylancl 592	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
7		df-br 5080	. . . 4 ⊢ (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ PolyAP )
8		df-vdwpc 16939	. . . . 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 2832	. . . 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 276	. . 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 1142	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → 𝑚 = 𝑀)
12	11	oveq2d 7379	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (1...𝑚) = (1...𝑀))
13		vdwpc.5	. . . . . . . 8 ⊢ 𝐽 = (1...𝑀)
14	12, 13	eqtr4di 2793	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (1...𝑚) = 𝐽)
15	14	oveq2d 7379	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (ℕ ↑m (1...𝑚)) = (ℕ ↑m 𝐽))
16		simp2 1143	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → 𝑘 = 𝐾)
17	16	fveq2d 6838	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (AP‘𝑘) = (AP‘𝐾))
18	17	oveqd 7380	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) = ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)))
19		simp3 1144	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
20	19	cnveqd 5824	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ◡𝑓 = ◡𝐹)
21	19	fveq1d 6836	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (𝑓‘(𝑎 + (𝑑‘𝑖))) = (𝐹‘(𝑎 + (𝑑‘𝑖))))
22	21	sneqd 4574	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → {(𝑓‘(𝑎 + (𝑑‘𝑖)))} = {(𝐹‘(𝑎 + (𝑑‘𝑖)))})
23	20, 22	imaeq12d 6020	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) = (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}))
24	18, 23	sseq12d 3955	. . . . . . . 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 3314	. . . . . . 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 5166	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) = (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖)))))
27	26	rneqd 5887	. . . . . . . . 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 6838	. . . . . . . 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 2756	. . . . . . 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 638	. . . . . 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 3315	. . . . 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 3164	. . . 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 7472	. . 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 284	. 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 1379	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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ⊆ wss 3890  {csn 4562  ⟨cop 4568   class class class wbr 5079   ↦ cmpt 5160  ^◡ccnv 5624  ran crn 5626   “ cima 5628  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  {coprab 7364   ↑_m cmap 8770  1c1 11037   + caddc 11039  ℕcn 12172  ℕ₀cn0 12435  ...cfz 13459  ♯chash 14290  APcvdwa 16934   PolyAP cvdwp 16936

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-vdwpc 16939

This theorem is referenced by:  vdwlem6  16955  vdwlem7  16956  vdwlem8  16957  vdwlem11  16960