Theorem vdwpc 16942

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 7174	. . 3 ⊢ ((𝐹:𝑋⟶𝑅 ∧ 𝑋 ∈ V) → 𝐹 ∈ V)
6	3, 4, 5	sylancl 587	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
7		df-br 5087	. . . 4 ⊢ (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ PolyAP )
8		df-vdwpc 16932	. . . . 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 2829	. . . 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 1137	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → 𝑚 = 𝑀)
12	11	oveq2d 7376	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (1...𝑚) = (1...𝑀))
13		vdwpc.5	. . . . . . . 8 ⊢ 𝐽 = (1...𝑀)
14	12, 13	eqtr4di 2790	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (1...𝑚) = 𝐽)
15	14	oveq2d 7376	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (ℕ ↑m (1...𝑚)) = (ℕ ↑m 𝐽))
16		simp2 1138	. . . . . . . . . . 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 7377	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) = ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)))
19		simp3 1139	. . . . . . . . . . 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 4580	. . . . . . . . . 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 3956	. . . . . . . 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 5173	. . . . . . . . . 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 2753	. . . . . . 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 633	. . . . . 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 3162	. . . 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 7469	. . 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 1374	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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  {csn 4568  ⟨cop 4574   class class class wbr 5086   ↦ cmpt 5167  ^◡ccnv 5623  ran crn 5625   “ cima 5627  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  {coprab 7361   ↑_m cmap 8766  1c1 11030   + caddc 11032  ℕcn 12165  ℕ₀cn0 12428  ...cfz 13452  ♯chash 14283  APcvdwa 16927   PolyAP cvdwp 16929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7363  df-oprab 7364  df-vdwpc 16932

This theorem is referenced by:  vdwlem6  16948  vdwlem7  16949  vdwlem8  16950  vdwlem11  16953