Theorem vdwpc 17078

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 7229	. . 3 ⊢ ((𝐹:𝑋⟶𝑅 ∧ 𝑋 ∈ V) → 𝐹 ∈ V)
6	3, 4, 5	sylancl 598	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
7		df-br 5108	. . . 4 ⊢ (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ PolyAP )
8		df-vdwpc 17068	. . . . 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 2854	. . . 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 278	. . 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 1154	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → 𝑚 = 𝑀)
12	11	oveq2d 7433	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (1...𝑚) = (1...𝑀))
13		vdwpc.5	. . . . . . . 8 ⊢ 𝐽 = (1...𝑀)
14	12, 13	eqtr4di 2815	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (1...𝑚) = 𝐽)
15	14	oveq2d 7433	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (ℕ ↑m (1...𝑚)) = (ℕ ↑m 𝐽))
16		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → 𝑘 = 𝐾)
17	16	fveq2d 6886	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (AP‘𝑘) = (AP‘𝐾))
18	17	oveqd 7434	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) = ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)))
19		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
20	19	cnveqd 5859	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ◡𝑓 = ◡𝐹)
21	19	fveq1d 6884	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (𝑓‘(𝑎 + (𝑑‘𝑖))) = (𝐹‘(𝑎 + (𝑑‘𝑖))))
22	21	sneqd 4599	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → {(𝑓‘(𝑎 + (𝑑‘𝑖)))} = {(𝐹‘(𝑎 + (𝑑‘𝑖)))})
23	20, 22	imaeq12d 6061	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) = (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}))
24	18, 23	sseq12d 3967	. . . . . . . 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 3336	. . . . . . 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 5196	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) = (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖)))))
27	26	rneqd 5926	. . . . . . . . 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 6886	. . . . . . . 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 2778	. . . . . . 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 644	. . . . . 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 3337	. . . . 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 3188	. . . 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 7526	. . 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 286	. 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 1398	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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ⊆ wss 3902  {csn 4587  ⟨cop 4593   class class class wbr 5107   ↦ cmpt 5190  ^◡ccnv 5658  ran crn 5660   “ cima 5662  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417  {coprab 7418   ↑_m cmap 8830  1c1 11129   + caddc 11131  ℕcn 12261  ℕ₀cn0 12532  ...cfz 13565  ♯chash 14398  APcvdwa 17063   PolyAP cvdwp 17065

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-vdwpc 17068

This theorem is used by:  vdwlem6  17084  vdwlem7  17085  vdwlem8  17086  vdwlem11  17089