Theorem vdwpc 17027

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 7263	. . 3 ⊢ ((𝐹:𝑋⟶𝑅 ∧ 𝑋 ∈ V) → 𝐹 ∈ V)
6	3, 4, 5	sylancl 585	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
7		df-br 5167	. . . 4 ⊢ (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ PolyAP )
8		df-vdwpc 17017	. . . . 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 2836	. . . 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 7464	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (1...𝑚) = (1...𝑀))
13		vdwpc.5	. . . . . . . 8 ⊢ 𝐽 = (1...𝑀)
14	12, 13	eqtr4di 2798	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (1...𝑚) = 𝐽)
15	14	oveq2d 7464	. . . . . 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 6924	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (AP‘𝑘) = (AP‘𝐾))
18	17	oveqd 7465	. . . . . . . . 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 5900	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ◡𝑓 = ◡𝐹)
21	19	fveq1d 6922	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (𝑓‘(𝑎 + (𝑑‘𝑖))) = (𝐹‘(𝑎 + (𝑑‘𝑖))))
22	21	sneqd 4660	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → {(𝑓‘(𝑎 + (𝑑‘𝑖)))} = {(𝐹‘(𝑎 + (𝑑‘𝑖)))})
23	20, 22	imaeq12d 6090	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) = (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}))
24	18, 23	sseq12d 4042	. . . . . . . 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 3354	. . . . . . 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 5257	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) = (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖)))))
27	26	rneqd 5963	. . . . . . . . 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 6924	. . . . . . . 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 631	. . . . . 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 3355	. . . . 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 3185	. . . 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 7558	. . 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 1371	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 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976  {csn 4648  ⟨cop 4654   class class class wbr 5166   ↦ cmpt 5249  ^◡ccnv 5699  ran crn 5701   “ cima 5703  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  {coprab 7449   ↑_m cmap 8884  1c1 11185   + caddc 11187  ℕcn 12293  ℕ₀cn0 12553  ...cfz 13567  ♯chash 14379  APcvdwa 17012   PolyAP cvdwp 17014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-vdwpc 17017

This theorem is referenced by:  vdwlem6  17033  vdwlem7  17034  vdwlem8  17035  vdwlem11  17038