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Theorem vdwpc 15627
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 (𝜑𝐾 ∈ ℕ0)
vdwmc.3 (𝜑𝐹:𝑋𝑅)
vdwpc.4 (𝜑𝑀 ∈ ℕ)
vdwpc.5 𝐽 = (1...𝑀)
Assertion
Ref Expression
vdwpc (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 𝐽)(∀𝑖𝐽 ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 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.
StepHypRef Expression
1 vdwpc.4 . 2 (𝜑𝑀 ∈ ℕ)
2 vdwmc.2 . 2 (𝜑𝐾 ∈ ℕ0)
3 vdwmc.3 . . 3 (𝜑𝐹:𝑋𝑅)
4 vdwmc.1 . . 3 𝑋 ∈ V
5 fex 6455 . . 3 ((𝐹:𝑋𝑅𝑋 ∈ V) → 𝐹 ∈ V)
63, 4, 5sylancl 693 . 2 (𝜑𝐹 ∈ V)
7 df-br 4624 . . . 4 (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ PolyAP )
8 df-vdwpc 15617 . . . . 5 PolyAP = {⟨⟨𝑚, 𝑘⟩, 𝑓⟩ ∣ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚)}
98eleq2i 2690 . . . 4 (⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ PolyAP ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ {⟨⟨𝑚, 𝑘⟩, 𝑓⟩ ∣ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚)})
107, 9bitri 264 . . 3 (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ {⟨⟨𝑚, 𝑘⟩, 𝑓⟩ ∣ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚)})
11 simp1 1059 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑚 = 𝑀)
1211oveq2d 6631 . . . . . . . 8 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (1...𝑚) = (1...𝑀))
13 vdwpc.5 . . . . . . . 8 𝐽 = (1...𝑀)
1412, 13syl6eqr 2673 . . . . . . 7 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (1...𝑚) = 𝐽)
1514oveq2d 6631 . . . . . 6 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (ℕ ↑𝑚 (1...𝑚)) = (ℕ ↑𝑚 𝐽))
16 simp2 1060 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑘 = 𝐾)
1716fveq2d 6162 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (AP‘𝑘) = (AP‘𝐾))
1817oveqd 6632 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) = ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
19 simp3 1061 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑓 = 𝐹)
2019cnveqd 5268 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑓 = 𝐹)
2119fveq1d 6160 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑓‘(𝑎 + (𝑑𝑖))) = (𝐹‘(𝑎 + (𝑑𝑖))))
2221sneqd 4167 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → {(𝑓‘(𝑎 + (𝑑𝑖)))} = {(𝐹‘(𝑎 + (𝑑𝑖)))})
2320, 22imaeq12d 5436 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) = (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}))
2418, 23sseq12d 3619 . . . . . . . 8 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ↔ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))})))
2514, 24raleqbidv 3145 . . . . . . 7 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ↔ ∀𝑖𝐽 ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))})))
2614, 21mpteq12dv 4703 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖)))))
2726rneqd 5323 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖)))))
2827fveq2d 6162 . . . . . . . 8 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (#‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = (#‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))))
2928, 11eqeq12d 2636 . . . . . . 7 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ((#‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚 ↔ (#‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀))
3025, 29anbi12d 746 . . . . . 6 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ((∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚) ↔ (∀𝑖𝐽 ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀)))
3115, 30rexeqbidv 3146 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (∃𝑑 ∈ (ℕ ↑𝑚 (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚) ↔ ∃𝑑 ∈ (ℕ ↑𝑚 𝐽)(∀𝑖𝐽 ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀)))
3231rexbidv 3047 . . . 4 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 𝐽)(∀𝑖𝐽 ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀)))
3332eloprabga 6712 . . 3 ((𝑀 ∈ ℕ ∧ 𝐾 ∈ ℕ0𝐹 ∈ V) → (⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ {⟨⟨𝑚, 𝑘⟩, 𝑓⟩ ∣ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚)} ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 𝐽)(∀𝑖𝐽 ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀)))
3410, 33syl5bb 272 . 2 ((𝑀 ∈ ℕ ∧ 𝐾 ∈ ℕ0𝐹 ∈ V) → (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 𝐽)(∀𝑖𝐽 ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀)))
351, 2, 6, 34syl3anc 1323 1 (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 𝐽)(∀𝑖𝐽 ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  wrex 2909  Vcvv 3190  wss 3560  {csn 4155  cop 4161   class class class wbr 4623  cmpt 4683  ccnv 5083  ran crn 5085  cima 5087  wf 5853  cfv 5857  (class class class)co 6615  {coprab 6616  𝑚 cmap 7817  1c1 9897   + caddc 9899  cn 10980  0cn0 11252  ...cfz 12284  #chash 13073  APcvdwa 15612   PolyAP cvdwp 15614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-vdwpc 15617
This theorem is referenced by:  vdwlem6  15633  vdwlem7  15634  vdwlem8  15635  vdwlem11  15638
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