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Theorem vdwpc 16852
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 7176 . . 3 ((𝐹:𝑋𝑅𝑋 ∈ V) → 𝐹 ∈ V)
63, 4, 5sylancl 586 . 2 (𝜑𝐹 ∈ V)
7 df-br 5106 . . . 4 (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ PolyAP )
8 df-vdwpc 16842 . . . . 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 2829 . . . 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 274 . . 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 1136 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑚 = 𝑀)
1211oveq2d 7373 . . . . . . . 8 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (1...𝑚) = (1...𝑀))
13 vdwpc.5 . . . . . . . 8 𝐽 = (1...𝑀)
1412, 13eqtr4di 2794 . . . . . . 7 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (1...𝑚) = 𝐽)
1514oveq2d 7373 . . . . . 6 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (ℕ ↑m (1...𝑚)) = (ℕ ↑m 𝐽))
16 simp2 1137 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑘 = 𝐾)
1716fveq2d 6846 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (AP‘𝑘) = (AP‘𝐾))
1817oveqd 7374 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) = ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
19 simp3 1138 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑓 = 𝐹)
2019cnveqd 5831 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑓 = 𝐹)
2119fveq1d 6844 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑓‘(𝑎 + (𝑑𝑖))) = (𝐹‘(𝑎 + (𝑑𝑖))))
2221sneqd 4598 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → {(𝑓‘(𝑎 + (𝑑𝑖)))} = {(𝐹‘(𝑎 + (𝑑𝑖)))})
2320, 22imaeq12d 6014 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) = (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}))
2418, 23sseq12d 3977 . . . . . . . 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 3319 . . . . . . 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 5196 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖)))))
2726rneqd 5893 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖)))))
2827fveq2d 6846 . . . . . . . 8 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = (♯‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))))
2928, 11eqeq12d 2752 . . . . . . 7 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ((♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚 ↔ (♯‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀))
3025, 29anbi12d 631 . . . . . 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 3320 . . . . 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 3175 . . . 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 7464 . . 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, 33bitrid 282 . 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 1371 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910  {csn 4586  cop 4592   class class class wbr 5105  cmpt 5188  ccnv 5632  ran crn 5634  cima 5636  wf 6492  cfv 6496  (class class class)co 7357  {coprab 7358  m cmap 8765  1c1 11052   + caddc 11054  cn 12153  0cn0 12413  ...cfz 13424  chash 14230  APcvdwa 16837   PolyAP cvdwp 16839
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-vdwpc 16842
This theorem is referenced by:  vdwlem6  16858  vdwlem7  16859  vdwlem8  16860  vdwlem11  16863
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