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Theorem vdwpc 15906
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 6654 . . 3 ((𝐹:𝑋𝑅𝑋 ∈ V) → 𝐹 ∈ V)
63, 4, 5sylancl 697 . 2 (𝜑𝐹 ∈ V)
7 df-br 4805 . . . 4 (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ PolyAP )
8 df-vdwpc 15896 . . . . 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 2831 . . . 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 1131 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑚 = 𝑀)
1211oveq2d 6830 . . . . . . . 8 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (1...𝑚) = (1...𝑀))
13 vdwpc.5 . . . . . . . 8 𝐽 = (1...𝑀)
1412, 13syl6eqr 2812 . . . . . . 7 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (1...𝑚) = 𝐽)
1514oveq2d 6830 . . . . . 6 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (ℕ ↑𝑚 (1...𝑚)) = (ℕ ↑𝑚 𝐽))
16 simp2 1132 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑘 = 𝐾)
1716fveq2d 6357 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (AP‘𝑘) = (AP‘𝐾))
1817oveqd 6831 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) = ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
19 simp3 1133 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑓 = 𝐹)
2019cnveqd 5453 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑓 = 𝐹)
2119fveq1d 6355 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑓‘(𝑎 + (𝑑𝑖))) = (𝐹‘(𝑎 + (𝑑𝑖))))
2221sneqd 4333 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → {(𝑓‘(𝑎 + (𝑑𝑖)))} = {(𝐹‘(𝑎 + (𝑑𝑖)))})
2320, 22imaeq12d 5625 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) = (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}))
2418, 23sseq12d 3775 . . . . . . . 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 3291 . . . . . . 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 4885 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖)))))
2726rneqd 5508 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖)))))
2827fveq2d 6357 . . . . . . . 8 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = (♯‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))))
2928, 11eqeq12d 2775 . . . . . . 7 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ((♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚 ↔ (♯‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀))
3025, 29anbi12d 749 . . . . . 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 3292 . . . . 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 3190 . . . 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 6913 . . 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 1477 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 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  wss 3715  {csn 4321  cop 4327   class class class wbr 4804  cmpt 4881  ccnv 5265  ran crn 5267  cima 5269  wf 6045  cfv 6049  (class class class)co 6814  {coprab 6815  𝑚 cmap 8025  1c1 10149   + caddc 10151  cn 11232  0cn0 11504  ...cfz 12539  chash 13331  APcvdwa 15891   PolyAP cvdwp 15893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-vdwpc 15896
This theorem is referenced by:  vdwlem6  15912  vdwlem7  15913  vdwlem8  15914  vdwlem11  15917
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