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Theorem vdwpc 15965
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 6682 . . 3 ((𝐹:𝑋𝑅𝑋 ∈ V) → 𝐹 ∈ V)
63, 4, 5sylancl 580 . 2 (𝜑𝐹 ∈ V)
7 df-br 4810 . . . 4 (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ PolyAP )
8 df-vdwpc 15955 . . . . 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 2836 . . . 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 266 . . 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 1166 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑚 = 𝑀)
1211oveq2d 6858 . . . . . . . 8 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (1...𝑚) = (1...𝑀))
13 vdwpc.5 . . . . . . . 8 𝐽 = (1...𝑀)
1412, 13syl6eqr 2817 . . . . . . 7 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (1...𝑚) = 𝐽)
1514oveq2d 6858 . . . . . 6 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (ℕ ↑𝑚 (1...𝑚)) = (ℕ ↑𝑚 𝐽))
16 simp2 1167 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑘 = 𝐾)
1716fveq2d 6379 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (AP‘𝑘) = (AP‘𝐾))
1817oveqd 6859 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) = ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
19 simp3 1168 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑓 = 𝐹)
2019cnveqd 5466 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑓 = 𝐹)
2119fveq1d 6377 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑓‘(𝑎 + (𝑑𝑖))) = (𝐹‘(𝑎 + (𝑑𝑖))))
2221sneqd 4346 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → {(𝑓‘(𝑎 + (𝑑𝑖)))} = {(𝐹‘(𝑎 + (𝑑𝑖)))})
2320, 22imaeq12d 5649 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) = (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}))
2418, 23sseq12d 3794 . . . . . . . 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 3300 . . . . . . 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 4892 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖)))))
2726rneqd 5521 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖)))))
2827fveq2d 6379 . . . . . . . 8 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = (♯‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))))
2928, 11eqeq12d 2780 . . . . . . 7 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ((♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚 ↔ (♯‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀))
3025, 29anbi12d 624 . . . . . 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 3301 . . . . 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 3199 . . . 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 6945 . . 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 274 . 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 1490 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  wss 3732  {csn 4334  cop 4340   class class class wbr 4809  cmpt 4888  ccnv 5276  ran crn 5278  cima 5280  wf 6064  cfv 6068  (class class class)co 6842  {coprab 6843  𝑚 cmap 8060  1c1 10190   + caddc 10192  cn 11274  0cn0 11538  ...cfz 12533  chash 13321  APcvdwa 15950   PolyAP cvdwp 15952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-vdwpc 15955
This theorem is referenced by:  vdwlem6  15971  vdwlem7  15972  vdwlem8  15973  vdwlem11  15976
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