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Theorem vdwpc 16306
 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 6966 . . 3 ((𝐹:𝑋𝑅𝑋 ∈ V) → 𝐹 ∈ V)
63, 4, 5sylancl 589 . 2 (𝜑𝐹 ∈ V)
7 df-br 5031 . . . 4 (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ⟨⟨𝑀, 𝐾⟩, 𝐹⟩ ∈ PolyAP )
8 df-vdwpc 16296 . . . . 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 2881 . . . 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 278 . . 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 1133 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑚 = 𝑀)
1211oveq2d 7151 . . . . . . . 8 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (1...𝑚) = (1...𝑀))
13 vdwpc.5 . . . . . . . 8 𝐽 = (1...𝑀)
1412, 13eqtr4di 2851 . . . . . . 7 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (1...𝑚) = 𝐽)
1514oveq2d 7151 . . . . . 6 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (ℕ ↑m (1...𝑚)) = (ℕ ↑m 𝐽))
16 simp2 1134 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑘 = 𝐾)
1716fveq2d 6649 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (AP‘𝑘) = (AP‘𝐾))
1817oveqd 7152 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) = ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
19 simp3 1135 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑓 = 𝐹)
2019cnveqd 5710 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → 𝑓 = 𝐹)
2119fveq1d 6647 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑓‘(𝑎 + (𝑑𝑖))) = (𝐹‘(𝑎 + (𝑑𝑖))))
2221sneqd 4537 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → {(𝑓‘(𝑎 + (𝑑𝑖)))} = {(𝐹‘(𝑎 + (𝑑𝑖)))})
2320, 22imaeq12d 5897 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) = (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}))
2418, 23sseq12d 3948 . . . . . . . 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 3354 . . . . . . 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 5115 . . . . . . . . . 10 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖)))))
2726rneqd 5772 . . . . . . . . 9 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖)))))
2827fveq2d 6649 . . . . . . . 8 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = (♯‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))))
2928, 11eqeq12d 2814 . . . . . . 7 ((𝑚 = 𝑀𝑘 = 𝐾𝑓 = 𝐹) → ((♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚 ↔ (♯‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀))
3025, 29anbi12d 633 . . . . . 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 3355 . . . . 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 3256 . . . 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 7240 . . 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 286 . 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 1368 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  {csn 4525  ⟨cop 4531   class class class wbr 5030   ↦ cmpt 5110  ◡ccnv 5518  ran crn 5520   “ cima 5522  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  {coprab 7136   ↑m cmap 8389  1c1 10527   + caddc 10529  ℕcn 11625  ℕ0cn0 11885  ...cfz 12885  ♯chash 13686  APcvdwa 16291   PolyAP cvdwp 16293 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-vdwpc 16296 This theorem is referenced by:  vdwlem6  16312  vdwlem7  16313  vdwlem8  16314  vdwlem11  16317
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