MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vdwpc Structured version   Visualization version   GIF version

Theorem vdwpc 16913
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 (πœ‘ β†’ 𝐾 ∈ β„•0)
vdwmc.3 (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘…)
vdwpc.4 (πœ‘ β†’ 𝑀 ∈ β„•)
vdwpc.5 𝐽 = (1...𝑀)
Assertion
Ref Expression
vdwpc (πœ‘ β†’ (βŸ¨π‘€, 𝐾⟩ PolyAP 𝐹 ↔ βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m 𝐽)(βˆ€π‘– ∈ 𝐽 ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐹 β€œ {(πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ 𝐽 ↦ (πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)))
Distinct variable groups:   π‘Ž,𝑑,𝑖,𝐹   𝐾,π‘Ž,𝑑,𝑖   𝐽,𝑑,𝑖   𝑀,π‘Ž,𝑑,𝑖
Allowed substitution hints:   πœ‘(𝑖,π‘Ž,𝑑)   𝑅(𝑖,π‘Ž,𝑑)   𝐽(π‘Ž)   𝑋(𝑖,π‘Ž,𝑑)

Proof of Theorem vdwpc
Dummy variables 𝑓 π‘˜ π‘š are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdwpc.4 . 2 (πœ‘ β†’ 𝑀 ∈ β„•)
2 vdwmc.2 . 2 (πœ‘ β†’ 𝐾 ∈ β„•0)
3 vdwmc.3 . . 3 (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘…)
4 vdwmc.1 . . 3 𝑋 ∈ V
5 fex 7228 . . 3 ((𝐹:π‘‹βŸΆπ‘… ∧ 𝑋 ∈ V) β†’ 𝐹 ∈ V)
63, 4, 5sylancl 587 . 2 (πœ‘ β†’ 𝐹 ∈ V)
7 df-br 5150 . . . 4 (βŸ¨π‘€, 𝐾⟩ PolyAP 𝐹 ↔ βŸ¨βŸ¨π‘€, 𝐾⟩, 𝐹⟩ ∈ PolyAP )
8 df-vdwpc 16903 . . . . 5 PolyAP = {βŸ¨βŸ¨π‘š, π‘˜βŸ©, π‘“βŸ© ∣ βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m (1...π‘š))(βˆ€π‘– ∈ (1...π‘š)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜π‘˜)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...π‘š) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = π‘š)}
98eleq2i 2826 . . . 4 (βŸ¨βŸ¨π‘€, 𝐾⟩, 𝐹⟩ ∈ PolyAP ↔ βŸ¨βŸ¨π‘€, 𝐾⟩, 𝐹⟩ ∈ {βŸ¨βŸ¨π‘š, π‘˜βŸ©, π‘“βŸ© ∣ βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m (1...π‘š))(βˆ€π‘– ∈ (1...π‘š)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜π‘˜)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...π‘š) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = π‘š)})
107, 9bitri 275 . . 3 (βŸ¨π‘€, 𝐾⟩ PolyAP 𝐹 ↔ βŸ¨βŸ¨π‘€, 𝐾⟩, 𝐹⟩ ∈ {βŸ¨βŸ¨π‘š, π‘˜βŸ©, π‘“βŸ© ∣ βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m (1...π‘š))(βˆ€π‘– ∈ (1...π‘š)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜π‘˜)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...π‘š) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = π‘š)})
11 simp1 1137 . . . . . . . . 9 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ π‘š = 𝑀)
1211oveq2d 7425 . . . . . . . 8 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ (1...π‘š) = (1...𝑀))
13 vdwpc.5 . . . . . . . 8 𝐽 = (1...𝑀)
1412, 13eqtr4di 2791 . . . . . . 7 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ (1...π‘š) = 𝐽)
1514oveq2d 7425 . . . . . 6 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ (β„• ↑m (1...π‘š)) = (β„• ↑m 𝐽))
16 simp2 1138 . . . . . . . . . . 11 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ π‘˜ = 𝐾)
1716fveq2d 6896 . . . . . . . . . 10 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ (APβ€˜π‘˜) = (APβ€˜πΎ))
1817oveqd 7426 . . . . . . . . 9 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜π‘˜)(π‘‘β€˜π‘–)) = ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)))
19 simp3 1139 . . . . . . . . . . 11 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ 𝑓 = 𝐹)
2019cnveqd 5876 . . . . . . . . . 10 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ ◑𝑓 = ◑𝐹)
2119fveq1d 6894 . . . . . . . . . . 11 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))) = (πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–))))
2221sneqd 4641 . . . . . . . . . 10 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))} = {(πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))})
2320, 22imaeq12d 6061 . . . . . . . . 9 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) = (◑𝐹 β€œ {(πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}))
2418, 23sseq12d 4016 . . . . . . . 8 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ (((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜π‘˜)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ↔ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐹 β€œ {(πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))})))
2514, 24raleqbidv 3343 . . . . . . 7 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ (βˆ€π‘– ∈ (1...π‘š)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜π‘˜)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ↔ βˆ€π‘– ∈ 𝐽 ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐹 β€œ {(πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))})))
2614, 21mpteq12dv 5240 . . . . . . . . . 10 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ (𝑖 ∈ (1...π‘š) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))) = (𝑖 ∈ 𝐽 ↦ (πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))))
2726rneqd 5938 . . . . . . . . 9 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ ran (𝑖 ∈ (1...π‘š) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))) = ran (𝑖 ∈ 𝐽 ↦ (πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))))
2827fveq2d 6896 . . . . . . . 8 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ (β™―β€˜ran (𝑖 ∈ (1...π‘š) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = (β™―β€˜ran (𝑖 ∈ 𝐽 ↦ (πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–))))))
2928, 11eqeq12d 2749 . . . . . . 7 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ ((β™―β€˜ran (𝑖 ∈ (1...π‘š) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = π‘š ↔ (β™―β€˜ran (𝑖 ∈ 𝐽 ↦ (πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀))
3025, 29anbi12d 632 . . . . . 6 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ ((βˆ€π‘– ∈ (1...π‘š)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜π‘˜)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...π‘š) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = π‘š) ↔ (βˆ€π‘– ∈ 𝐽 ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐹 β€œ {(πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ 𝐽 ↦ (πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)))
3115, 30rexeqbidv 3344 . . . . 5 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ (βˆƒπ‘‘ ∈ (β„• ↑m (1...π‘š))(βˆ€π‘– ∈ (1...π‘š)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜π‘˜)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...π‘š) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = π‘š) ↔ βˆƒπ‘‘ ∈ (β„• ↑m 𝐽)(βˆ€π‘– ∈ 𝐽 ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐹 β€œ {(πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ 𝐽 ↦ (πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)))
3231rexbidv 3179 . . . 4 ((π‘š = 𝑀 ∧ π‘˜ = 𝐾 ∧ 𝑓 = 𝐹) β†’ (βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m (1...π‘š))(βˆ€π‘– ∈ (1...π‘š)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜π‘˜)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...π‘š) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = π‘š) ↔ βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m 𝐽)(βˆ€π‘– ∈ 𝐽 ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐹 β€œ {(πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ 𝐽 ↦ (πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)))
3332eloprabga 7516 . . 3 ((𝑀 ∈ β„• ∧ 𝐾 ∈ β„•0 ∧ 𝐹 ∈ V) β†’ (βŸ¨βŸ¨π‘€, 𝐾⟩, 𝐹⟩ ∈ {βŸ¨βŸ¨π‘š, π‘˜βŸ©, π‘“βŸ© ∣ βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m (1...π‘š))(βˆ€π‘– ∈ (1...π‘š)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜π‘˜)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...π‘š) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = π‘š)} ↔ βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m 𝐽)(βˆ€π‘– ∈ 𝐽 ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐹 β€œ {(πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ 𝐽 ↦ (πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)))
3410, 33bitrid 283 . 2 ((𝑀 ∈ β„• ∧ 𝐾 ∈ β„•0 ∧ 𝐹 ∈ V) β†’ (βŸ¨π‘€, 𝐾⟩ PolyAP 𝐹 ↔ βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m 𝐽)(βˆ€π‘– ∈ 𝐽 ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐹 β€œ {(πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ 𝐽 ↦ (πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)))
351, 2, 6, 34syl3anc 1372 1 (πœ‘ β†’ (βŸ¨π‘€, 𝐾⟩ PolyAP 𝐹 ↔ βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m 𝐽)(βˆ€π‘– ∈ 𝐽 ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐹 β€œ {(πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ 𝐽 ↦ (πΉβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949  {csn 4629  βŸ¨cop 4635   class class class wbr 5149   ↦ cmpt 5232  β—‘ccnv 5676  ran crn 5678   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  {coprab 7410   ↑m cmap 8820  1c1 11111   + caddc 11113  β„•cn 12212  β„•0cn0 12472  ...cfz 13484  β™―chash 14290  APcvdwa 16898   PolyAP cvdwp 16900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-vdwpc 16903
This theorem is referenced by:  vdwlem6  16919  vdwlem7  16920  vdwlem8  16921  vdwlem11  16924
  Copyright terms: Public domain W3C validator