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

Theorem vdwnn 16389
 Description: Van der Waerden's theorem, infinitary version. For any finite coloring 𝐹 of the positive integers, there is a color 𝑐 that contains arbitrarily long arithmetic progressions. (Contributed by Mario Carneiro, 13-Sep-2014.)
Assertion
Ref Expression
vdwnn ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) → ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
Distinct variable groups:   𝑎,𝑐,𝑑,𝑘,𝑚,𝐹   𝑅,𝑐
Allowed substitution hints:   𝑅(𝑘,𝑚,𝑎,𝑑)

Proof of Theorem vdwnn
Dummy variables 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 766 . . 3 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})) → 𝑅 ∈ Fin)
2 simplr 768 . . 3 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})) → 𝐹:ℕ⟶𝑅)
3 oveq1 7157 . . . . . . . . . . 11 (𝑚 = 𝑤 → (𝑚 · 𝑑) = (𝑤 · 𝑑))
43oveq2d 7166 . . . . . . . . . 10 (𝑚 = 𝑤 → (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑤 · 𝑑)))
54eleq1d 2836 . . . . . . . . 9 (𝑚 = 𝑤 → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}) ↔ (𝑎 + (𝑤 · 𝑑)) ∈ (𝐹 “ {𝑢})))
65cbvralvw 3361 . . . . . . . 8 (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑎 + (𝑤 · 𝑑)) ∈ (𝐹 “ {𝑢}))
7 oveq1 7157 . . . . . . . . . 10 (𝑎 = 𝑦 → (𝑎 + (𝑤 · 𝑑)) = (𝑦 + (𝑤 · 𝑑)))
87eleq1d 2836 . . . . . . . . 9 (𝑎 = 𝑦 → ((𝑎 + (𝑤 · 𝑑)) ∈ (𝐹 “ {𝑢}) ↔ (𝑦 + (𝑤 · 𝑑)) ∈ (𝐹 “ {𝑢})))
98ralbidv 3126 . . . . . . . 8 (𝑎 = 𝑦 → (∀𝑤 ∈ (0...(𝑘 − 1))(𝑎 + (𝑤 · 𝑑)) ∈ (𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑑)) ∈ (𝐹 “ {𝑢})))
106, 9syl5bb 286 . . . . . . 7 (𝑎 = 𝑦 → (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑑)) ∈ (𝐹 “ {𝑢})))
11 oveq2 7158 . . . . . . . . . 10 (𝑑 = 𝑧 → (𝑤 · 𝑑) = (𝑤 · 𝑧))
1211oveq2d 7166 . . . . . . . . 9 (𝑑 = 𝑧 → (𝑦 + (𝑤 · 𝑑)) = (𝑦 + (𝑤 · 𝑧)))
1312eleq1d 2836 . . . . . . . 8 (𝑑 = 𝑧 → ((𝑦 + (𝑤 · 𝑑)) ∈ (𝐹 “ {𝑢}) ↔ (𝑦 + (𝑤 · 𝑧)) ∈ (𝐹 “ {𝑢})))
1413ralbidv 3126 . . . . . . 7 (𝑑 = 𝑧 → (∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑑)) ∈ (𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (𝐹 “ {𝑢})))
1510, 14cbvrex2vw 3374 . . . . . 6 (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}) ↔ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (𝐹 “ {𝑢}))
16 oveq1 7157 . . . . . . . . 9 (𝑘 = 𝑥 → (𝑘 − 1) = (𝑥 − 1))
1716oveq2d 7166 . . . . . . . 8 (𝑘 = 𝑥 → (0...(𝑘 − 1)) = (0...(𝑥 − 1)))
1817raleqdv 3329 . . . . . . 7 (𝑘 = 𝑥 → (∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (𝐹 “ {𝑢})))
19182rexbidv 3224 . . . . . 6 (𝑘 = 𝑥 → (∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (𝐹 “ {𝑢}) ↔ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (𝐹 “ {𝑢})))
2015, 19syl5bb 286 . . . . 5 (𝑘 = 𝑥 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}) ↔ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (𝐹 “ {𝑢})))
2120notbid 321 . . . 4 (𝑘 = 𝑥 → (¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}) ↔ ¬ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (𝐹 “ {𝑢})))
2221cbvrabv 3404 . . 3 {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢})} = {𝑥 ∈ ℕ ∣ ¬ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (𝐹 “ {𝑢})}
23 simpr 488 . . . . 5 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})) → ¬ ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
24 sneq 4532 . . . . . . . . . . 11 (𝑐 = 𝑢 → {𝑐} = {𝑢})
2524imaeq2d 5901 . . . . . . . . . 10 (𝑐 = 𝑢 → (𝐹 “ {𝑐}) = (𝐹 “ {𝑢}))
2625eleq2d 2837 . . . . . . . . 9 (𝑐 = 𝑢 → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}) ↔ (𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢})))
2726ralbidv 3126 . . . . . . . 8 (𝑐 = 𝑢 → (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}) ↔ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢})))
28272rexbidv 3224 . . . . . . 7 (𝑐 = 𝑢 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢})))
2928ralbidv 3126 . . . . . 6 (𝑐 = 𝑢 → (∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}) ↔ ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢})))
3029cbvrexvw 3362 . . . . 5 (∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}) ↔ ∃𝑢𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}))
3123, 30sylnib 331 . . . 4 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})) → ¬ ∃𝑢𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}))
32 rabn0 4281 . . . . . . 7 ({𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢})} ≠ ∅ ↔ ∃𝑘 ∈ ℕ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}))
33 rexnal 3165 . . . . . . 7 (∃𝑘 ∈ ℕ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}) ↔ ¬ ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}))
3432, 33bitri 278 . . . . . 6 ({𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢})} ≠ ∅ ↔ ¬ ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}))
3534ralbii 3097 . . . . 5 (∀𝑢𝑅 {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢})} ≠ ∅ ↔ ∀𝑢𝑅 ¬ ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}))
36 ralnex 3163 . . . . 5 (∀𝑢𝑅 ¬ ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}) ↔ ¬ ∃𝑢𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}))
3735, 36bitri 278 . . . 4 (∀𝑢𝑅 {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢})} ≠ ∅ ↔ ¬ ∃𝑢𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢}))
3831, 37sylibr 237 . . 3 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})) → ∀𝑢𝑅 {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑢})} ≠ ∅)
391, 2, 22, 38vdwnnlem3 16388 . 2 ¬ ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
40 iman 405 . 2 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) → ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})) ↔ ¬ ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})))
4139, 40mpbir 234 1 ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) → ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  {crab 3074  ∅c0 4225  {csn 4522  ◡ccnv 5523   “ cima 5527  ⟶wf 6331  (class class class)co 7150  Fincfn 8527  0cc0 10575  1c1 10576   + caddc 10578   · cmul 10580   − cmin 10908  ℕcn 11674  ...cfz 12939 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 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-oadd 8116  df-er 8299  df-map 8418  df-pm 8419  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-sup 8939  df-inf 8940  df-dju 9363  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-n0 11935  df-xnn0 12007  df-z 12021  df-uz 12283  df-rp 12431  df-fz 12940  df-fl 13211  df-hash 13741  df-vdwap 16359  df-vdwmc 16360  df-vdwpc 16361 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator