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Mirrors > Home > MPE Home > Th. List > vdw | Structured version Visualization version GIF version |
Description: Van der Waerden's theorem. For any finite coloring 𝑅 and integer 𝐾, there is an 𝑁 such that every coloring function from 1...𝑁 to 𝑅 contains a monochromatic arithmetic progression (which written out in full means that there is a color 𝑐 and base, increment values 𝑎, 𝑑 such that all the numbers 𝑎, 𝑎 + 𝑑, ..., 𝑎 + (𝑘 − 1)𝑑 lie in the preimage of {𝑐}, i.e. they are all in 1...𝑁 and 𝑓 evaluated at each one yields 𝑐). (Contributed by Mario Carneiro, 13-Sep-2014.) |
Ref | Expression |
---|---|
vdw | ⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . 3 ⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Fin) | |
2 | simpr 484 | . . 3 ⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0) | |
3 | 1, 2 | vdwlem13 17027 | . 2 ⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓) |
4 | ovex 7464 | . . . . 5 ⊢ (1...𝑛) ∈ V | |
5 | simpllr 776 | . . . . 5 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑m (1...𝑛))) → 𝐾 ∈ ℕ0) | |
6 | simpll 767 | . . . . . . 7 ⊢ (((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ Fin) | |
7 | elmapg 8878 | . . . . . . 7 ⊢ ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → (𝑓 ∈ (𝑅 ↑m (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅)) | |
8 | 6, 4, 7 | sylancl 586 | . . . . . 6 ⊢ (((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → (𝑓 ∈ (𝑅 ↑m (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅)) |
9 | 8 | biimpa 476 | . . . . 5 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑m (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅) |
10 | simplr 769 | . . . . . . 7 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑m (1...𝑛))) → 𝑛 ∈ ℕ) | |
11 | nnuz 12919 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
12 | 10, 11 | eleqtrdi 2849 | . . . . . 6 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑m (1...𝑛))) → 𝑛 ∈ (ℤ≥‘1)) |
13 | eluzfz1 13568 | . . . . . 6 ⊢ (𝑛 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑛)) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑m (1...𝑛))) → 1 ∈ (1...𝑛)) |
15 | 4, 5, 9, 14 | vdwmc2 17013 | . . . 4 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑m (1...𝑛))) → (𝐾 MonoAP 𝑓 ↔ ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐}))) |
16 | 15 | ralbidva 3174 | . . 3 ⊢ (((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → (∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐}))) |
17 | 16 | rexbidva 3175 | . 2 ⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐}))) |
18 | 3, 17 | mpbid 232 | 1 ⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 Vcvv 3478 {csn 4631 class class class wbr 5148 ◡ccnv 5688 “ cima 5692 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Fincfn 8984 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 ℕcn 12264 ℕ0cn0 12524 ℤ≥cuz 12876 ...cfz 13544 MonoAP cvdwm 17000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-hash 14367 df-vdwap 17002 df-vdwmc 17003 df-vdwpc 17004 |
This theorem is referenced by: vdwnnlem1 17029 |
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