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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 475 | . . 3 ⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Fin) | |
2 | simpr 478 | . . 3 ⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0) | |
3 | 1, 2 | vdwlem13 16027 | . 2 ⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓) |
4 | ovex 6909 | . . . . 5 ⊢ (1...𝑛) ∈ V | |
5 | simpllr 794 | . . . . 5 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))) → 𝐾 ∈ ℕ0) | |
6 | simpll 784 | . . . . . . 7 ⊢ (((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ Fin) | |
7 | elmapg 8107 | . . . . . . 7 ⊢ ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → (𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅)) | |
8 | 6, 4, 7 | sylancl 581 | . . . . . 6 ⊢ (((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → (𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅)) |
9 | 8 | biimpa 469 | . . . . 5 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅) |
10 | simplr 786 | . . . . . . 7 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))) → 𝑛 ∈ ℕ) | |
11 | nnuz 11964 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
12 | 10, 11 | syl6eleq 2887 | . . . . . 6 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))) → 𝑛 ∈ (ℤ≥‘1)) |
13 | eluzfz1 12599 | . . . . . 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 16013 | . . . 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 3165 | . . 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 3229 | . 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 224 | 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 198 ∧ wa 385 ∈ wcel 2157 ∀wral 3088 ∃wrex 3089 Vcvv 3384 {csn 4367 class class class wbr 4842 ◡ccnv 5310 “ cima 5314 ⟶wf 6096 ‘cfv 6100 (class class class)co 6877 ↑𝑚 cmap 8094 Fincfn 8194 0cc0 10223 1c1 10224 + caddc 10226 · cmul 10228 − cmin 10555 ℕcn 11311 ℕ0cn0 11577 ℤ≥cuz 11927 ...cfz 12577 MonoAP cvdwm 16000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-1st 7400 df-2nd 7401 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-2o 7799 df-oadd 7802 df-er 7981 df-map 8096 df-pm 8097 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-card 9050 df-cda 9277 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-nn 11312 df-2 11373 df-n0 11578 df-xnn0 11650 df-z 11664 df-uz 11928 df-rp 12072 df-fz 12578 df-hash 13368 df-vdwap 16002 df-vdwmc 16003 df-vdwpc 16004 |
This theorem is referenced by: vdwnnlem1 16029 |
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