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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 473 | . . 3 ⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Fin) | |
2 | simpr 477 | . . 3 ⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0) | |
3 | 1, 2 | vdwlem13 15697 | . 2 ⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓) |
4 | ovex 6678 | . . . . 5 ⊢ (1...𝑛) ∈ V | |
5 | simpllr 799 | . . . . 5 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))) → 𝐾 ∈ ℕ0) | |
6 | simpll 790 | . . . . . . 7 ⊢ (((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ Fin) | |
7 | elmapg 7870 | . . . . . . 7 ⊢ ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → (𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅)) | |
8 | 6, 4, 7 | sylancl 694 | . . . . . 6 ⊢ (((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → (𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅)) |
9 | 8 | biimpa 501 | . . . . 5 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅) |
10 | simplr 792 | . . . . . . 7 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))) → 𝑛 ∈ ℕ) | |
11 | nnuz 11723 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
12 | 10, 11 | syl6eleq 2711 | . . . . . 6 ⊢ ((((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))) → 𝑛 ∈ (ℤ≥‘1)) |
13 | eluzfz1 12348 | . . . . . 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 15683 | . . . 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 2985 | . . 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 3049 | . 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 222 | 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 196 ∧ wa 384 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 Vcvv 3200 {csn 4177 class class class wbr 4653 ◡ccnv 5113 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Fincfn 7955 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 ℕcn 11020 ℕ0cn0 11292 ℤ≥cuz 11687 ...cfz 12326 MonoAP cvdwm 15670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-hash 13118 df-vdwap 15672 df-vdwmc 15673 df-vdwpc 15674 |
This theorem is referenced by: vdwnnlem1 15699 |
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