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Theorem vdwnnlem1 15537
 Description: Corollary of vdw 15536, and lemma for vdwnn 15540. If 𝐹 is a coloring of the integers, then there are arbitrarily long monochromatic APs in 𝐹. (Contributed by Mario Carneiro, 13-Sep-2014.)
Assertion
Ref Expression
vdwnnlem1 ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) → ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
Distinct variable groups:   𝑎,𝑑,𝑚,𝑐,𝐾   𝑅,𝑎,𝑐,𝑑   𝐹,𝑎,𝑐,𝑑,𝑚
Allowed substitution hint:   𝑅(𝑚)

Proof of Theorem vdwnnlem1
Dummy variables 𝑓 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdw 15536 . . 3 ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝑓 “ {𝑐}))
213adant2 1073 . 2 ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝑓 “ {𝑐}))
3 simpl2 1058 . . . . . . 7 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶𝑅)
4 fzssuz 12253 . . . . . . . 8 (1...𝑛) ⊆ (ℤ‘1)
5 nnuz 11599 . . . . . . . 8 ℕ = (ℤ‘1)
64, 5sseqtr4i 3601 . . . . . . 7 (1...𝑛) ⊆ ℕ
7 fssres 5983 . . . . . . 7 ((𝐹:ℕ⟶𝑅 ∧ (1...𝑛) ⊆ ℕ) → (𝐹 ↾ (1...𝑛)):(1...𝑛)⟶𝑅)
83, 6, 7sylancl 693 . . . . . 6 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → (𝐹 ↾ (1...𝑛)):(1...𝑛)⟶𝑅)
9 simpl1 1057 . . . . . . 7 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ Fin)
10 ovex 6577 . . . . . . 7 (1...𝑛) ∈ V
11 elmapg 7757 . . . . . . 7 ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → ((𝐹 ↾ (1...𝑛)) ∈ (𝑅𝑚 (1...𝑛)) ↔ (𝐹 ↾ (1...𝑛)):(1...𝑛)⟶𝑅))
129, 10, 11sylancl 693 . . . . . 6 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → ((𝐹 ↾ (1...𝑛)) ∈ (𝑅𝑚 (1...𝑛)) ↔ (𝐹 ↾ (1...𝑛)):(1...𝑛)⟶𝑅))
138, 12mpbird 246 . . . . 5 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → (𝐹 ↾ (1...𝑛)) ∈ (𝑅𝑚 (1...𝑛)))
14 cnveq 5218 . . . . . . . . . . 11 (𝑓 = (𝐹 ↾ (1...𝑛)) → 𝑓 = (𝐹 ↾ (1...𝑛)))
1514imaeq1d 5384 . . . . . . . . . 10 (𝑓 = (𝐹 ↾ (1...𝑛)) → (𝑓 “ {𝑐}) = ((𝐹 ↾ (1...𝑛)) “ {𝑐}))
1615eleq2d 2673 . . . . . . . . 9 (𝑓 = (𝐹 ↾ (1...𝑛)) → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝑓 “ {𝑐}) ↔ (𝑎 + (𝑚 · 𝑑)) ∈ ((𝐹 ↾ (1...𝑛)) “ {𝑐})))
1716ralbidv 2969 . . . . . . . 8 (𝑓 = (𝐹 ↾ (1...𝑛)) → (∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝑓 “ {𝑐}) ↔ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ ((𝐹 ↾ (1...𝑛)) “ {𝑐})))
18172rexbidv 3039 . . . . . . 7 (𝑓 = (𝐹 ↾ (1...𝑛)) → (∃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 ↾ (1...𝑛)) “ {𝑐})))
1918rexbidv 3034 . . . . . 6 (𝑓 = (𝐹 ↾ (1...𝑛)) → (∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝑓 “ {𝑐}) ↔ ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ ((𝐹 ↾ (1...𝑛)) “ {𝑐})))
2019rspcv 3278 . . . . 5 ((𝐹 ↾ (1...𝑛)) ∈ (𝑅𝑚 (1...𝑛)) → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝑓 “ {𝑐}) → ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ ((𝐹 ↾ (1...𝑛)) “ {𝑐})))
2113, 20syl 17 . . . 4 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝑓 “ {𝑐}) → ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ ((𝐹 ↾ (1...𝑛)) “ {𝑐})))
22 resss 5342 . . . . . . . . . 10 (𝐹 ↾ (1...𝑛)) ⊆ 𝐹
23 cnvss 5216 . . . . . . . . . 10 ((𝐹 ↾ (1...𝑛)) ⊆ 𝐹(𝐹 ↾ (1...𝑛)) ⊆ 𝐹)
24 imass1 5419 . . . . . . . . . 10 ((𝐹 ↾ (1...𝑛)) ⊆ 𝐹 → ((𝐹 ↾ (1...𝑛)) “ {𝑐}) ⊆ (𝐹 “ {𝑐}))
2522, 23, 24mp2b 10 . . . . . . . . 9 ((𝐹 ↾ (1...𝑛)) “ {𝑐}) ⊆ (𝐹 “ {𝑐})
2625sseli 3564 . . . . . . . 8 ((𝑎 + (𝑚 · 𝑑)) ∈ ((𝐹 ↾ (1...𝑛)) “ {𝑐}) → (𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
2726ralimi 2936 . . . . . . 7 (∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ ((𝐹 ↾ (1...𝑛)) “ {𝑐}) → ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
2827reximi 2994 . . . . . 6 (∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ ((𝐹 ↾ (1...𝑛)) “ {𝑐}) → ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
2928reximi 2994 . . . . 5 (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ ((𝐹 ↾ (1...𝑛)) “ {𝑐}) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
3029reximi 2994 . . . 4 (∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ ((𝐹 ↾ (1...𝑛)) “ {𝑐}) → ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
3121, 30syl6 34 . . 3 (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝑓 “ {𝑐}) → ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})))
3231rexlimdva 3013 . 2 ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝑓 “ {𝑐}) → ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})))
332, 32mpd 15 1 ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) → ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125  ◡ccnv 5037   ↾ cres 5040   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Fincfn 7841  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ℤ≥cuz 11563  ...cfz 12197 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-hash 12980  df-vdwap 15510  df-vdwmc 15511  df-vdwpc 15512 This theorem is referenced by:  vdwnnlem3  15539
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