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Theorem ramz2 15775
 Description: The Ramsey number when 𝐹 has value zero for some color 𝐶. (Contributed by Mario Carneiro, 22-Apr-2015.)
Assertion
Ref Expression
ramz2 (((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) → (𝑀 Ramsey 𝐹) = 0)

Proof of Theorem ramz2
Dummy variables 𝑏 𝑓 𝑐 𝑠 𝑥 𝑎 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . 3 (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖}) = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
2 simpl1 1084 . . . 4 (((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) → 𝑀 ∈ ℕ)
32nnnn0d 11389 . . 3 (((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) → 𝑀 ∈ ℕ0)
4 simpl2 1085 . . 3 (((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) → 𝑅𝑉)
5 simpl3 1086 . . 3 (((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) → 𝐹:𝑅⟶ℕ0)
6 0nn0 11345 . . . 4 0 ∈ ℕ0
76a1i 11 . . 3 (((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) → 0 ∈ ℕ0)
8 simplrl 817 . . . 4 ((((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) ∧ (0 ≤ (#‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶𝑅)) → 𝐶𝑅)
9 0elpw 4864 . . . . 5 ∅ ∈ 𝒫 𝑠
109a1i 11 . . . 4 ((((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) ∧ (0 ≤ (#‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶𝑅)) → ∅ ∈ 𝒫 𝑠)
11 simplrr 818 . . . . 5 ((((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) ∧ (0 ≤ (#‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶𝑅)) → (𝐹𝐶) = 0)
12 0le0 11148 . . . . 5 0 ≤ 0
1311, 12syl6eqbr 4724 . . . 4 ((((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) ∧ (0 ≤ (#‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶𝑅)) → (𝐹𝐶) ≤ 0)
14 simpll1 1120 . . . . . 6 ((((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) ∧ (0 ≤ (#‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶𝑅)) → 𝑀 ∈ ℕ)
1510hashbc 15758 . . . . . 6 (𝑀 ∈ ℕ → (∅(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) = ∅)
1614, 15syl 17 . . . . 5 ((((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) ∧ (0 ≤ (#‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶𝑅)) → (∅(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) = ∅)
17 0ss 4005 . . . . 5 ∅ ⊆ (𝑓 “ {𝐶})
1816, 17syl6eqss 3688 . . . 4 ((((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) ∧ (0 ≤ (#‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶𝑅)) → (∅(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝐶}))
19 fveq2 6229 . . . . . . 7 (𝑐 = 𝐶 → (𝐹𝑐) = (𝐹𝐶))
2019breq1d 4695 . . . . . 6 (𝑐 = 𝐶 → ((𝐹𝑐) ≤ (#‘𝑥) ↔ (𝐹𝐶) ≤ (#‘𝑥)))
21 sneq 4220 . . . . . . . 8 (𝑐 = 𝐶 → {𝑐} = {𝐶})
2221imaeq2d 5501 . . . . . . 7 (𝑐 = 𝐶 → (𝑓 “ {𝑐}) = (𝑓 “ {𝐶}))
2322sseq2d 3666 . . . . . 6 (𝑐 = 𝐶 → ((𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐}) ↔ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝐶})))
2420, 23anbi12d 747 . . . . 5 (𝑐 = 𝐶 → (((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})) ↔ ((𝐹𝐶) ≤ (#‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝐶}))))
25 fveq2 6229 . . . . . . . 8 (𝑥 = ∅ → (#‘𝑥) = (#‘∅))
26 hash0 13196 . . . . . . . 8 (#‘∅) = 0
2725, 26syl6eq 2701 . . . . . . 7 (𝑥 = ∅ → (#‘𝑥) = 0)
2827breq2d 4697 . . . . . 6 (𝑥 = ∅ → ((𝐹𝐶) ≤ (#‘𝑥) ↔ (𝐹𝐶) ≤ 0))
29 oveq1 6697 . . . . . . 7 (𝑥 = ∅ → (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) = (∅(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀))
3029sseq1d 3665 . . . . . 6 (𝑥 = ∅ → ((𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝐶}) ↔ (∅(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝐶})))
3128, 30anbi12d 747 . . . . 5 (𝑥 = ∅ → (((𝐹𝐶) ≤ (#‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝐶})) ↔ ((𝐹𝐶) ≤ 0 ∧ (∅(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝐶}))))
3224, 31rspc2ev 3355 . . . 4 ((𝐶𝑅 ∧ ∅ ∈ 𝒫 𝑠 ∧ ((𝐹𝐶) ≤ 0 ∧ (∅(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝐶}))) → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))
338, 10, 13, 18, 32syl112anc 1370 . . 3 ((((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) ∧ (0 ≤ (#‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶𝑅)) → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))
341, 3, 4, 5, 7, 33ramub 15764 . 2 (((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) → (𝑀 Ramsey 𝐹) ≤ 0)
35 ramubcl 15769 . . . 4 (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (0 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 0)) → (𝑀 Ramsey 𝐹) ∈ ℕ0)
363, 4, 5, 7, 34, 35syl32anc 1374 . . 3 (((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) → (𝑀 Ramsey 𝐹) ∈ ℕ0)
37 nn0le0eq0 11359 . . 3 ((𝑀 Ramsey 𝐹) ∈ ℕ0 → ((𝑀 Ramsey 𝐹) ≤ 0 ↔ (𝑀 Ramsey 𝐹) = 0))
3836, 37syl 17 . 2 (((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) → ((𝑀 Ramsey 𝐹) ≤ 0 ↔ (𝑀 Ramsey 𝐹) = 0))
3934, 38mpbid 222 1 (((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) → (𝑀 Ramsey 𝐹) = 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∃wrex 2942  {crab 2945  Vcvv 3231   ⊆ wss 3607  ∅c0 3948  𝒫 cpw 4191  {csn 4210   class class class wbr 4685  ◡ccnv 5142   “ cima 5146  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  0cc0 9974   ≤ cle 10113  ℕcn 11058  ℕ0cn0 11330  #chash 13157   Ramsey cram 15750 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-seq 12842  df-fac 13101  df-bc 13130  df-hash 13158  df-ram 15752 This theorem is referenced by:  ramz  15776  ramcl  15780
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