Theorem ramlb 15561

Description: Establish a lower bound on a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)

Hypotheses

Ref	Expression
ramlb.c	⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
ramlb.m	⊢ (𝜑 → 𝑀 ∈ ℕ0)
ramlb.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)
ramlb.f	⊢ (𝜑 → 𝐹:𝑅⟶ℕ0)
ramlb.s	⊢ (𝜑 → 𝑁 ∈ ℕ0)
ramlb.g	⊢ (𝜑 → 𝐺:((1...𝑁)𝐶𝑀)⟶𝑅)
ramlb.i	⊢ ((𝜑 ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → ((𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐}) → (#‘𝑥) < (𝐹‘𝑐)))

Assertion

Ref	Expression
ramlb	⊢ (𝜑 → 𝑁 < (𝑀 Ramsey 𝐹))

Distinct variable groups:   𝑥,𝑐,𝐶   𝐹,𝑐,𝑥   𝐺,𝑐,𝑥   𝑎,𝑏,𝑐,𝑖,𝑥,𝑀   𝜑,𝑐,𝑥   𝑁,𝑐,𝑥   𝑅,𝑐,𝑥   𝑉,𝑐,𝑥

Allowed substitution hints:   𝜑(𝑖,𝑎,𝑏)   𝐶(𝑖,𝑎,𝑏)   𝑅(𝑖,𝑎,𝑏)   𝐹(𝑖,𝑎,𝑏)   𝐺(𝑖,𝑎,𝑏)   𝑁(𝑖,𝑎,𝑏)   𝑉(𝑖,𝑎,𝑏)

Proof of Theorem ramlb

Step	Hyp	Ref	Expression
1		ramlb.c	. . . . 5 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
2		ramlb.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
3	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → 𝑀 ∈ ℕ0)
4		ramlb.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → 𝑅 ∈ 𝑉)
6		ramlb.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑅⟶ℕ0)
7	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → 𝐹:𝑅⟶ℕ0)
8		ramlb.s	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
9	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → 𝑁 ∈ ℕ0)
10		simpr 476	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → (𝑀 Ramsey 𝐹) ≤ 𝑁)
11		ramubcl 15560	. . . . . 6 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑁 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁)) → (𝑀 Ramsey 𝐹) ∈ ℕ0)
12	3, 5, 7, 9, 10, 11	syl32anc 1326	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → (𝑀 Ramsey 𝐹) ∈ ℕ0)
13		fzfid 12634	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → (1...𝑁) ∈ Fin)
14		hashfz1 12996	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)
15	8, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (#‘(1...𝑁)) = 𝑁)
16	15	breq2d 4595	. . . . . 6 ⊢ (𝜑 → ((𝑀 Ramsey 𝐹) ≤ (#‘(1...𝑁)) ↔ (𝑀 Ramsey 𝐹) ≤ 𝑁))
17	16	biimpar 501	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → (𝑀 Ramsey 𝐹) ≤ (#‘(1...𝑁)))
18		ramlb.g	. . . . . 6 ⊢ (𝜑 → 𝐺:((1...𝑁)𝐶𝑀)⟶𝑅)
19	18	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → 𝐺:((1...𝑁)𝐶𝑀)⟶𝑅)
20	1, 3, 5, 7, 12, 13, 17, 19	rami 15557	. . . 4 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 (1...𝑁)((𝐹‘𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})))
21		elpwi 4117	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (1...𝑁) → 𝑥 ⊆ (1...𝑁))
22		ramlb.i	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → ((𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐}) → (#‘𝑥) < (𝐹‘𝑐)))
23	22	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → ((𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐}) → (#‘𝑥) < (𝐹‘𝑐)))
24		fzfid 12634	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → (1...𝑁) ∈ Fin)
25		simprr 792	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → 𝑥 ⊆ (1...𝑁))
26		ssfi 8065	. . . . . . . . . . . . . 14 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑥 ⊆ (1...𝑁)) → 𝑥 ∈ Fin)
27	24, 25, 26	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → 𝑥 ∈ Fin)
28		hashcl 13009	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Fin → (#‘𝑥) ∈ ℕ0)
29	27, 28	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → (#‘𝑥) ∈ ℕ0)
30	29	nn0red 11229	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → (#‘𝑥) ∈ ℝ)
31		simpl 472	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁)) → 𝑐 ∈ 𝑅)
32		ffvelrn 6265	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑅⟶ℕ0 ∧ 𝑐 ∈ 𝑅) → (𝐹‘𝑐) ∈ ℕ0)
33	7, 31, 32	syl2an 493	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → (𝐹‘𝑐) ∈ ℕ0)
34	33	nn0red 11229	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → (𝐹‘𝑐) ∈ ℝ)
35	30, 34	ltnled 10063	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → ((#‘𝑥) < (𝐹‘𝑐) ↔ ¬ (𝐹‘𝑐) ≤ (#‘𝑥)))
36	23, 35	sylibd 228	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → ((𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐}) → ¬ (𝐹‘𝑐) ≤ (#‘𝑥)))
37	21, 36	sylanr2 683	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ∈ 𝒫 (1...𝑁))) → ((𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐}) → ¬ (𝐹‘𝑐) ≤ (#‘𝑥)))
38	37	con2d 128	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ∈ 𝒫 (1...𝑁))) → ((𝐹‘𝑐) ≤ (#‘𝑥) → ¬ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})))
39		imnan 437	. . . . . . 7 ⊢ (((𝐹‘𝑐) ≤ (#‘𝑥) → ¬ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})) ↔ ¬ ((𝐹‘𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})))
40	38, 39	sylib 207	. . . . . 6 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ∈ 𝒫 (1...𝑁))) → ¬ ((𝐹‘𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})))
41	40	pm2.21d 117	. . . . 5 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ∈ 𝒫 (1...𝑁))) → (((𝐹‘𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})) → ¬ (𝑀 Ramsey 𝐹) ≤ 𝑁))
42	41	rexlimdvva 3020	. . . 4 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → (∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 (1...𝑁)((𝐹‘𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})) → ¬ (𝑀 Ramsey 𝐹) ≤ 𝑁))
43	20, 42	mpd 15	. . 3 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → ¬ (𝑀 Ramsey 𝐹) ≤ 𝑁)
44	43	pm2.01da 457	. 2 ⊢ (𝜑 → ¬ (𝑀 Ramsey 𝐹) ≤ 𝑁)
45	8	nn0red 11229	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
46	45	rexrd 9968	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ*)
47		ramxrcl 15559	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℝ*)
48	2, 4, 6, 47	syl3anc 1318	. . 3 ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ∈ ℝ*)
49		xrltnle 9984	. . 3 ⊢ ((𝑁 ∈ ℝ* ∧ (𝑀 Ramsey 𝐹) ∈ ℝ*) → (𝑁 < (𝑀 Ramsey 𝐹) ↔ ¬ (𝑀 Ramsey 𝐹) ≤ 𝑁))
50	46, 48, 49	syl2anc 691	. 2 ⊢ (𝜑 → (𝑁 < (𝑀 Ramsey 𝐹) ↔ ¬ (𝑀 Ramsey 𝐹) ≤ 𝑁))
51	44, 50	mpbird 246	1 ⊢ (𝜑 → 𝑁 < (𝑀 Ramsey 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  ^◡ccnv 5037   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  1c1 9816  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954  ℕ₀cn0 11169  ...cfz 12197  #chash 12979   Ramsey cram 15541

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-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-ram 15543

This theorem is referenced by:  0ram  15562  ram0  15564