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Theorem ramlb 16958

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 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → 𝑀 ∈ ℕ₀)
4		ramlb.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
5	4	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → 𝑅 ∈ 𝑉)
6		ramlb.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑅⟶ℕ₀)
7	6	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → 𝐹:𝑅⟶ℕ₀)
8		ramlb.s	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
9	8	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → 𝑁 ∈ ℕ₀)
10		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → (𝑀 Ramsey 𝐹) ≤ 𝑁)
11		ramubcl 16957	. . . . . 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 1376	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → (𝑀 Ramsey 𝐹) ∈ ℕ₀)
13		fzfid 13944	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → (1...𝑁) ∈ Fin)
14		hashfz1 14312	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
15	8, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
16	15	breq2d 5161	. . . . . 6 ⊢ (𝜑 → ((𝑀 Ramsey 𝐹) ≤ (♯‘(1...𝑁)) ↔ (𝑀 Ramsey 𝐹) ≤ 𝑁))
17	16	biimpar 476	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → (𝑀 Ramsey 𝐹) ≤ (♯‘(1...𝑁)))
18		ramlb.g	. . . . . 6 ⊢ (𝜑 → 𝐺:((1...𝑁)𝐶𝑀)⟶𝑅)
19	18	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → 𝐺:((1...𝑁)𝐶𝑀)⟶𝑅)
20	1, 3, 5, 7, 12, 13, 17, 19	rami 16954	. . . 4 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 (1...𝑁)((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝐺 “ {𝑐})))
21		elpwi 4610	. . . . . . . . 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 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → ((𝑥𝐶𝑀) ⊆ (^◡𝐺 “ {𝑐}) → (♯‘𝑥) < (𝐹‘𝑐)))
24		fzfid 13944	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → (1...𝑁) ∈ Fin)
25		simprr 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → 𝑥 ⊆ (1...𝑁))
26	24, 25	ssfid 9271	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → 𝑥 ∈ Fin)
27		hashcl 14322	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
28	26, 27	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → (♯‘𝑥) ∈ ℕ₀)
29	28	nn0red 12539	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → (♯‘𝑥) ∈ ℝ)
30		simpl 481	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁)) → 𝑐 ∈ 𝑅)
31		ffvelcdm 7084	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑅⟶ℕ₀ ∧ 𝑐 ∈ 𝑅) → (𝐹‘𝑐) ∈ ℕ₀)
32	7, 30, 31	syl2an 594	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → (𝐹‘𝑐) ∈ ℕ₀)
33	32	nn0red 12539	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → (𝐹‘𝑐) ∈ ℝ)
34	29, 33	ltnled 11367	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → ((♯‘𝑥) < (𝐹‘𝑐) ↔ ¬ (𝐹‘𝑐) ≤ (♯‘𝑥)))
35	23, 34	sylibd 238	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → ((𝑥𝐶𝑀) ⊆ (^◡𝐺 “ {𝑐}) → ¬ (𝐹‘𝑐) ≤ (♯‘𝑥)))
36	21, 35	sylanr2 679	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ∈ 𝒫 (1...𝑁))) → ((𝑥𝐶𝑀) ⊆ (^◡𝐺 “ {𝑐}) → ¬ (𝐹‘𝑐) ≤ (♯‘𝑥)))
37	36	con2d 134	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ∈ 𝒫 (1...𝑁))) → ((𝐹‘𝑐) ≤ (♯‘𝑥) → ¬ (𝑥𝐶𝑀) ⊆ (^◡𝐺 “ {𝑐})))
38		imnan 398	. . . . . . 7 ⊢ (((𝐹‘𝑐) ≤ (♯‘𝑥) → ¬ (𝑥𝐶𝑀) ⊆ (^◡𝐺 “ {𝑐})) ↔ ¬ ((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝐺 “ {𝑐})))
39	37, 38	sylib 217	. . . . . 6 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ∈ 𝒫 (1...𝑁))) → ¬ ((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝐺 “ {𝑐})))
40	39	pm2.21d 121	. . . . 5 ⊢ (((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ∈ 𝒫 (1...𝑁))) → (((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝐺 “ {𝑐})) → ¬ (𝑀 Ramsey 𝐹) ≤ 𝑁))
41	40	rexlimdvva 3209	. . . 4 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → (∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 (1...𝑁)((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝐺 “ {𝑐})) → ¬ (𝑀 Ramsey 𝐹) ≤ 𝑁))
42	20, 41	mpd 15	. . 3 ⊢ ((𝜑 ∧ (𝑀 Ramsey 𝐹) ≤ 𝑁) → ¬ (𝑀 Ramsey 𝐹) ≤ 𝑁)
43	42	pm2.01da 795	. 2 ⊢ (𝜑 → ¬ (𝑀 Ramsey 𝐹) ≤ 𝑁)
44	8	nn0red 12539	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
45	44	rexrd 11270	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ^*)
46		ramxrcl 16956	. . . 4 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) → (𝑀 Ramsey 𝐹) ∈ ℝ^*)
47	2, 4, 6, 46	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ∈ ℝ^*)
48		xrltnle 11287	. . 3 ⊢ ((𝑁 ∈ ℝ^* ∧ (𝑀 Ramsey 𝐹) ∈ ℝ^*) → (𝑁 < (𝑀 Ramsey 𝐹) ↔ ¬ (𝑀 Ramsey 𝐹) ≤ 𝑁))
49	45, 47, 48	syl2anc 582	. 2 ⊢ (𝜑 → (𝑁 < (𝑀 Ramsey 𝐹) ↔ ¬ (𝑀 Ramsey 𝐹) ≤ 𝑁))
50	43, 49	mpbird 256	1 ⊢ (𝜑 → 𝑁 < (𝑀 Ramsey 𝐹))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∃wrex 3068 {crab 3430 Vcvv 3472 ⊆ wss 3949 𝒫 cpw 4603 {csn 4629 class class class wbr 5149 ^◡ccnv 5676 “ cima 5680 ⟶wf 6540 ‘cfv 6544 (class class class)co 7413 ∈ cmpo 7415 Fincfn 8943 1c1 11115 ℝ^*cxr 11253 < clt 11254 ≤ cle 11255 ℕ₀cn0 12478 ...cfz 13490 ♯chash 14296 Ramsey cram 16938

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-n0 12479 df-z 12565 df-uz 12829 df-fz 13491 df-hash 14297 df-ram 16940

This theorem is referenced by: 0ram 16959 ram0 16961

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