erdszelem10 - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > erdszelem10		Structured version Visualization version GIF version

Theorem erdszelem10 35183

Description: Lemma for erdsze 35185. (Contributed by Mario Carneiro, 22-Jan-2015.)

**Hypotheses**
Ref	Expression
erdsze.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
erdsze.f	⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)
erdszelem.i	⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
erdszelem.j	⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ^◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
erdszelem.t	⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩)
erdszelem.r	⊢ (𝜑 → 𝑅 ∈ ℕ)
erdszelem.s	⊢ (𝜑 → 𝑆 ∈ ℕ)
erdszelem.m	⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)

**Assertion**
Ref	Expression
erdszelem10	⊢ (𝜑 → ∃𝑚 ∈ (1...𝑁)(¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))))

Distinct variable groups: 𝑥,𝑦 𝑚,𝑛,𝑥,𝑦,𝐹 𝑛,𝐼,𝑥,𝑦 𝑛,𝐽,𝑥,𝑦 𝑅,𝑚,𝑥,𝑦 𝑚,𝑁,𝑛,𝑥,𝑦 𝜑,𝑚,𝑛,𝑥,𝑦 𝑆,𝑚,𝑥,𝑦 𝑇,𝑚 Allowed substitution hints: 𝑅(𝑛) 𝑆(𝑛) 𝑇(𝑥,𝑦,𝑛) 𝐼(𝑚) 𝐽(𝑚)

Proof of Theorem erdszelem10 Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfi 13879	. . . . . . . 8 ⊢ (1...(𝑅 − 1)) ∈ Fin
2		fzfi 13879	. . . . . . . 8 ⊢ (1...(𝑆 − 1)) ∈ Fin
3		xpfi 9209	. . . . . . . 8 ⊢ (((1...(𝑅 − 1)) ∈ Fin ∧ (1...(𝑆 − 1)) ∈ Fin) → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin)
4	1, 2, 3	mp2an 692	. . . . . . 7 ⊢ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin
5		ssdomg 8925	. . . . . . 7 ⊢ (((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin → (ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) → ran 𝑇 ≼ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
6	4, 5	ax-mp 5	. . . . . 6 ⊢ (ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) → ran 𝑇 ≼ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
7		domnsym 9020	. . . . . 6 ⊢ (ran 𝑇 ≼ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) → ¬ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ ran 𝑇)
8	6, 7	syl 17	. . . . 5 ⊢ (ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) → ¬ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ ran 𝑇)
9		erdszelem.m	. . . . . . . 8 ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)
10		hashxp 14341	. . . . . . . . . 10 ⊢ (((1...(𝑅 − 1)) ∈ Fin ∧ (1...(𝑆 − 1)) ∈ Fin) → (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((♯‘(1...(𝑅 − 1))) · (♯‘(1...(𝑆 − 1)))))
11	1, 2, 10	mp2an 692	. . . . . . . . 9 ⊢ (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((♯‘(1...(𝑅 − 1))) · (♯‘(1...(𝑆 − 1))))
12		erdszelem.r	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
13		nnm1nn0 12425	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ₀)
14		hashfz1 14253	. . . . . . . . . . 11 ⊢ ((𝑅 − 1) ∈ ℕ₀ → (♯‘(1...(𝑅 − 1))) = (𝑅 − 1))
15	12, 13, 14	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(1...(𝑅 − 1))) = (𝑅 − 1))
16		erdszelem.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℕ)
17		nnm1nn0 12425	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ₀)
18		hashfz1 14253	. . . . . . . . . . 11 ⊢ ((𝑆 − 1) ∈ ℕ₀ → (♯‘(1...(𝑆 − 1))) = (𝑆 − 1))
19	16, 17, 18	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(1...(𝑆 − 1))) = (𝑆 − 1))
20	15, 19	oveq12d 7367	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘(1...(𝑅 − 1))) · (♯‘(1...(𝑆 − 1)))) = ((𝑅 − 1) · (𝑆 − 1)))
21	11, 20	eqtrid 2776	. . . . . . . 8 ⊢ (𝜑 → (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((𝑅 − 1) · (𝑆 − 1)))
22		erdsze.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
23	22	nnnn0d 12445	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
24		hashfz1 14253	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
26	9, 21, 25	3brtr4d 5124	. . . . . . 7 ⊢ (𝜑 → (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (♯‘(1...𝑁)))
27		fzfid 13880	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
28		hashsdom 14288	. . . . . . . 8 ⊢ ((((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (♯‘(1...𝑁)) ↔ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ (1...𝑁)))
29	4, 27, 28	sylancr 587	. . . . . . 7 ⊢ (𝜑 → ((♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (♯‘(1...𝑁)) ↔ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ (1...𝑁)))
30	26, 29	mpbid 232	. . . . . 6 ⊢ (𝜑 → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ (1...𝑁))
31		erdsze.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)
32		erdszelem.i	. . . . . . . 8 ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
33		erdszelem.j	. . . . . . . 8 ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ^◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
34		erdszelem.t	. . . . . . . 8 ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩)
35	22, 31, 32, 33, 34	erdszelem9 35182	. . . . . . 7 ⊢ (𝜑 → 𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
36		f1f1orn 6775	. . . . . . 7 ⊢ (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇:(1...𝑁)–1-1-onto→ran 𝑇)
37		ovex 7382	. . . . . . . 8 ⊢ (1...𝑁) ∈ V
38	37	f1oen 8898	. . . . . . 7 ⊢ (𝑇:(1...𝑁)–1-1-onto→ran 𝑇 → (1...𝑁) ≈ ran 𝑇)
39	35, 36, 38	3syl 18	. . . . . 6 ⊢ (𝜑 → (1...𝑁) ≈ ran 𝑇)
40		sdomentr 9028	. . . . . 6 ⊢ ((((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ (1...𝑁) ∧ (1...𝑁) ≈ ran 𝑇) → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ ran 𝑇)
41	30, 39, 40	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ ran 𝑇)
42	8, 41	nsyl3 138	. . . 4 ⊢ (𝜑 → ¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
43		nss 4000	. . . . 5 ⊢ (¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠(𝑠 ∈ ran 𝑇 ∧ ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
44		df-rex 3054	. . . . 5 ⊢ (∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠(𝑠 ∈ ran 𝑇 ∧ ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
45	43, 44	bitr4i 278	. . . 4 ⊢ (¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
46	42, 45	sylib 218	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
47		f1fn 6721	. . . 4 ⊢ (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇 Fn (1...𝑁))
48		eleq1 2816	. . . . . 6 ⊢ (𝑠 = (𝑇‘𝑚) → (𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
49	48	notbid 318	. . . . 5 ⊢ (𝑠 = (𝑇‘𝑚) → (¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ¬ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
50	49	rexrn 7021	. . . 4 ⊢ (𝑇 Fn (1...𝑁) → (∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑚 ∈ (1...𝑁) ¬ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
51	35, 47, 50	3syl 18	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑚 ∈ (1...𝑁) ¬ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
52	46, 51	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑚 ∈ (1...𝑁) ¬ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
53		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐼‘𝑛) = (𝐼‘𝑚))
54		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐽‘𝑛) = (𝐽‘𝑚))
55	53, 54	opeq12d 4832	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
56		opex 5407	. . . . . . . . 9 ⊢ ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩ ∈ V
57	55, 34, 56	fvmpt 6930	. . . . . . . 8 ⊢ (𝑚 ∈ (1...𝑁) → (𝑇‘𝑚) = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
58	57	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (𝑇‘𝑚) = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
59	58	eleq1d 2813	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → ((𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩ ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
60		opelxp 5655	. . . . . 6 ⊢ (⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩ ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ((𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∧ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))))
61	59, 60	bitrdi 287	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → ((𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ((𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∧ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))))
62	61	notbid 318	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (¬ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ¬ ((𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∧ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))))
63		ianor 983	. . . 4 ⊢ (¬ ((𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∧ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))) ↔ (¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))))
64	62, 63	bitrdi 287	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (¬ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ (¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))))
65	64	rexbidva 3151	. 2 ⊢ (𝜑 → (∃𝑚 ∈ (1...𝑁) ¬ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑚 ∈ (1...𝑁)(¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))))
66	52, 65	mpbid 232	1 ⊢ (𝜑 → ∃𝑚 ∈ (1...𝑁)(¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ∃wrex 3053 {crab 3394 ⊆ wss 3903 𝒫 cpw 4551 ⟨cop 4583 class class class wbr 5092 ↦ cmpt 5173 × cxp 5617 ^◡ccnv 5618 ran crn 5620 ↾ cres 5621 “ cima 5622 Fn wfn 6477 –1-1→wf1 6479 –1-1-onto→wf1o 6481 ‘cfv 6482 Isom wiso 6483 (class class class)co 7349 ≈ cen 8869 ≼ cdom 8870 ≺ csdm 8871 Fincfn 8872 supcsup 9330 ℝcr 11008 1c1 11010 · cmul 11014 < clt 11149 − cmin 11347 ℕcn 12128 ℕ₀cn0 12384 ...cfz 13410 ♯chash 14237

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-oadd 8392 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-dju 9797 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-n0 12385 df-xnn0 12458 df-z 12472 df-uz 12736 df-fz 13411 df-hash 14238

This theorem is referenced by: erdszelem11 35184

W3C validator