erdszelem10 - Mathbox for Mario Carneiro

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Theorem erdszelem10 35187

Description: Lemma for erdsze 35189. (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 13937	. . . . . . . 8 ⊢ (1...(𝑅 − 1)) ∈ Fin
2		fzfi 13937	. . . . . . . 8 ⊢ (1...(𝑆 − 1)) ∈ Fin
3		xpfi 9269	. . . . . . . 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 8971	. . . . . . 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 9067	. . . . . 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 14399	. . . . . . . . . 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 12483	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ₀)
14		hashfz1 14311	. . . . . . . . . . 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 12483	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ₀)
18		hashfz1 14311	. . . . . . . . . . 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 7405	. . . . . . . . 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 12503	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
24		hashfz1 14311	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
26	9, 21, 25	3brtr4d 5139	. . . . . . 7 ⊢ (𝜑 → (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (♯‘(1...𝑁)))
27		fzfid 13938	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
28		hashsdom 14346	. . . . . . . 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 35186	. . . . . . 7 ⊢ (𝜑 → 𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
36		f1f1orn 6811	. . . . . . 7 ⊢ (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇:(1...𝑁)–1-1-onto→ran 𝑇)
37		ovex 7420	. . . . . . . 8 ⊢ (1...𝑁) ∈ V
38	37	f1oen 8944	. . . . . . 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 9075	. . . . . 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 4011	. . . . 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 6757	. . . 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 7059	. . . 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 6858	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐼‘𝑛) = (𝐼‘𝑚))
54		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐽‘𝑛) = (𝐽‘𝑚))
55	53, 54	opeq12d 4845	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
56		opex 5424	. . . . . . . . 9 ⊢ ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩ ∈ V
57	55, 34, 56	fvmpt 6968	. . . . . . . 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 5674	. . . . . 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 3155	. 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 3405 ⊆ wss 3914 𝒫 cpw 4563 ⟨cop 4595 class class class wbr 5107 ↦ cmpt 5188 × cxp 5636 ^◡ccnv 5637 ran crn 5639 ↾ cres 5640 “ cima 5641 Fn wfn 6506 –1-1→wf1 6508 –1-1-onto→wf1o 6510 ‘cfv 6511 Isom wiso 6512 (class class class)co 7387 ≈ cen 8915 ≼ cdom 8916 ≺ csdm 8917 Fincfn 8918 supcsup 9391 ℝcr 11067 1c1 11069 · cmul 11073 < clt 11208 − cmin 11405 ℕcn 12186 ℕ₀cn0 12442 ...cfz 13468 ♯chash 14295

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-oadd 8438 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-fz 13469 df-hash 14296

This theorem is referenced by: erdszelem11 35188

W3C validator