erdszelem10 - Mathbox for Mario Carneiro

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Theorem erdszelem10 33794

Description: Lemma for erdsze 33796. (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 13877	. . . . . . . 8 ⊢ (1...(𝑅 − 1)) ∈ Fin
2		fzfi 13877	. . . . . . . 8 ⊢ (1...(𝑆 − 1)) ∈ Fin
3		xpfi 9261	. . . . . . . 8 ⊢ (((1...(𝑅 − 1)) ∈ Fin ∧ (1...(𝑆 − 1)) ∈ Fin) → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin)
4	1, 2, 3	mp2an 690	. . . . . . 7 ⊢ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin
5		ssdomg 8940	. . . . . . 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 9043	. . . . . 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 14334	. . . . . . . . . 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 690	. . . . . . . . 9 ⊢ (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((♯‘(1...(𝑅 − 1))) · (♯‘(1...(𝑆 − 1))))
12		erdszelem.r	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
13		nnm1nn0 12454	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ₀)
14		hashfz1 14246	. . . . . . . . . . 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 12454	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ₀)
18		hashfz1 14246	. . . . . . . . . . 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 7375	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘(1...(𝑅 − 1))) · (♯‘(1...(𝑆 − 1)))) = ((𝑅 − 1) · (𝑆 − 1)))
21	11, 20	eqtrid 2788	. . . . . . . 8 ⊢ (𝜑 → (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((𝑅 − 1) · (𝑆 − 1)))
22		erdsze.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
23	22	nnnn0d 12473	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
24		hashfz1 14246	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
26	9, 21, 25	3brtr4d 5137	. . . . . . 7 ⊢ (𝜑 → (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (♯‘(1...𝑁)))
27		fzfid 13878	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
28		hashsdom 14281	. . . . . . . 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 231	. . . . . 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 33793	. . . . . . 7 ⊢ (𝜑 → 𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
36		f1f1orn 6795	. . . . . . 7 ⊢ (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇:(1...𝑁)–1-1-onto→ran 𝑇)
37		ovex 7390	. . . . . . . 8 ⊢ (1...𝑁) ∈ V
38	37	f1oen 8913	. . . . . . 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 9055	. . . . . 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 4006	. . . . 5 ⊢ (¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠(𝑠 ∈ ran 𝑇 ∧ ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
44		df-rex 3074	. . . . 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 277	. . . 4 ⊢ (¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
46	42, 45	sylib 217	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
47		f1fn 6739	. . . 4 ⊢ (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇 Fn (1...𝑁))
48		eleq1 2825	. . . . . 6 ⊢ (𝑠 = (𝑇‘𝑚) → (𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
49	48	notbid 317	. . . . 5 ⊢ (𝑠 = (𝑇‘𝑚) → (¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ¬ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
50	49	rexrn 7037	. . . 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 231	. 2 ⊢ (𝜑 → ∃𝑚 ∈ (1...𝑁) ¬ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
53		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐼‘𝑛) = (𝐼‘𝑚))
54		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐽‘𝑛) = (𝐽‘𝑚))
55	53, 54	opeq12d 4838	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
56		opex 5421	. . . . . . . . 9 ⊢ ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩ ∈ V
57	55, 34, 56	fvmpt 6948	. . . . . . . 8 ⊢ (𝑚 ∈ (1...𝑁) → (𝑇‘𝑚) = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
58	57	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (𝑇‘𝑚) = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
59	58	eleq1d 2822	. . . . . 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 5669	. . . . . 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 286	. . . . 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 317	. . . 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 980	. . . 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 286	. . 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 3173	. 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 231	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 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∃wrex 3073 {crab 3407 ⊆ wss 3910 𝒫 cpw 4560 ⟨cop 4592 class class class wbr 5105 ↦ cmpt 5188 × cxp 5631 ^◡ccnv 5632 ran crn 5634 ↾ cres 5635 “ cima 5636 Fn wfn 6491 –1-1→wf1 6493 –1-1-onto→wf1o 6495 ‘cfv 6496 Isom wiso 6497 (class class class)co 7357 ≈ cen 8880 ≼ cdom 8881 ≺ csdm 8882 Fincfn 8883 supcsup 9376 ℝcr 11050 1c1 11052 · cmul 11056 < clt 11189 − cmin 11385 ℕcn 12153 ℕ₀cn0 12413 ...cfz 13424 ♯chash 14230

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-fz 13425 df-hash 14231

This theorem is referenced by: erdszelem11 33795

W3C validator