erdszelem10 - Mathbox for Mario Carneiro

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Theorem erdszelem10 34489

Description: Lemma for erdsze 34491. (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 13941	. . . . . . . 8 ⊢ (1...(𝑅 − 1)) ∈ Fin
2		fzfi 13941	. . . . . . . 8 ⊢ (1...(𝑆 − 1)) ∈ Fin
3		xpfi 9319	. . . . . . . 8 ⊢ (((1...(𝑅 − 1)) ∈ Fin ∧ (1...(𝑆 − 1)) ∈ Fin) → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin)
4	1, 2, 3	mp2an 688	. . . . . . 7 ⊢ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin
5		ssdomg 8998	. . . . . . 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 9101	. . . . . 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 14398	. . . . . . . . . 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 688	. . . . . . . . 9 ⊢ (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((♯‘(1...(𝑅 − 1))) · (♯‘(1...(𝑆 − 1))))
12		erdszelem.r	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
13		nnm1nn0 12517	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ₀)
14		hashfz1 14310	. . . . . . . . . . 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 12517	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ₀)
18		hashfz1 14310	. . . . . . . . . . 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 7429	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘(1...(𝑅 − 1))) · (♯‘(1...(𝑆 − 1)))) = ((𝑅 − 1) · (𝑆 − 1)))
21	11, 20	eqtrid 2782	. . . . . . . 8 ⊢ (𝜑 → (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((𝑅 − 1) · (𝑆 − 1)))
22		erdsze.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
23	22	nnnn0d 12536	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
24		hashfz1 14310	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
26	9, 21, 25	3brtr4d 5179	. . . . . . 7 ⊢ (𝜑 → (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (♯‘(1...𝑁)))
27		fzfid 13942	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
28		hashsdom 14345	. . . . . . . 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 585	. . . . . . 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 34488	. . . . . . 7 ⊢ (𝜑 → 𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
36		f1f1orn 6843	. . . . . . 7 ⊢ (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇:(1...𝑁)–1-1-onto→ran 𝑇)
37		ovex 7444	. . . . . . . 8 ⊢ (1...𝑁) ∈ V
38	37	f1oen 8971	. . . . . . 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 9113	. . . . . 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 582	. . . . 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 4045	. . . . 5 ⊢ (¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠(𝑠 ∈ ran 𝑇 ∧ ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
44		df-rex 3069	. . . . 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 6787	. . . 4 ⊢ (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇 Fn (1...𝑁))
48		eleq1 2819	. . . . . 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 7087	. . . 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 6890	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐼‘𝑛) = (𝐼‘𝑚))
54		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐽‘𝑛) = (𝐽‘𝑚))
55	53, 54	opeq12d 4880	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
56		opex 5463	. . . . . . . . 9 ⊢ ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩ ∈ V
57	55, 34, 56	fvmpt 6997	. . . . . . . 8 ⊢ (𝑚 ∈ (1...𝑁) → (𝑇‘𝑚) = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
58	57	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (𝑇‘𝑚) = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
59	58	eleq1d 2816	. . . . . 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 5711	. . . . . 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 978	. . . 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 3174	. 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 394 ∨ wo 843 = wceq 1539 ∃wex 1779 ∈ wcel 2104 ∃wrex 3068 {crab 3430 ⊆ wss 3947 𝒫 cpw 4601 ⟨cop 4633 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ^◡ccnv 5674 ran crn 5676 ↾ cres 5677 “ cima 5678 Fn wfn 6537 –1-1→wf1 6539 –1-1-onto→wf1o 6541 ‘cfv 6542 Isom wiso 6543 (class class class)co 7411 ≈ cen 8938 ≼ cdom 8939 ≺ csdm 8940 Fincfn 8941 supcsup 9437 ℝcr 11111 1c1 11113 · cmul 11117 < clt 11252 − cmin 11448 ℕcn 12216 ℕ₀cn0 12476 ...cfz 13488 ♯chash 14294

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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13489 df-hash 14295

This theorem is referenced by: erdszelem11 34490

W3C validator