erdszelem10 - Mathbox for Mario Carneiro

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Theorem erdszelem10 35700

Description: Lemma for erdsze 35702. (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 14013	. . . . . . . 8 ⊢ (1...(𝑅 − 1)) ∈ Fin
2		fzfi 14013	. . . . . . . 8 ⊢ (1...(𝑆 − 1)) ∈ Fin
3		xpfi 9275	. . . . . . . 8 ⊢ (((1...(𝑅 − 1)) ∈ Fin ∧ (1...(𝑆 − 1)) ∈ Fin) → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin)
4	1, 2, 3	mp2an 704	. . . . . . 7 ⊢ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin
5		ssdomg 8993	. . . . . . 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 9087	. . . . . 6 ⊢ (ran 𝑇 ≼ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) → ¬ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ ran 𝑇)
8	6, 7	syl 18	. . . . 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 14476	. . . . . . . . . 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 704	. . . . . . . . 9 ⊢ (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((♯‘(1...(𝑅 − 1))) · (♯‘(1...(𝑆 − 1))))
12		erdszelem.r	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
13		nnm1nn0 12549	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ₀)
14		hashfz1 14387	. . . . . . . . . . 11 ⊢ ((𝑅 − 1) ∈ ℕ₀ → (♯‘(1...(𝑅 − 1))) = (𝑅 − 1))
15	12, 13, 14	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(1...(𝑅 − 1))) = (𝑅 − 1))
16		erdszelem.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℕ)
17		nnm1nn0 12549	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ₀)
18		hashfz1 14387	. . . . . . . . . . 11 ⊢ ((𝑆 − 1) ∈ ℕ₀ → (♯‘(1...(𝑆 − 1))) = (𝑆 − 1))
19	16, 17, 18	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(1...(𝑆 − 1))) = (𝑆 − 1))
20	15, 19	oveq12d 7428	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘(1...(𝑅 − 1))) · (♯‘(1...(𝑆 − 1)))) = ((𝑅 − 1) · (𝑆 − 1)))
21	11, 20	eqtrid 2810	. . . . . . . 8 ⊢ (𝜑 → (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((𝑅 − 1) · (𝑆 − 1)))
22		erdsze.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
23	22	nnnn0d 12569	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
24		hashfz1 14387	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
25	23, 24	syl 18	. . . . . . . 8 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
26	9, 21, 25	3brtr4d 5143	. . . . . . 7 ⊢ (𝜑 → (♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (♯‘(1...𝑁)))
27		fzfid 14014	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
28		hashsdom 14422	. . . . . . . 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 598	. . . . . . 7 ⊢ (𝜑 → ((♯‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (♯‘(1...𝑁)) ↔ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ (1...𝑁)))
30	26, 29	mpbid 235	. . . . . 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 35699	. . . . . . 7 ⊢ (𝜑 → 𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
36		f1f1orn 6832	. . . . . . 7 ⊢ (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇:(1...𝑁)–1-1-onto→ran 𝑇)
37		ovex 7443	. . . . . . . 8 ⊢ (1...𝑁) ∈ V
38	37	f1oen 8965	. . . . . . 7 ⊢ (𝑇:(1...𝑁)–1-1-onto→ran 𝑇 → (1...𝑁) ≈ ran 𝑇)
39	35, 36, 38	3syl 19	. . . . . 6 ⊢ (𝜑 → (1...𝑁) ≈ ran 𝑇)
40		sdomentr 9095	. . . . . 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 595	. . . . 5 ⊢ (𝜑 → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ ran 𝑇)
42	8, 41	nsyl3 139	. . . 4 ⊢ (𝜑 → ¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
43		nss 4001	. . . . 5 ⊢ (¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠(𝑠 ∈ ran 𝑇 ∧ ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
44		df-rex 3090	. . . . 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 281	. . . 4 ⊢ (¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
46	42, 45	sylib 221	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
47		f1fn 6775	. . . 4 ⊢ (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇 Fn (1...𝑁))
48		eleq1 2851	. . . . . 6 ⊢ (𝑠 = (𝑇‘𝑚) → (𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
49	48	notbid 321	. . . . 5 ⊢ (𝑠 = (𝑇‘𝑚) → (¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ¬ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
50	49	rexrn 7082	. . . 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 19	. . 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 235	. 2 ⊢ (𝜑 → ∃𝑚 ∈ (1...𝑁) ¬ (𝑇‘𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
53		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐼‘𝑛) = (𝐼‘𝑚))
54		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐽‘𝑛) = (𝐽‘𝑚))
55	53, 54	opeq12d 4846	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
56		opex 5445	. . . . . . . . 9 ⊢ ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩ ∈ V
57	55, 34, 56	fvmpt 6989	. . . . . . . 8 ⊢ (𝑚 ∈ (1...𝑁) → (𝑇‘𝑚) = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
58	57	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (𝑇‘𝑚) = ⟨(𝐼‘𝑚), (𝐽‘𝑚)⟩)
59	58	eleq1d 2848	. . . . . 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 5697	. . . . . 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 290	. . . . 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 321	. . . 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 997	. . . 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 290	. . 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 3187	. 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 235	1 ⊢ (𝜑 → ∃𝑚 ∈ (1...𝑁)(¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∃wex 1809 ∈ wcel 2143 ∃wrex 3089 {crab 3416 ⊆ wss 3905 𝒫 cpw 4562 ⟨cop 4595 class class class wbr 5109 ↦ cmpt 5192 × cxp 5659 ^◡ccnv 5660 ran crn 5662 ↾ cres 5663 “ cima 5664 Fn wfn 6531 –1-1→wf1 6533 –1-1-onto→wf1o 6535 ‘cfv 6536 Isom wiso 6537 (class class class)co 7410 ≈ cen 8936 ≼ cdom 8937 ≺ csdm 8938 Fincfn 8939 supcsup 9396 ℝcr 11103 1c1 11105 · cmul 11109 < clt 11247 − cmin 11445 ℕcn 12237 ℕ₀cn0 12508 ...cfz 13539 ♯chash 14371

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-dju 9892 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-n0 12509 df-xnn0 12582 df-z 12596 df-uz 12867 df-fz 13540 df-hash 14372

This theorem is used by: erdszelem11 35701

W3C validator