erdszelem7 - Mathbox for Mario Carneiro

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Theorem erdszelem7 33619

Description: Lemma for erdsze 33624. (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.k	⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
erdszelem.o	⊢ 𝑂 Or ℝ
erdszelem.a	⊢ (𝜑 → 𝐴 ∈ (1...𝑁))
erdszelem7.r	⊢ (𝜑 → 𝑅 ∈ ℕ)
erdszelem7.m	⊢ (𝜑 → ¬ (𝐾‘𝐴) ∈ (1...(𝑅 − 1)))

**Assertion**
Ref	Expression
erdszelem7	⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠))))

Distinct variable groups: 𝑥,𝑦,𝑠,𝐹 𝐾,𝑠 𝐴,𝑠,𝑥,𝑦 𝑂,𝑠,𝑥,𝑦 𝑅,𝑠,𝑥,𝑦 𝑁,𝑠,𝑥,𝑦 𝜑,𝑠,𝑥,𝑦 Allowed substitution hints: 𝐾(𝑥,𝑦)

**Proof of Theorem erdszelem7**
Step	Hyp	Ref	Expression
1		hashf 14192	. . . 4 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
2		ffun 6668	. . . 4 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → Fun ♯)
3	1, 2	ax-mp 5	. . 3 ⊢ Fun ♯
4		erdszelem.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (1...𝑁))
5		erdsze.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6		erdsze.f	. . . . 5 ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)
7		erdszelem.k	. . . . 5 ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
8		erdszelem.o	. . . . 5 ⊢ 𝑂 Or ℝ
9	5, 6, 7, 8	erdszelem5 33617	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...𝑁)) → (𝐾‘𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
10	4, 9	mpdan 685	. . 3 ⊢ (𝜑 → (𝐾‘𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
11		fvelima 6905	. . 3 ⊢ ((Fun ♯ ∧ (𝐾‘𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)})) → ∃𝑠 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} (♯‘𝑠) = (𝐾‘𝐴))
12	3, 10, 11	sylancr 587	. 2 ⊢ (𝜑 → ∃𝑠 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} (♯‘𝑠) = (𝐾‘𝐴))
13		eqid 2736	. . . . . 6 ⊢ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}
14	13	erdszelem1 33613	. . . . 5 ⊢ (𝑠 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} ↔ (𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠))
15		simprl1 1218	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑠 ⊆ (1...𝐴))
16		elfzuz3 13392	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (1...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐴))
17		fzss2 13435	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝐴) → (1...𝐴) ⊆ (1...𝑁))
18	4, 16, 17	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝐴) ⊆ (1...𝑁))
19	18	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → (1...𝐴) ⊆ (1...𝑁))
20	15, 19	sstrd 3952	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑠 ⊆ (1...𝑁))
21		velpw 4563	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 (1...𝑁) ↔ 𝑠 ⊆ (1...𝑁))
22	20, 21	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑠 ∈ 𝒫 (1...𝑁))
23		erdszelem7.m	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝐾‘𝐴) ∈ (1...(𝑅 − 1)))
24	5, 6, 7, 8	erdszelem6 33618	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶ℕ)
25	24, 4	ffvelcdmd 7032	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘𝐴) ∈ ℕ)
26		nnuz 12760	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
27	25, 26	eleqtrdi 2848	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘𝐴) ∈ (ℤ_≥‘1))
28		erdszelem7.r	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℕ)
29		nnz 12478	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℕ → 𝑅 ∈ ℤ)
30		peano2zm 12504	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℤ → (𝑅 − 1) ∈ ℤ)
31	28, 29, 30	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 − 1) ∈ ℤ)
32		elfz5 13387	. . . . . . . . . . . . 13 ⊢ (((𝐾‘𝐴) ∈ (ℤ_≥‘1) ∧ (𝑅 − 1) ∈ ℤ) → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
33	27, 31, 32	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
34		nnltlem1 12528	. . . . . . . . . . . . 13 ⊢ (((𝐾‘𝐴) ∈ ℕ ∧ 𝑅 ∈ ℕ) → ((𝐾‘𝐴) < 𝑅 ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
35	25, 28, 34	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾‘𝐴) < 𝑅 ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
36	33, 35	bitr4d 281	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) < 𝑅))
37	23, 36	mtbid 323	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝐾‘𝐴) < 𝑅)
38	28	nnred 12126	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℝ)
39	13	erdszelem2 33614	. . . . . . . . . . . . . 14 ⊢ ((♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) ∈ Fin ∧ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) ⊆ ℕ)
40	39	simpri 486	. . . . . . . . . . . . 13 ⊢ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) ⊆ ℕ
41		nnssre 12115	. . . . . . . . . . . . 13 ⊢ ℕ ⊆ ℝ
42	40, 41	sstri 3951	. . . . . . . . . . . 12 ⊢ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) ⊆ ℝ
43	42, 10	sselid 3940	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘𝐴) ∈ ℝ)
44	38, 43	lenltd 11259	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ≤ (𝐾‘𝐴) ↔ ¬ (𝐾‘𝐴) < 𝑅))
45	37, 44	mpbird 256	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ≤ (𝐾‘𝐴))
46	45	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑅 ≤ (𝐾‘𝐴))
47		simprr 771	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → (♯‘𝑠) = (𝐾‘𝐴))
48	46, 47	breqtrrd 5131	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑅 ≤ (♯‘𝑠))
49		simprl2 1219	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)))
50	22, 48, 49	jca32 516	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → (𝑠 ∈ 𝒫 (1...𝑁) ∧ (𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)))))
51	50	expr 457	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠)) → ((♯‘𝑠) = (𝐾‘𝐴) → (𝑠 ∈ 𝒫 (1...𝑁) ∧ (𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠))))))
52	14, 51	sylan2b 594	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) → ((♯‘𝑠) = (𝐾‘𝐴) → (𝑠 ∈ 𝒫 (1...𝑁) ∧ (𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠))))))
53	52	expimpd 454	. . 3 ⊢ (𝜑 → ((𝑠 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} ∧ (♯‘𝑠) = (𝐾‘𝐴)) → (𝑠 ∈ 𝒫 (1...𝑁) ∧ (𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠))))))
54	53	reximdv2 3159	. 2 ⊢ (𝜑 → (∃𝑠 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} (♯‘𝑠) = (𝐾‘𝐴) → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)))))
55	12, 54	mpd 15	1 ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3071 {crab 3405 Vcvv 3443 ∪ cun 3906 ⊆ wss 3908 𝒫 cpw 4558 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 Or wor 5542 ↾ cres 5633 “ cima 5634 Fun wfun 6487 ⟶wf 6489 –1-1→wf1 6490 ‘cfv 6493 Isom wiso 6494 (class class class)co 7351 Fincfn 8841 supcsup 9334 ℝcr 11008 1c1 11010 +∞cpnf 11144 < clt 11147 ≤ cle 11148 − cmin 11343 ℕcn 12111 ℕ₀cn0 12371 ℤcz 12457 ℤ_≥cuz 12721 ...cfz 13378 ♯chash 14184

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-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 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

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 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 df-xnn0 12444 df-z 12458 df-uz 12722 df-fz 13379 df-hash 14185

This theorem is referenced by: erdszelem11 33623

W3C validator