erdszelem7 - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > erdszelem7		Structured version Visualization version GIF version

Theorem erdszelem7 35419

Description: Lemma for erdsze 35424. (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 14275	. . . 4 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
2		ffun 6675	. . . 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 35417	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...𝑁)) → (𝐾‘𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
10	4, 9	mpdan 688	. . 3 ⊢ (𝜑 → (𝐾‘𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
11		fvelima 6909	. . 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 588	. 2 ⊢ (𝜑 → ∃𝑠 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} (♯‘𝑠) = (𝐾‘𝐴))
13		eqid 2737	. . . . . 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 35413	. . . . 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 1220	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑠 ⊆ (1...𝐴))
16		elfzuz3 13451	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (1...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐴))
17		fzss2 13494	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝐴) → (1...𝐴) ⊆ (1...𝑁))
18	4, 16, 17	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝐴) ⊆ (1...𝑁))
19	18	adantr 480	. . . . . . . . 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 3946	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑠 ⊆ (1...𝑁))
21		velpw 4561	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 (1...𝑁) ↔ 𝑠 ⊆ (1...𝑁))
22	20, 21	sylibr 234	. . . . . . 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 35418	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶ℕ)
25	24, 4	ffvelcdmd 7041	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘𝐴) ∈ ℕ)
26		nnuz 12804	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
27	25, 26	eleqtrdi 2847	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘𝐴) ∈ (ℤ_≥‘1))
28		erdszelem7.r	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℕ)
29		nnz 12523	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℕ → 𝑅 ∈ ℤ)
30		peano2zm 12548	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℤ → (𝑅 − 1) ∈ ℤ)
31	28, 29, 30	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 − 1) ∈ ℤ)
32		elfz5 13446	. . . . . . . . . . . . 13 ⊢ (((𝐾‘𝐴) ∈ (ℤ_≥‘1) ∧ (𝑅 − 1) ∈ ℤ) → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
33	27, 31, 32	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
34		nnltlem1 12573	. . . . . . . . . . . . 13 ⊢ (((𝐾‘𝐴) ∈ ℕ ∧ 𝑅 ∈ ℕ) → ((𝐾‘𝐴) < 𝑅 ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
35	25, 28, 34	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾‘𝐴) < 𝑅 ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
36	33, 35	bitr4d 282	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) < 𝑅))
37	23, 36	mtbid 324	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝐾‘𝐴) < 𝑅)
38	28	nnred 12174	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℝ)
39	13	erdszelem2 35414	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) ⊆ ℕ
41		nnssre 12163	. . . . . . . . . . . . 13 ⊢ ℕ ⊆ ℝ
42	40, 41	sstri 3945	. . . . . . . . . . . 12 ⊢ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) ⊆ ℝ
43	42, 10	sselid 3933	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘𝐴) ∈ ℝ)
44	38, 43	lenltd 11293	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ≤ (𝐾‘𝐴) ↔ ¬ (𝐾‘𝐴) < 𝑅))
45	37, 44	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ≤ (𝐾‘𝐴))
46	45	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑅 ≤ (𝐾‘𝐴))
47		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → (♯‘𝑠) = (𝐾‘𝐴))
48	46, 47	breqtrrd 5128	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑅 ≤ (♯‘𝑠))
49		simprl2 1221	. . . . . . 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 515	. . . . . 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 456	. . . . 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 595	. . . 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 453	. . 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 3148	. 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∃wrex 3062 {crab 3401 Vcvv 3442 ∪ cun 3901 ⊆ wss 3903 𝒫 cpw 4556 {csn 4582 class class class wbr 5100 ↦ cmpt 5181 Or wor 5541 ↾ cres 5636 “ cima 5637 Fun wfun 6496 ⟶wf 6498 –1-1→wf1 6499 ‘cfv 6502 Isom wiso 6503 (class class class)co 7370 Fincfn 8897 supcsup 9357 ℝcr 11039 1c1 11041 +∞cpnf 11177 < clt 11180 ≤ cle 11181 − cmin 11378 ℕcn 12159 ℕ₀cn0 12415 ℤcz 12502 ℤ_≥cuz 12765 ...cfz 13437 ♯chash 14267

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-om 7821 df-1st 7945 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-er 8647 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-sup 9359 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-nn 12160 df-n0 12416 df-xnn0 12489 df-z 12503 df-uz 12766 df-fz 13438 df-hash 14268

This theorem is referenced by: erdszelem11 35423

W3C validator