erdszelem7 - Mathbox for Mario Carneiro

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Theorem erdszelem7 32872

Description: Lemma for erdsze 32877. (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 13904	. . . 4 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
2		ffun 6548	. . . 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 32870	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...𝑁)) → (𝐾‘𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
10	4, 9	mpdan 687	. . 3 ⊢ (𝜑 → (𝐾‘𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
11		fvelima 6778	. . 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 590	. 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 32866	. . . . 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 13109	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (1...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐴))
17		fzss2 13152	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝐴) → (1...𝐴) ⊆ (1...𝑁))
18	4, 16, 17	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝐴) ⊆ (1...𝑁))
19	18	adantr 484	. . . . . . . . 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 3911	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑠 ⊆ (1...𝑁))
21		velpw 4518	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 (1...𝑁) ↔ 𝑠 ⊆ (1...𝑁))
22	20, 21	sylibr 237	. . . . . . 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 32871	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶ℕ)
25	24, 4	ffvelrnd 6905	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘𝐴) ∈ ℕ)
26		nnuz 12477	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
27	25, 26	eleqtrdi 2848	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘𝐴) ∈ (ℤ_≥‘1))
28		erdszelem7.r	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℕ)
29		nnz 12199	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℕ → 𝑅 ∈ ℤ)
30		peano2zm 12220	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℤ → (𝑅 − 1) ∈ ℤ)
31	28, 29, 30	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 − 1) ∈ ℤ)
32		elfz5 13104	. . . . . . . . . . . . 13 ⊢ (((𝐾‘𝐴) ∈ (ℤ_≥‘1) ∧ (𝑅 − 1) ∈ ℤ) → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
33	27, 31, 32	syl2anc 587	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
34		nnltlem1 12244	. . . . . . . . . . . . 13 ⊢ (((𝐾‘𝐴) ∈ ℕ ∧ 𝑅 ∈ ℕ) → ((𝐾‘𝐴) < 𝑅 ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
35	25, 28, 34	syl2anc 587	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾‘𝐴) < 𝑅 ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
36	33, 35	bitr4d 285	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) < 𝑅))
37	23, 36	mtbid 327	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝐾‘𝐴) < 𝑅)
38	28	nnred 11845	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℝ)
39	13	erdszelem2 32867	. . . . . . . . . . . . . 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 489	. . . . . . . . . . . . 13 ⊢ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) ⊆ ℕ
41		nnssre 11834	. . . . . . . . . . . . 13 ⊢ ℕ ⊆ ℝ
42	40, 41	sstri 3910	. . . . . . . . . . . 12 ⊢ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) ⊆ ℝ
43	42, 10	sseldi 3899	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘𝐴) ∈ ℝ)
44	38, 43	lenltd 10978	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ≤ (𝐾‘𝐴) ↔ ¬ (𝐾‘𝐴) < 𝑅))
45	37, 44	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ≤ (𝐾‘𝐴))
46	45	adantr 484	. . . . . . . 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 5081	. . . . . . 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 519	. . . . . 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 460	. . . . 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 597	. . . 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 457	. . 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 3190	. 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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∃wrex 3062 {crab 3065 Vcvv 3408 ∪ cun 3864 ⊆ wss 3866 𝒫 cpw 4513 {csn 4541 class class class wbr 5053 ↦ cmpt 5135 Or wor 5467 ↾ cres 5553 “ cima 5554 Fun wfun 6374 ⟶wf 6376 –1-1→wf1 6377 ‘cfv 6380 Isom wiso 6381 (class class class)co 7213 Fincfn 8626 supcsup 9056 ℝcr 10728 1c1 10730 +∞cpnf 10864 < clt 10867 ≤ cle 10868 − cmin 11062 ℕcn 11830 ℕ₀cn0 12090 ℤcz 12176 ℤ_≥cuz 12438 ...cfz 13095 ♯chash 13896

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-xnn0 12163 df-z 12177 df-uz 12439 df-fz 13096 df-hash 13897

This theorem is referenced by: erdszelem11 32876

W3C validator