erdszelem7 - Mathbox for Mario Carneiro

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Theorem erdszelem7 30433

Description: Lemma for erdsze 30438. (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 12987	. . . 4 ⊢ #:V⟶(ℕ₀ ∪ {+∞})
2		ffun 5961	. . . 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 30431	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...𝑁)) → (𝐾‘𝐴) ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
10	4, 9	mpdan 699	. . 3 ⊢ (𝜑 → (𝐾‘𝐴) ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
11		fvelima 6158	. . 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 694	. 2 ⊢ (𝜑 → ∃𝑠 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} (#‘𝑠) = (𝐾‘𝐴))
13		eqid 2610	. . . . . 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 30427	. . . . 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 1099	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (#‘𝑠) = (𝐾‘𝐴))) → 𝑠 ⊆ (1...𝐴))
16		elfzuz3 12210	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (1...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐴))
17		fzss2 12252	. . . . . . . . . . 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 3578	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (#‘𝑠) = (𝐾‘𝐴))) → 𝑠 ⊆ (1...𝑁))
21		selpw 4115	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 (1...𝑁) ↔ 𝑠 ⊆ (1...𝑁))
22	20, 21	sylibr 223	. . . . . . 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 30432	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶ℕ)
25	24, 4	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘𝐴) ∈ ℕ)
26		nnuz 11599	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
27	25, 26	syl6eleq 2698	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘𝐴) ∈ (ℤ_≥‘1))
28		erdszelem7.r	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℕ)
29		nnz 11276	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℕ → 𝑅 ∈ ℤ)
30		peano2zm 11297	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℤ → (𝑅 − 1) ∈ ℤ)
31	28, 29, 30	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 − 1) ∈ ℤ)
32		elfz5 12205	. . . . . . . . . . . . 13 ⊢ (((𝐾‘𝐴) ∈ (ℤ_≥‘1) ∧ (𝑅 − 1) ∈ ℤ) → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
33	27, 31, 32	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
34		nnltlem1 11320	. . . . . . . . . . . . 13 ⊢ (((𝐾‘𝐴) ∈ ℕ ∧ 𝑅 ∈ ℕ) → ((𝐾‘𝐴) < 𝑅 ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
35	25, 28, 34	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾‘𝐴) < 𝑅 ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
36	33, 35	bitr4d 270	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) < 𝑅))
37	23, 36	mtbid 313	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝐾‘𝐴) < 𝑅)
38	28	nnred 10912	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℝ)
39	13	erdszelem2 30428	. . . . . . . . . . . . . 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 477	. . . . . . . . . . . . 13 ⊢ (# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) ⊆ ℕ
41		nnssre 10901	. . . . . . . . . . . . 13 ⊢ ℕ ⊆ ℝ
42	40, 41	sstri 3577	. . . . . . . . . . . 12 ⊢ (# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) ⊆ ℝ
43	42, 10	sseldi 3566	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘𝐴) ∈ ℝ)
44	38, 43	lenltd 10062	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ≤ (𝐾‘𝐴) ↔ ¬ (𝐾‘𝐴) < 𝑅))
45	37, 44	mpbird 246	. . . . . . . . 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 792	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (#‘𝑠) = (𝐾‘𝐴))) → (#‘𝑠) = (𝐾‘𝐴))
48	46, 47	breqtrrd 4611	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (#‘𝑠) = (𝐾‘𝐴))) → 𝑅 ≤ (#‘𝑠))
49		simprl2 1100	. . . . . . 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 556	. . . . . 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 641	. . . . 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 491	. . . 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 627	. . 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 2997	. 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 Or wor 4958 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 Fincfn 7841 supcsup 8229 ℝcr 9814 1c1 9816 +∞cpnf 9950 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 #chash 12979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980

This theorem is referenced by: erdszelem11 30437

W3C validator