erdszelem7 - Mathbox for Mario Carneiro

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Theorem erdszelem7 35434

Description: Lemma for erdsze 35439. (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 14292	. . . 4 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
2		ffun 6659	. . . 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 35432	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...𝑁)) → (𝐾‘𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
10	4, 9	mpdan 693	. . 3 ⊢ (𝜑 → (𝐾‘𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
11		fvelima 6893	. . 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 593	. 2 ⊢ (𝜑 → ∃𝑠 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} (♯‘𝑠) = (𝐾‘𝐴))
13		eqid 2739	. . . . . 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 35428	. . . . 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 1225	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑠 ⊆ (1...𝐴))
16		elfzuz3 13467	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (1...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐴))
17		fzss2 13510	. . . . . . . . . . 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 3925	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑠 ⊆ (1...𝑁))
21		velpw 4535	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 (1...𝑁) ↔ 𝑠 ⊆ (1...𝑁))
22	20, 21	sylibr 235	. . . . . . 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 35433	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶ℕ)
25	24, 4	ffvelcdmd 7027	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘𝐴) ∈ ℕ)
26		nnuz 12819	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
27	25, 26	eleqtrdi 2849	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘𝐴) ∈ (ℤ_≥‘1))
28		erdszelem7.r	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℕ)
29		nnz 12537	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℕ → 𝑅 ∈ ℤ)
30		peano2zm 12562	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℤ → (𝑅 − 1) ∈ ℤ)
31	28, 29, 30	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 − 1) ∈ ℤ)
32		elfz5 13462	. . . . . . . . . . . . 13 ⊢ (((𝐾‘𝐴) ∈ (ℤ_≥‘1) ∧ (𝑅 − 1) ∈ ℤ) → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
33	27, 31, 32	syl2anc 590	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
34		nnltlem1 12588	. . . . . . . . . . . . 13 ⊢ (((𝐾‘𝐴) ∈ ℕ ∧ 𝑅 ∈ ℕ) → ((𝐾‘𝐴) < 𝑅 ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
35	25, 28, 34	syl2anc 590	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾‘𝐴) < 𝑅 ↔ (𝐾‘𝐴) ≤ (𝑅 − 1)))
36	33, 35	bitr4d 283	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾‘𝐴) ∈ (1...(𝑅 − 1)) ↔ (𝐾‘𝐴) < 𝑅))
37	23, 36	mtbid 325	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝐾‘𝐴) < 𝑅)
38	28	nnred 12181	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℝ)
39	13	erdszelem2 35429	. . . . . . . . . . . . . 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 12170	. . . . . . . . . . . . 13 ⊢ ℕ ⊆ ℝ
42	40, 41	sstri 3924	. . . . . . . . . . . 12 ⊢ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) ⊆ ℝ
43	42, 10	sselid 3913	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘𝐴) ∈ ℝ)
44	38, 43	lenltd 11284	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ≤ (𝐾‘𝐴) ↔ ¬ (𝐾‘𝐴) < 𝑅))
45	37, 44	mpbird 258	. . . . . . . . 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 778	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → (♯‘𝑠) = (𝐾‘𝐴))
48	46, 47	breqtrrd 5101	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)) ∧ 𝐴 ∈ 𝑠) ∧ (♯‘𝑠) = (𝐾‘𝐴))) → 𝑅 ≤ (♯‘𝑠))
49		simprl2 1226	. . . . . . 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 520	. . . . . 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 600	. . . 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 3149	. 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 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∃wrex 3063 {crab 3391 Vcvv 3431 ∪ cun 3881 ⊆ wss 3883 𝒫 cpw 4530 {csn 4556 class class class wbr 5073 ↦ cmpt 5154 Or wor 5526 ↾ cres 5621 “ cima 5622 Fun wfun 6480 ⟶wf 6482 –1-1→wf1 6483 ‘cfv 6486 Isom wiso 6487 (class class class)co 7357 Fincfn 8884 supcsup 9344 ℝcr 11029 1c1 11031 +∞cpnf 11168 < clt 11171 ≤ cle 11172 − cmin 11369 ℕcn 12166 ℕ₀cn0 12429 ℤcz 12516 ℤ_≥cuz 12780 ...cfz 13453 ♯chash 14284

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7808 df-1st 7932 df-2nd 7933 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-er 8634 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9346 df-card 9855 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12167 df-n0 12430 df-xnn0 12503 df-z 12517 df-uz 12781 df-fz 13454 df-hash 14285

This theorem is referenced by: erdszelem11 35438

W3C validator