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Theorem caucvgrlem2 15617

Description: Lemma for caucvgr 15618. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Mario Carneiro, 8-May-2016.)

**Hypotheses**
Ref	Expression
caucvgr.1	⊢ (𝜑 → 𝐴 ⊆ ℝ)
caucvgr.2	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
caucvgr.3	⊢ (𝜑 → sup(𝐴, ℝ^*, < ) = +∞)
caucvgr.4	⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
caucvgrlem2.5	⊢ 𝐻:ℂ⟶ℝ
caucvgrlem2.6	⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))))

**Assertion**
Ref	Expression
caucvgrlem2	⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑛))) ⇝_𝑟 ( ⇝_𝑟 ‘(𝐻 ∘ 𝐹)))

Distinct variable groups: 𝑗,𝑘,𝑛,𝑥,𝐴 𝑗,𝐹,𝑘,𝑛,𝑥 𝑗,𝐻,𝑘,𝑛,𝑥 𝜑,𝑗,𝑘,𝑛,𝑥

**Proof of Theorem caucvgrlem2**
Step	Hyp	Ref	Expression
1		caucvgrlem2.5	. . 3 ⊢ 𝐻:ℂ⟶ℝ
2		caucvgr.2	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
3		fcompt 7127	. . 3 ⊢ ((𝐻:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (𝐻 ∘ 𝐹) = (𝑛 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑛))))
4	1, 2, 3	sylancr 587	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) = (𝑛 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑛))))
5		caucvgr.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
6		fco 6738	. . . . . 6 ⊢ ((𝐻:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (𝐻 ∘ 𝐹):𝐴⟶ℝ)
7	1, 2, 6	sylancr 587	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝐹):𝐴⟶ℝ)
8		caucvgr.3	. . . . 5 ⊢ (𝜑 → sup(𝐴, ℝ^*, < ) = +∞)
9		caucvgr.4	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
10	2	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → 𝐹:𝐴⟶ℂ)
11		simprr 771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → 𝑘 ∈ 𝐴)
12	10, 11	ffvelcdmd 7084	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (𝐹‘𝑘) ∈ ℂ)
13		simprl 769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → 𝑗 ∈ 𝐴)
14	10, 13	ffvelcdmd 7084	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (𝐹‘𝑗) ∈ ℂ)
15		caucvgrlem2.6	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))))
16	12, 14, 15	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))))
17	1	ffvelcdmi 7082	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑘) ∈ ℂ → (𝐻‘(𝐹‘𝑘)) ∈ ℝ)
18	12, 17	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (𝐻‘(𝐹‘𝑘)) ∈ ℝ)
19	1	ffvelcdmi 7082	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑗) ∈ ℂ → (𝐻‘(𝐹‘𝑗)) ∈ ℝ)
20	14, 19	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (𝐻‘(𝐹‘𝑗)) ∈ ℝ)
21	18, 20	resubcld 11638	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗))) ∈ ℝ)
22	21	recnd 11238	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗))) ∈ ℂ)
23	22	abscld 15379	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ∈ ℝ)
24	12, 14	subcld 11567	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((𝐹‘𝑘) − (𝐹‘𝑗)) ∈ ℂ)
25	24	abscld 15379	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∈ ℝ)
26		rpre 12978	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
27	26	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → 𝑥 ∈ ℝ)
28		lelttr 11300	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) < 𝑥))
29	23, 25, 27, 28	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (((abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) < 𝑥))
30	16, 29	mpand 693	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) < 𝑥))
31		fvco3 6987	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝑘 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
32	10, 11, 31	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
33		fvco3 6987	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝑗 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑗) = (𝐻‘(𝐹‘𝑗)))
34	10, 13, 33	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((𝐻 ∘ 𝐹)‘𝑗) = (𝐻‘(𝐹‘𝑗)))
35	32, 34	oveq12d 7423	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗)) = ((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗))))
36	35	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) = (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))))
37	36	breq1d 5157	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥 ↔ (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) < 𝑥))
38	30, 37	sylibrd 258	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥))
39	38	imim2d 57	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (𝑗 ≤ 𝑘 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥)))
40	39	anassrs 468	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐴) → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (𝑗 ≤ 𝑘 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥)))
41	40	ralimdva 3167	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝐴) → (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥)))
42	41	reximdva 3168	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥)))
43	42	ralimdva 3167	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥)))
44	9, 43	mpd 15	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥))
45	5, 7, 8, 44	caurcvgr 15616	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝_𝑟 (lim sup‘(𝐻 ∘ 𝐹)))
46		rlimrel 15433	. . . . 5 ⊢ Rel ⇝_𝑟
47	46	releldmi 5945	. . . 4 ⊢ ((𝐻 ∘ 𝐹) ⇝_𝑟 (lim sup‘(𝐻 ∘ 𝐹)) → (𝐻 ∘ 𝐹) ∈ dom ⇝_𝑟 )
48	45, 47	syl 17	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ dom ⇝_𝑟 )
49		ax-resscn 11163	. . . . 5 ⊢ ℝ ⊆ ℂ
50		fss 6731	. . . . 5 ⊢ (((𝐻 ∘ 𝐹):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝐻 ∘ 𝐹):𝐴⟶ℂ)
51	7, 49, 50	sylancl 586	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹):𝐴⟶ℂ)
52	51, 8	rlimdm 15491	. . 3 ⊢ (𝜑 → ((𝐻 ∘ 𝐹) ∈ dom ⇝_𝑟 ↔ (𝐻 ∘ 𝐹) ⇝_𝑟 ( ⇝_𝑟 ‘(𝐻 ∘ 𝐹))))
53	48, 52	mpbid 231	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝_𝑟 ( ⇝_𝑟 ‘(𝐻 ∘ 𝐹)))
54	4, 53	eqbrtrrd 5171	1 ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑛))) ⇝_𝑟 ( ⇝_𝑟 ‘(𝐻 ∘ 𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ∘ ccom 5679 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 supcsup 9431 ℂcc 11104 ℝcr 11105 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11440 ℝ⁺crp 12970 abscabs 15177 lim supclsp 15410 ⇝_𝑟 crli 15425

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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-ico 13326 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-rlim 15429

This theorem is referenced by: caucvgr 15618

W3C validator