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Theorem psercn2 25487

Description: Since by pserulm 25486 the series converges uniformly, it is also continuous by ulmcn 25463. (Contributed by Mario Carneiro, 3-Mar-2015.)

**Hypotheses**
Ref	Expression
pserf.g	⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))
pserf.f	⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ₀ ((𝐺‘𝑦)‘𝑗))
pserf.a	⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
pserf.r	⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ^*, < )
pserulm.h	⊢ 𝐻 = (𝑖 ∈ ℕ₀ ↦ (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
pserulm.m	⊢ (𝜑 → 𝑀 ∈ ℝ)
pserulm.l	⊢ (𝜑 → 𝑀 < 𝑅)
pserulm.y	⊢ (𝜑 → 𝑆 ⊆ (^◡abs “ (0[,]𝑀)))

**Assertion**
Ref	Expression
psercn2	⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Distinct variable groups: 𝑗,𝑛,𝑟,𝑥,𝑦,𝐴 𝑖,𝑗,𝑦,𝐻 𝑖,𝑀,𝑗,𝑦 𝑥,𝑖,𝑟 𝑖,𝐺,𝑗,𝑟,𝑦 𝑆,𝑖,𝑗,𝑦 𝜑,𝑖,𝑗,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑛,𝑟) 𝐴(𝑖) 𝑅(𝑥,𝑦,𝑖,𝑗,𝑛,𝑟) 𝑆(𝑥,𝑛,𝑟) 𝐹(𝑥,𝑦,𝑖,𝑗,𝑛,𝑟) 𝐺(𝑥,𝑛) 𝐻(𝑥,𝑛,𝑟) 𝑀(𝑥,𝑛,𝑟)

Proof of Theorem psercn2 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 12549	. 2 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12261	. 2 ⊢ (𝜑 → 0 ∈ ℤ)
3		pserulm.y	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (^◡abs “ (0[,]𝑀)))
4		cnvimass 5978	. . . . . . . 8 ⊢ (^◡abs “ (0[,]𝑀)) ⊆ dom abs
5		absf 14977	. . . . . . . . 9 ⊢ abs:ℂ⟶ℝ
6	5	fdmi 6596	. . . . . . . 8 ⊢ dom abs = ℂ
7	4, 6	sseqtri 3953	. . . . . . 7 ⊢ (^◡abs “ (0[,]𝑀)) ⊆ ℂ
8	3, 7	sstrdi 3929	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑆 ⊆ ℂ)
10	9	resmptd 5937	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
11		simplr 765	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → 𝑦 ∈ ℂ)
12		elfznn0 13278	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ₀)
13	12	adantl 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ₀)
14		pserf.g	. . . . . . . . . 10 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))
15	14	pserval2 25475	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑦)‘𝑘) = ((𝐴‘𝑘) · (𝑦↑𝑘)))
16	11, 13, 15	syl2anc 583	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐺‘𝑦)‘𝑘) = ((𝐴‘𝑘) · (𝑦↑𝑘)))
17		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℕ₀)
18	17, 1	eleqtrdi 2849	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ (ℤ_≥‘0))
19	18	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) → 𝑖 ∈ (ℤ_≥‘0))
20		pserf.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
21	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝐴:ℕ₀⟶ℂ)
22	21	ffvelrnda 6943	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
23	22	adantlr 711	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
24		expcl 13728	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑦↑𝑘) ∈ ℂ)
25	24	adantll 710	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝑦↑𝑘) ∈ ℂ)
26	23, 25	mulcld 10926	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ)
27	12, 26	sylan2 592	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ)
28	16, 19, 27	fsumser 15370	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) → Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘)) = (seq0( + , (𝐺‘𝑦))‘𝑖))
29	28	mpteq2dva 5170	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) = (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
30		eqid 2738	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
31	30	cnfldtopon 23852	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
33		fzfid 13621	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (0...𝑖) ∈ Fin)
34	31	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑖)) → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
35		ffvelrn 6941	. . . . . . . . . . 11 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
36	21, 12, 35	syl2an 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑖)) → (𝐴‘𝑘) ∈ ℂ)
37	34, 34, 36	cnmptc 22721	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑖)) → (𝑦 ∈ ℂ ↦ (𝐴‘𝑘)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
38	12	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ₀)
39	30	expcn 23941	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ → (𝑦 ∈ ℂ ↦ (𝑦↑𝑘)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
40	38, 39	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑖)) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑘)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
41	30	mulcn 23936	. . . . . . . . . 10 ⊢ · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
42	41	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑖)) → · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
43	34, 37, 40, 42	cnmpt12f 22725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑖)) → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
44	30, 32, 33, 43	fsumcn 23939	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
45	30	cncfcn1 23980	. . . . . . 7 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
46	44, 45	eleqtrrdi 2850	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ (ℂ–cn→ℂ))
47	29, 46	eqeltrrd 2840	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ∈ (ℂ–cn→ℂ))
48		rescncf 23966	. . . . 5 ⊢ (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ∈ (ℂ–cn→ℂ) → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ↾ 𝑆) ∈ (𝑆–cn→ℂ)))
49	9, 47, 48	sylc 65	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ↾ 𝑆) ∈ (𝑆–cn→ℂ))
50	10, 49	eqeltrrd 2840	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ∈ (𝑆–cn→ℂ))
51		pserulm.h	. . 3 ⊢ 𝐻 = (𝑖 ∈ ℕ₀ ↦ (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
52	50, 51	fmptd 6970	. 2 ⊢ (𝜑 → 𝐻:ℕ₀⟶(𝑆–cn→ℂ))
53		pserf.f	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ₀ ((𝐺‘𝑦)‘𝑗))
54		pserf.r	. . 3 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ^*, < )
55		pserulm.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
56		pserulm.l	. . 3 ⊢ (𝜑 → 𝑀 < 𝑅)
57	14, 53, 20, 54, 51, 55, 56, 3	pserulm 25486	. 2 ⊢ (𝜑 → 𝐻(⇝_𝑢‘𝑆)𝐹)
58	1, 2, 52, 57	ulmcn 25463	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 {crab 3067 ⊆ wss 3883 class class class wbr 5070 ↦ cmpt 5153 ^◡ccnv 5579 dom cdm 5580 ↾ cres 5582 “ cima 5583 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 supcsup 9129 ℂcc 10800 ℝcr 10801 0cc0 10802 + caddc 10805 · cmul 10807 ℝ^*cxr 10939 < clt 10940 ℕ₀cn0 12163 ℤ_≥cuz 12511 [,]cicc 13011 ...cfz 13168 seqcseq 13649 ↑cexp 13710 abscabs 14873 ⇝ cli 15121 Σcsu 15325 TopOpenctopn 17049 ℂ_fldccnfld 20510 TopOnctopon 21967 Cn ccn 22283 ×_t ctx 22619 –cn→ccncf 23945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cn 22286 df-cnp 22287 df-tx 22621 df-hmeo 22814 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-ulm 25441

This theorem is referenced by: psercn 25490

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