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Theorem psercn2 25580

Description: Since by pserulm 25579 the series converges uniformly, it is also continuous by ulmcn 25556. (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 12618	. 2 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12329	. 2 ⊢ (𝜑 → 0 ∈ ℤ)
3		pserulm.y	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (^◡abs “ (0[,]𝑀)))
4		cnvimass 5991	. . . . . . . 8 ⊢ (^◡abs “ (0[,]𝑀)) ⊆ dom abs
5		absf 15047	. . . . . . . . 9 ⊢ abs:ℂ⟶ℝ
6	5	fdmi 6614	. . . . . . . 8 ⊢ dom abs = ℂ
7	4, 6	sseqtri 3958	. . . . . . 7 ⊢ (^◡abs “ (0[,]𝑀)) ⊆ ℂ
8	3, 7	sstrdi 3934	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
9	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑆 ⊆ ℂ)
10	9	resmptd 5950	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
11		simplr 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → 𝑦 ∈ ℂ)
12		elfznn0 13347	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ₀)
13	12	adantl 482	. . . . . . . . 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 25568	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑦)‘𝑘) = ((𝐴‘𝑘) · (𝑦↑𝑘)))
16	11, 13, 15	syl2anc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐺‘𝑦)‘𝑘) = ((𝐴‘𝑘) · (𝑦↑𝑘)))
17		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℕ₀)
18	17, 1	eleqtrdi 2849	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ (ℤ_≥‘0))
19	18	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) → 𝑖 ∈ (ℤ_≥‘0))
20		pserf.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
21	20	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝐴:ℕ₀⟶ℂ)
22	21	ffvelrnda 6963	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
23	22	adantlr 712	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
24		expcl 13798	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑦↑𝑘) ∈ ℂ)
25	24	adantll 711	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝑦↑𝑘) ∈ ℂ)
26	23, 25	mulcld 10993	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ)
27	12, 26	sylan2 593	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ)
28	16, 19, 27	fsumser 15440	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) → Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘)) = (seq0( + , (𝐺‘𝑦))‘𝑖))
29	28	mpteq2dva 5176	. . . . . 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 23944	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
33		fzfid 13691	. . . . . . . 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 6961	. . . . . . . . . . 11 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
36	21, 12, 35	syl2an 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑖)) → (𝐴‘𝑘) ∈ ℂ)
37	34, 34, 36	cnmptc 22811	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑖)) → (𝑦 ∈ ℂ ↦ (𝐴‘𝑘)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
38	12	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ₀)
39	30	expcn 24033	. . . . . . . . . 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 24028	. . . . . . . . . 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 22815	. . . . . . . 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 24031	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
45	30	cncfcn1 24072	. . . . . . 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 24058	. . . . 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 6990	. 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 25579	. 2 ⊢ (𝜑 → 𝐻(⇝_𝑢‘𝑆)𝐹)
58	1, 2, 52, 57	ulmcn 25556	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 ⊆ wss 3888 class class class wbr 5076 ↦ cmpt 5159 ^◡ccnv 5590 dom cdm 5591 ↾ cres 5593 “ cima 5594 ⟶wf 6431 ‘cfv 6435 (class class class)co 7277 supcsup 9197 ℂcc 10867 ℝcr 10868 0cc0 10869 + caddc 10872 · cmul 10874 ℝ^*cxr 11006 < clt 11007 ℕ₀cn0 12231 ℤ_≥cuz 12580 [,]cicc 13080 ...cfz 13237 seqcseq 13719 ↑cexp 13780 abscabs 14943 ⇝ cli 15191 Σcsu 15395 TopOpenctopn 17130 ℂ_fldccnfld 20595 TopOnctopon 22057 Cn ccn 22373 ×_t ctx 22709 –cn→ccncf 24037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-inf2 9397 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-pre-sup 10947 ax-addf 10948 ax-mulf 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-isom 6444 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7976 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-2o 8296 df-er 8496 df-map 8615 df-pm 8616 df-ixp 8684 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-fsupp 9127 df-fi 9168 df-sup 9199 df-inf 9200 df-oi 9267 df-card 9695 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12436 df-uz 12581 df-q 12687 df-rp 12729 df-xneg 12846 df-xadd 12847 df-xmul 12848 df-ico 13083 df-icc 13084 df-fz 13238 df-fzo 13381 df-fl 13510 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-limsup 15178 df-clim 15195 df-rlim 15196 df-sum 15396 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cn 22376 df-cnp 22377 df-tx 22711 df-hmeo 22904 df-xms 23471 df-ms 23472 df-tms 23473 df-cncf 24039 df-ulm 25534

This theorem is referenced by: psercn 25583

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