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Theorem psercn2OLD 26333

Description: Obsolete version of psercn2 26332 as of 16-Apr-2025. (Contributed by Mario Carneiro, 3-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

**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
psercn2OLD	⊢ (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 psercn2OLD Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 12835	. 2 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12541	. 2 ⊢ (𝜑 → 0 ∈ ℤ)
3		pserulm.y	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (^◡abs “ (0[,]𝑀)))
4		cnvimass 6053	. . . . . . . 8 ⊢ (^◡abs “ (0[,]𝑀)) ⊆ dom abs
5		absf 15304	. . . . . . . . 9 ⊢ abs:ℂ⟶ℝ
6	5	fdmi 6699	. . . . . . . 8 ⊢ dom abs = ℂ
7	4, 6	sseqtri 3995	. . . . . . 7 ⊢ (^◡abs “ (0[,]𝑀)) ⊆ ℂ
8	3, 7	sstrdi 3959	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑆 ⊆ ℂ)
10	9	resmptd 6011	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
11		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → 𝑦 ∈ ℂ)
12		elfznn0 13581	. . . . . . . . . 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 26320	. . . . . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℕ₀)
18	17, 1	eleqtrdi 2838	. . . . . . . . 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	ffvelcdmda 7056	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
23	22	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
24		expcl 14044	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑦↑𝑘) ∈ ℂ)
25	24	adantll 714	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝑦↑𝑘) ∈ ℂ)
26	23, 25	mulcld 11194	. . . . . . . . 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 15696	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) → Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘)) = (seq0( + , (𝐺‘𝑦))‘𝑖))
29	28	mpteq2dva 5200	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) = (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
30		eqid 2729	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
31	30	cnfldtopon 24670	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
33		fzfid 13938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (0...𝑖) ∈ Fin)
34	31	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑖)) → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
35		ffvelcdm 7053	. . . . . . . . . . 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 23549	. . . . . . . . 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 24763	. . . . . . . . . 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 24756	. . . . . . . . . 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 23553	. . . . . . . 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 24761	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
45	30	cncfcn1 24804	. . . . . . 7 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
46	44, 45	eleqtrrdi 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ (ℂ–cn→ℂ))
47	29, 46	eqeltrrd 2829	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ∈ (ℂ–cn→ℂ))
48		rescncf 24790	. . . . 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 2829	. . 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 7086	. 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 26331	. 2 ⊢ (𝜑 → 𝐻(⇝_𝑢‘𝑆)𝐹)
58	1, 2, 52, 57	ulmcn 26308	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3405 ⊆ wss 3914 class class class wbr 5107 ↦ cmpt 5188 ^◡ccnv 5637 dom cdm 5638 ↾ cres 5640 “ cima 5641 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 supcsup 9391 ℂcc 11066 ℝcr 11067 0cc0 11068 + caddc 11071 · cmul 11073 ℝ^*cxr 11207 < clt 11208 ℕ₀cn0 12442 ℤ_≥cuz 12793 [,]cicc 13309 ...cfz 13468 seqcseq 13966 ↑cexp 14026 abscabs 15200 ⇝ cli 15450 Σcsu 15652 TopOpenctopn 17384 ℂ_fldccnfld 21264 TopOnctopon 22797 Cn ccn 23111 ×_t ctx 23447 –cn→ccncf 24769

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cn 23114 df-cnp 23115 df-tx 23449 df-hmeo 23642 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-ulm 26286

This theorem is referenced by: (None)

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