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Theorem psercn2OLD 26393

Description: Obsolete version of psercn2 26392 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 12793	. 2 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12504	. 2 ⊢ (𝜑 → 0 ∈ ℤ)
3		pserulm.y	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (^◡abs “ (0[,]𝑀)))
4		cnvimass 6042	. . . . . . . 8 ⊢ (^◡abs “ (0[,]𝑀)) ⊆ dom abs
5		absf 15265	. . . . . . . . 9 ⊢ abs:ℂ⟶ℝ
6	5	fdmi 6674	. . . . . . . 8 ⊢ dom abs = ℂ
7	4, 6	sseqtri 3983	. . . . . . 7 ⊢ (^◡abs “ (0[,]𝑀)) ⊆ ℂ
8	3, 7	sstrdi 3947	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑆 ⊆ ℂ)
10	9	resmptd 6000	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
11		simplr 769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → 𝑦 ∈ ℂ)
12		elfznn0 13540	. . . . . . . . . 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 26380	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑦)‘𝑘) = ((𝐴‘𝑘) · (𝑦↑𝑘)))
16	11, 13, 15	syl2anc 585	. . . . . . . 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 2847	. . . . . . . . 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 7031	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
23	22	adantlr 716	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
24		expcl 14006	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑦↑𝑘) ∈ ℂ)
25	24	adantll 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝑦↑𝑘) ∈ ℂ)
26	23, 25	mulcld 11156	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ)
27	12, 26	sylan2 594	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ)
28	16, 19, 27	fsumser 15657	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑦 ∈ ℂ) → Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘)) = (seq0( + , (𝐺‘𝑦))‘𝑖))
29	28	mpteq2dva 5192	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) = (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))
30		eqid 2737	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
31	30	cnfldtopon 24730	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
33		fzfid 13900	. . . . . . . 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 7028	. . . . . . . . . . 11 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
36	21, 12, 35	syl2an 597	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑖)) → (𝐴‘𝑘) ∈ ℂ)
37	34, 34, 36	cnmptc 23610	. . . . . . . . 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 24823	. . . . . . . . . 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 24816	. . . . . . . . . 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 23614	. . . . . . . 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 24821	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
45	30	cncfcn1 24864	. . . . . . 7 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
46	44, 45	eleqtrrdi 2848	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ (ℂ–cn→ℂ))
47	29, 46	eqeltrrd 2838	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ∈ (ℂ–cn→ℂ))
48		rescncf 24850	. . . . 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 2838	. . 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 7061	. 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 26391	. 2 ⊢ (𝜑 → 𝐻(⇝_𝑢‘𝑆)𝐹)
58	1, 2, 52, 57	ulmcn 26368	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {crab 3400 ⊆ wss 3902 class class class wbr 5099 ↦ cmpt 5180 ^◡ccnv 5624 dom cdm 5625 ↾ cres 5627 “ cima 5628 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 supcsup 9347 ℂcc 11028 ℝcr 11029 0cc0 11030 + caddc 11033 · cmul 11035 ℝ^*cxr 11169 < clt 11170 ℕ₀cn0 12405 ℤ_≥cuz 12755 [,]cicc 13268 ...cfz 13427 seqcseq 13928 ↑cexp 13988 abscabs 15161 ⇝ cli 15411 Σcsu 15613 TopOpenctopn 17345 ℂ_fldccnfld 21313 TopOnctopon 22858 Cn ccn 23172 ×_t ctx 23508 –cn→ccncf 24829

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 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 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-limsup 15398 df-clim 15415 df-rlim 15416 df-sum 15614 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cn 23175 df-cnp 23176 df-tx 23510 df-hmeo 23703 df-xms 24268 df-ms 24269 df-tms 24270 df-cncf 24831 df-ulm 26346

This theorem is referenced by: (None)

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