etransclem43 - Mathbox for Glauco Siliprandi

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Theorem etransclem43 46268

Description: 𝐺 is a continuous function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem43.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
etransclem43.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
etransclem43.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem43.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem43.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
etransclem43.g	⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (0...𝑅)(((𝑆 D^𝑛 𝐹)‘𝑖)‘𝑥))

**Assertion**
Ref	Expression
etransclem43	⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ))

Distinct variable groups: 𝑥,𝐹 𝑗,𝑀,𝑥 𝑃,𝑗,𝑥 𝑅,𝑖,𝑗,𝑥 𝑆,𝑗,𝑥 𝑖,𝑋,𝑗,𝑥 𝜑,𝑖,𝑗,𝑥 Allowed substitution hints: 𝑃(𝑖) 𝑆(𝑖) 𝐹(𝑖,𝑗) 𝐺(𝑥,𝑖,𝑗) 𝑀(𝑖)

**Proof of Theorem etransclem43**
Step	Hyp	Ref	Expression
1		etransclem43.g	. 2 ⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (0...𝑅)(((𝑆 D^𝑛 𝐹)‘𝑖)‘𝑥))
2		etransclem43.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
3		etransclem43.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
4	2, 3	dvdmsscn 45927	. . 3 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
5		fzfid 13880	. . 3 ⊢ (𝜑 → (0...𝑅) ∈ Fin)
6	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑆 ∈ {ℝ, ℂ})
7	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
8		etransclem43.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ)
9	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑃 ∈ ℕ)
10		etransclem43.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑀 ∈ ℕ₀)
12		etransclem43.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
13		elfznn0 13523	. . . . . . 7 ⊢ (𝑖 ∈ (0...𝑅) → 𝑖 ∈ ℕ₀)
14	13	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑖 ∈ ℕ₀)
15	6, 7, 9, 11, 12, 14	etransclem33 46258	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((𝑆 D^𝑛 𝐹)‘𝑖):𝑋⟶ℂ)
16	15	feqmptd 6891	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((𝑆 D^𝑛 𝐹)‘𝑖) = (𝑥 ∈ 𝑋 ↦ (((𝑆 D^𝑛 𝐹)‘𝑖)‘𝑥)))
17	6, 7, 9, 11, 12, 14	etransclem40 46265	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((𝑆 D^𝑛 𝐹)‘𝑖) ∈ (𝑋–cn→ℂ))
18	16, 17	eqeltrrd 2829	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (𝑥 ∈ 𝑋 ↦ (((𝑆 D^𝑛 𝐹)‘𝑖)‘𝑥)) ∈ (𝑋–cn→ℂ))
19	4, 5, 18	fsumcncf 45869	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (0...𝑅)(((𝑆 D^𝑛 𝐹)‘𝑖)‘𝑥)) ∈ (𝑋–cn→ℂ))
20	1, 19	eqeltrid 2832	1 ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {cpr 4579 ↦ cmpt 5173 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 · cmul 11014 − cmin 11347 ℕcn 12128 ℕ₀cn0 12384 ...cfz 13410 ↑cexp 13968 Σcsu 15593 ∏cprod 15810 ↾_t crest 17324 TopOpenctopn 17325 ℂ_fldccnfld 21261 –cn→ccncf 24767 D^𝑛 cdvn 25763

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-prod 15811 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766 df-dvn 25767

This theorem is referenced by: etransclem46 46271

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