etransclem43 - Mathbox for Glauco Siliprandi

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Theorem etransclem43 46542

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 46201	. . 3 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
5		fzfid 13898	. . 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 13538	. . . . . . 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 46532	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((𝑆 D^𝑛 𝐹)‘𝑖):𝑋⟶ℂ)
16	15	feqmptd 6902	. . . 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 46539	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((𝑆 D^𝑛 𝐹)‘𝑖) ∈ (𝑋–cn→ℂ))
18	16, 17	eqeltrrd 2837	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (𝑥 ∈ 𝑋 ↦ (((𝑆 D^𝑛 𝐹)‘𝑖)‘𝑥)) ∈ (𝑋–cn→ℂ))
19	4, 5, 18	fsumcncf 46143	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (0...𝑅)(((𝑆 D^𝑛 𝐹)‘𝑖)‘𝑥)) ∈ (𝑋–cn→ℂ))
20	1, 19	eqeltrid 2840	1 ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {cpr 4582 ↦ cmpt 5179 ‘cfv 6492 (class class class)co 7358 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 · cmul 11033 − cmin 11366 ℕcn 12147 ℕ₀cn0 12403 ...cfz 13425 ↑cexp 13986 Σcsu 15611 ∏cprod 15828 ↾_t crest 17342 TopOpenctopn 17343 ℂ_fldccnfld 21311 –cn→ccncf 24827 D^𝑛 cdvn 25823

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9552 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-pm 8768 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ico 13269 df-icc 13270 df-fz 13426 df-fzo 13573 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-sum 15612 df-prod 15829 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19248 df-cmn 19713 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cld 22965 df-ntr 22966 df-cls 22967 df-nei 23044 df-lp 23082 df-perf 23083 df-cn 23173 df-cnp 23174 df-haus 23261 df-tx 23508 df-hmeo 23701 df-fil 23792 df-fm 23884 df-flim 23885 df-flf 23886 df-xms 24266 df-ms 24267 df-tms 24268 df-cncf 24829 df-limc 25825 df-dv 25826 df-dvn 25827

This theorem is referenced by: etransclem46 46545

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