etransclem43 - Mathbox for Glauco Siliprandi

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Theorem etransclem43 47108

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 46767	. . 3 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
5		fzfid 14040	. . 3 ⊢ (𝜑 → (0...𝑅) ∈ Fin)
6	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑆 ∈ {ℝ, ℂ})
7	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
8		etransclem43.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ)
9	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑃 ∈ ℕ)
10		etransclem43.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
11	10	adantr 486	. . . . . 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 13678	. . . . . . 7 ⊢ (𝑖 ∈ (0...𝑅) → 𝑖 ∈ ℕ₀)
14	13	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑖 ∈ ℕ₀)
15	6, 7, 9, 11, 12, 14	etransclem33 47098	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((𝑆 D^𝑛 𝐹)‘𝑖):𝑋⟶ℂ)
16	15	feqmptd 6947	. . . 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 47105	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((𝑆 D^𝑛 𝐹)‘𝑖) ∈ (𝑋–cn→ℂ))
18	16, 17	eqeltrrd 2861	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (𝑥 ∈ 𝑋 ↦ (((𝑆 D^𝑛 𝐹)‘𝑖)‘𝑥)) ∈ (𝑋–cn→ℂ))
19	4, 5, 18	fsumcncf 46709	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (0...𝑅)(((𝑆 D^𝑛 𝐹)‘𝑖)‘𝑥)) ∈ (𝑋–cn→ℂ))
20	1, 19	eqeltrid 2864	1 ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {cpr 4586 ↦ cmpt 5186 ‘cfv 6533 (class class class)co 7414 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 · cmul 11132 − cmin 11468 ℕcn 12260 ℕ₀cn0 12531 ...cfz 13564 ↑cexp 14128 Σcsu 15776 ∏cprod 15995 ↾_t crest 17508 TopOpenctopn 17509 ℂ_fldccnfld 21588 –cn→ccncf 25107 D^𝑛 cdvn 26094

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13046 df-xneg 13166 df-xadd 13167 df-xmul 13168 df-ico 13407 df-icc 13408 df-fz 13565 df-fzo 13713 df-seq 14069 df-exp 14129 df-fac 14341 df-bc 14370 df-hash 14398 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-clim 15578 df-sum 15777 df-prod 15996 df-struct 17242 df-sets 17259 df-slot 17277 df-ndx 17289 df-base 17305 df-ress 17326 df-plusg 17358 df-mulr 17359 df-starv 17360 df-sca 17361 df-vsca 17362 df-ip 17363 df-tset 17364 df-ple 17365 df-ds 17367 df-unif 17368 df-hom 17369 df-cco 17370 df-rest 17510 df-topn 17511 df-0g 17529 df-gsum 17530 df-topgen 17531 df-pt 17532 df-prds 17535 df-xrs 17591 df-qtop 17596 df-imas 17597 df-xps 17599 df-mre 17673 df-mrc 17674 df-acs 17676 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-submnd 18895 df-mulg 19194 df-cntz 19447 df-cmn 19912 df-psmet 21580 df-xmet 21581 df-met 21582 df-bl 21583 df-mopn 21584 df-fbas 21585 df-fg 21586 df-cnfld 21589 df-top 23122 df-topon 23139 df-topsp 23161 df-bases 23174 df-cld 23247 df-ntr 23248 df-cls 23249 df-nei 23326 df-lp 23364 df-perf 23365 df-cn 23455 df-cnp 23456 df-haus 23543 df-tx 23791 df-hmeo 23984 df-fil 24075 df-fm 24167 df-flim 24168 df-flf 24169 df-xms 24549 df-ms 24550 df-tms 24551 df-cncf 25109 df-limc 26096 df-dv 26097 df-dvn 26098

This theorem is used by: etransclem46 47111

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