etransclem43 - Mathbox for Glauco Siliprandi

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Theorem etransclem43 47429

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 47088	. . 3 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
5		fzfid 14129	. . 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 13767	. . . . . . 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 47419	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((𝑆 D^𝑛 𝐹)‘𝑖):𝑋⟶ℂ)
16	15	feqmptd 6964	. . . 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 47426	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((𝑆 D^𝑛 𝐹)‘𝑖) ∈ (𝑋–cn→ℂ))
18	16, 17	eqeltrrd 2865	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (𝑥 ∈ 𝑋 ↦ (((𝑆 D^𝑛 𝐹)‘𝑖)‘𝑥)) ∈ (𝑋–cn→ℂ))
19	4, 5, 18	fsumcncf 47030	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (0...𝑅)(((𝑆 D^𝑛 𝐹)‘𝑖)‘𝑥)) ∈ (𝑋–cn→ℂ))
20	1, 19	eqeltrid 2868	1 ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2146 {cpr 4590 ↦ cmpt 5189 ‘cfv 6547 (class class class)co 7430 ℂcc 11211 ℝcr 11212 0cc0 11213 1c1 11214 · cmul 11218 − cmin 11554 ℕcn 12348 ℕ₀cn0 12619 ...cfz 13652 ↑cexp 14217 Σcsu 15868 ∏cprod 16087 ↾_t crest 17606 TopOpenctopn 17607 ℂ_fldccnfld 21693 –cn→ccncf 25212 D^𝑛 cdvn 26199

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-inf2 9655 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-pre-sup 11291 ax-addf 11292

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 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 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7703 df-om 7888 df-1st 8011 df-2nd 8012 df-supp 8191 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-2o 8491 df-er 8731 df-map 8863 df-pm 8864 df-ixp 8940 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-fsupp 9368 df-fi 9416 df-sup 9447 df-inf 9448 df-oi 9517 df-card 10033 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-div 11987 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-7 12423 df-8 12424 df-9 12425 df-n0 12620 df-z 12707 df-dec 12828 df-uz 12979 df-q 13089 df-rp 13134 df-xneg 13254 df-xadd 13255 df-xmul 13256 df-ico 13495 df-icc 13496 df-fz 13653 df-fzo 13802 df-seq 14158 df-exp 14218 df-fac 14431 df-bc 14460 df-hash 14488 df-cj 15281 df-re 15282 df-im 15283 df-sqrt 15417 df-abs 15418 df-clim 15670 df-sum 15869 df-prod 16088 df-struct 17340 df-sets 17357 df-slot 17375 df-ndx 17387 df-base 17403 df-ress 17424 df-plusg 17456 df-mulr 17457 df-starv 17458 df-sca 17459 df-vsca 17460 df-ip 17461 df-tset 17462 df-ple 17463 df-ds 17465 df-unif 17466 df-hom 17467 df-cco 17468 df-rest 17608 df-topn 17609 df-0g 17627 df-gsum 17628 df-topgen 17629 df-pt 17630 df-prds 17633 df-xrs 17689 df-qtop 17694 df-imas 17695 df-xps 17697 df-mre 17771 df-mrc 17772 df-acs 17774 df-mgm 18831 df-sgrp 18923 df-mnd 18939 df-submnd 18994 df-mulg 19293 df-cntz 19546 df-cmn 20011 df-psmet 21685 df-xmet 21686 df-met 21687 df-bl 21688 df-mopn 21689 df-fbas 21690 df-fg 21691 df-cnfld 21694 df-top 23227 df-topon 23244 df-topsp 23266 df-bases 23279 df-cld 23352 df-ntr 23353 df-cls 23354 df-nei 23431 df-lp 23469 df-perf 23470 df-cn 23560 df-cnp 23561 df-haus 23648 df-tx 23896 df-hmeo 24089 df-fil 24180 df-fm 24272 df-flim 24273 df-flf 24274 df-xms 24654 df-ms 24655 df-tms 24656 df-cncf 25214 df-limc 26201 df-dv 26202 df-dvn 26203

This theorem is used by: etransclem46 47432

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