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Mirrors > Home > MPE Home > Th. List > taylfvallem | Structured version Visualization version GIF version |
Description: Lemma for taylfval 25529. (Contributed by Mario Carneiro, 30-Dec-2016.) |
Ref | Expression |
---|---|
taylfval.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
taylfval.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
taylfval.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
taylfval.n | ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) |
taylfval.b | ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
Ref | Expression |
---|---|
taylfvallem | ⊢ ((𝜑 ∧ 𝑋 ∈ ℂ) → (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))) ⊆ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 20612 | . 2 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnring 20631 | . . 3 ⊢ ℂfld ∈ Ring | |
3 | ringcmn 19831 | . . 3 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd) | |
4 | 2, 3 | mp1i 13 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ ℂ) → ℂfld ∈ CMnd) |
5 | cnfldtps 23952 | . . 3 ⊢ ℂfld ∈ TopSp | |
6 | 5 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ ℂ) → ℂfld ∈ TopSp) |
7 | ovex 7305 | . . . 4 ⊢ (0[,]𝑁) ∈ V | |
8 | 7 | inex1 5245 | . . 3 ⊢ ((0[,]𝑁) ∩ ℤ) ∈ V |
9 | 8 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ ℂ) → ((0[,]𝑁) ∩ ℤ) ∈ V) |
10 | taylfval.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
11 | taylfval.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
12 | taylfval.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
13 | taylfval.n | . . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) | |
14 | taylfval.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) | |
15 | 10, 11, 12, 13, 14 | taylfvallem1 25527 | . . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)) ∈ ℂ) |
16 | 15 | fmpttd 6986 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ ℂ) → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘))):((0[,]𝑁) ∩ ℤ)⟶ℂ) |
17 | 1, 4, 6, 9, 16 | tsmscl 23297 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ ℂ) → (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))) ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 844 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ∩ cin 3891 ⊆ wss 3892 {cpr 4569 ↦ cmpt 5162 dom cdm 5590 ⟶wf 6428 ‘cfv 6432 (class class class)co 7272 ℂcc 10880 ℝcr 10881 0cc0 10882 · cmul 10887 +∞cpnf 11017 − cmin 11216 / cdiv 11643 ℕ0cn0 12244 ℤcz 12330 [,]cicc 13093 ↑cexp 13793 !cfa 13998 CMndccmn 19397 Ringcrg 19794 ℂfldccnfld 20608 TopSpctps 22092 tsums ctsu 23288 D𝑛 cdvn 25039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-inf2 9387 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 ax-addf 10961 ax-mulf 10962 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-supp 7970 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-er 8490 df-map 8609 df-pm 8610 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-fsupp 9117 df-fi 9158 df-sup 9189 df-inf 9190 df-oi 9257 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-uz 12594 df-q 12700 df-rp 12742 df-xneg 12859 df-xadd 12860 df-xmul 12861 df-icc 13097 df-fz 13251 df-fzo 13394 df-seq 13733 df-exp 13794 df-fac 13999 df-hash 14056 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-plusg 16986 df-mulr 16987 df-starv 16988 df-tset 16992 df-ple 16993 df-ds 16995 df-unif 16996 df-rest 17144 df-topn 17145 df-0g 17163 df-gsum 17164 df-topgen 17165 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-grp 18591 df-minusg 18592 df-cntz 18934 df-cmn 19399 df-abl 19400 df-mgp 19732 df-ur 19749 df-ring 19796 df-cring 19797 df-psmet 20600 df-xmet 20601 df-met 20602 df-bl 20603 df-mopn 20604 df-fbas 20605 df-fg 20606 df-cnfld 20609 df-top 22054 df-topon 22071 df-topsp 22093 df-bases 22107 df-cld 22181 df-ntr 22182 df-cls 22183 df-nei 22260 df-lp 22298 df-perf 22299 df-cnp 22390 df-haus 22477 df-fil 23008 df-fm 23100 df-flim 23101 df-flf 23102 df-tsms 23289 df-xms 23484 df-ms 23485 df-limc 25041 df-dv 25042 df-dvn 25043 |
This theorem is referenced by: taylfval 25529 taylf 25531 |
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