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Mirrors > Home > MPE Home > Th. List > taylplem2 | Structured version Visualization version GIF version |
Description: Lemma for taylpfval 24952 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.) |
Ref | Expression |
---|---|
taylpfval.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
taylpfval.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
taylpfval.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
taylpfval.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
taylpfval.b | ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
Ref | Expression |
---|---|
taylplem2 | ⊢ (((𝜑 ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 11991 | . . . . . 6 ⊢ 0 ∈ ℤ | |
2 | taylpfval.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
3 | 2 | nn0zd 12084 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
4 | fzval2 12894 | . . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0...𝑁) = ((0[,]𝑁) ∩ ℤ)) | |
5 | 1, 3, 4 | sylancr 589 | . . . . 5 ⊢ (𝜑 → (0...𝑁) = ((0[,]𝑁) ∩ ℤ)) |
6 | 5 | eleq2d 2898 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ))) |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ ℂ) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ))) |
8 | 7 | biimpa 479 | . 2 ⊢ (((𝜑 ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) |
9 | taylpfval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
10 | taylpfval.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
11 | taylpfval.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
12 | 2 | orcd 869 | . . 3 ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) |
13 | taylpfval.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) | |
14 | 9, 10, 11, 2, 13 | taylplem1 24950 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
15 | 9, 10, 11, 12, 14 | taylfvallem1 24944 | . 2 ⊢ (((𝜑 ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)) ∈ ℂ) |
16 | 8, 15 | syldan 593 | 1 ⊢ (((𝜑 ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 {cpr 4568 dom cdm 5554 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 0cc0 10536 · cmul 10541 +∞cpnf 10671 − cmin 10869 / cdiv 11296 ℕ0cn0 11896 ℤcz 11980 [,]cicc 12740 ...cfz 12891 ↑cexp 13428 !cfa 13632 D𝑛 cdvn 24461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fi 8874 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-icc 12744 df-fz 12892 df-seq 13369 df-exp 13429 df-fac 13633 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-plusg 16577 df-mulr 16578 df-starv 16579 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-rest 16695 df-topn 16696 df-topgen 16716 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-lp 21743 df-perf 21744 df-cnp 21835 df-haus 21922 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-xms 22929 df-ms 22930 df-limc 24463 df-dv 24464 df-dvn 24465 |
This theorem is referenced by: taylpf 24953 dvtaylp 24957 |
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