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Theorem eltayl 26744

Description: Value of the Taylor series as a relation (elementhood in the domain here expresses that the series is convergent). (Contributed by Mario Carneiro, 30-Dec-2016.)

**Hypotheses**
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 D^𝑛 𝐹)‘𝑘))
taylfval.t	⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)

**Assertion**
Ref	Expression
eltayl	⊢ (𝜑 → (𝑋𝑇𝑌 ↔ (𝑋 ∈ ℂ ∧ 𝑌 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))))))

Distinct variable groups: 𝐵,𝑘 𝑘,𝐹 𝜑,𝑘 𝑘,𝑁 𝑆,𝑘 𝑘,𝑋 Allowed substitution hints: 𝐴(𝑘) 𝑇(𝑘) 𝑌(𝑘)

Proof of Theorem eltayl Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		taylfval.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		taylfval.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
3		taylfval.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
4		taylfval.n	. . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ₀ ∨ 𝑁 = +∞))
5		taylfval.b	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
6		taylfval.t	. . . 4 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)
7	1, 2, 3, 4, 5, 6	taylfval 26743	. . 3 ⊢ (𝜑 → 𝑇 = ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))))
8	7	eleq2d 2850	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ 𝑇 ↔ ⟨𝑋, 𝑌⟩ ∈ ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))))
9		df-br 5109	. . 3 ⊢ (𝑋𝑇𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑇)
10	9	bicomi 227	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ 𝑇 ↔ 𝑋𝑇𝑌)
11		oveq1 7458	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 − 𝐵) = (𝑋 − 𝐵))
12	11	oveq1d 7466	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 − 𝐵)↑𝑘) = ((𝑋 − 𝐵)↑𝑘))
13	12	oveq2d 7467	. . . . 5 ⊢ (𝑥 = 𝑋 → (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) = (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))
14	13	mpteq2dv 5205	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))) = (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘))))
15	14	oveq2d 7467	. . 3 ⊢ (𝑥 = 𝑋 → (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) = (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))))
16	15	opeliunxp2 5825	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ↔ (𝑋 ∈ ℂ ∧ 𝑌 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘))))))
17	8, 10, 16	3bitr3g 316	1 ⊢ (𝜑 → (𝑋𝑇𝑌 ↔ (𝑋 ∈ ℂ ∧ 𝑌 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ⊆ wss 3900 ∩ cin 4092 {csn 4588 {cpr 4590 ⟨cop 4840 ∪ ciun 4956 class class class wbr 5108 ↦ cmpt 5192 × cxp 5656 dom cdm 5658 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 · cmul 11244 +∞cpnf 11379 − cmin 11580 / cdiv 12013 ℕ₀cn0 12646 ℤcz 12733 [,]cicc 13519 ↑cexp 14244 !cfa 14458 ℂ_fldccnfld 21729 tsums ctsu 24500 D^𝑛 cdvn 26239 Tayl ctayl 26737

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

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 2102 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 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-map 8887 df-pm 8888 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-icc 13523 df-fz 13680 df-fzo 13829 df-seq 14185 df-exp 14245 df-fac 14459 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-plusg 17486 df-mulr 17487 df-starv 17488 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-grp 19197 df-minusg 19198 df-cntz 19582 df-cmn 20047 df-abl 20048 df-mgp 20412 df-ur 20459 df-ring 20512 df-cring 20513 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cnp 23601 df-haus 23688 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-tsms 24501 df-xms 24694 df-ms 24695 df-limc 26241 df-dv 26242 df-dvn 26243 df-tayl 26739

This theorem is used by: taylf 26745 tayl0 26746

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