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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fwddifn0 | Structured version Visualization version GIF version |
Description: The value of the n-iterated forward difference operator at zero is just the function value. (Contributed by Scott Fenton, 28-May-2020.) |
Ref | Expression |
---|---|
fwddifn0.1 | ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
fwddifn0.2 | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
fwddifn0.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Ref | Expression |
---|---|
fwddifn0 |
⊢ (𝜑 → ((0
△ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 11724 | . . . 4 ⊢ 0 ∈ ℕ0 | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℕ0) |
3 | fwddifn0.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
4 | fwddifn0.2 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
5 | fwddifn0.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
6 | 3, 5 | sseldd 3859 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
7 | 0z 11804 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
8 | fzsn 12765 | . . . . . . 7 ⊢ (0 ∈ ℤ → (0...0) = {0}) | |
9 | 7, 8 | ax-mp 5 | . . . . . 6 ⊢ (0...0) = {0} |
10 | 9 | eleq2i 2857 | . . . . 5 ⊢ (𝑘 ∈ (0...0) ↔ 𝑘 ∈ {0}) |
11 | velsn 4457 | . . . . 5 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0) | |
12 | 10, 11 | bitri 267 | . . . 4 ⊢ (𝑘 ∈ (0...0) ↔ 𝑘 = 0) |
13 | oveq2 6984 | . . . . . 6 ⊢ (𝑘 = 0 → (𝑋 + 𝑘) = (𝑋 + 0)) | |
14 | 13 | adantl 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑋 + 𝑘) = (𝑋 + 0)) |
15 | 6 | addid1d 10640 | . . . . . . 7 ⊢ (𝜑 → (𝑋 + 0) = 𝑋) |
16 | 15, 5 | eqeltrd 2866 | . . . . . 6 ⊢ (𝜑 → (𝑋 + 0) ∈ 𝐴) |
17 | 16 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑋 + 0) ∈ 𝐴) |
18 | 14, 17 | eqeltrd 2866 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑋 + 𝑘) ∈ 𝐴) |
19 | 12, 18 | sylan2b 584 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...0)) → (𝑋 + 𝑘) ∈ 𝐴) |
20 | 2, 3, 4, 6, 19 | fwddifnval 33151 |
. 2
⊢ (𝜑 → ((0
△ |
21 | 15 | fveq2d 6503 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝑋 + 0)) = (𝐹‘𝑋)) |
22 | 21 | oveq2d 6992 | . . . . . . . 8 ⊢ (𝜑 → (1 · (𝐹‘(𝑋 + 0))) = (1 · (𝐹‘𝑋))) |
23 | 4, 5 | ffvelrnd 6677 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℂ) |
24 | 23 | mulid2d 10458 | . . . . . . . 8 ⊢ (𝜑 → (1 · (𝐹‘𝑋)) = (𝐹‘𝑋)) |
25 | 22, 24 | eqtrd 2814 | . . . . . . 7 ⊢ (𝜑 → (1 · (𝐹‘(𝑋 + 0))) = (𝐹‘𝑋)) |
26 | 25 | oveq2d 6992 | . . . . . 6 ⊢ (𝜑 → (1 · (1 · (𝐹‘(𝑋 + 0)))) = (1 · (𝐹‘𝑋))) |
27 | 26, 24 | eqtrd 2814 | . . . . 5 ⊢ (𝜑 → (1 · (1 · (𝐹‘(𝑋 + 0)))) = (𝐹‘𝑋)) |
28 | 27, 23 | eqeltrd 2866 | . . . 4 ⊢ (𝜑 → (1 · (1 · (𝐹‘(𝑋 + 0)))) ∈ ℂ) |
29 | oveq2 6984 | . . . . . . 7 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0)) | |
30 | bcnn 13487 | . . . . . . . 8 ⊢ (0 ∈ ℕ0 → (0C0) = 1) | |
31 | 1, 30 | ax-mp 5 | . . . . . . 7 ⊢ (0C0) = 1 |
32 | 29, 31 | syl6eq 2830 | . . . . . 6 ⊢ (𝑘 = 0 → (0C𝑘) = 1) |
33 | oveq2 6984 | . . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0)) | |
34 | 0m0e0 11567 | . . . . . . . . . 10 ⊢ (0 − 0) = 0 | |
35 | 33, 34 | syl6eq 2830 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0) |
36 | 35 | oveq2d 6992 | . . . . . . . 8 ⊢ (𝑘 = 0 → (-1↑(0 − 𝑘)) = (-1↑0)) |
37 | neg1cn 11561 | . . . . . . . . 9 ⊢ -1 ∈ ℂ | |
38 | exp0 13248 | . . . . . . . . 9 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
39 | 37, 38 | ax-mp 5 | . . . . . . . 8 ⊢ (-1↑0) = 1 |
40 | 36, 39 | syl6eq 2830 | . . . . . . 7 ⊢ (𝑘 = 0 → (-1↑(0 − 𝑘)) = 1) |
41 | 13 | fveq2d 6503 | . . . . . . 7 ⊢ (𝑘 = 0 → (𝐹‘(𝑋 + 𝑘)) = (𝐹‘(𝑋 + 0))) |
42 | 40, 41 | oveq12d 6994 | . . . . . 6 ⊢ (𝑘 = 0 → ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘))) = (1 · (𝐹‘(𝑋 + 0)))) |
43 | 32, 42 | oveq12d 6994 | . . . . 5 ⊢ (𝑘 = 0 → ((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (1 · (1 · (𝐹‘(𝑋 + 0))))) |
44 | 43 | fsum1 14962 | . . . 4 ⊢ ((0 ∈ ℤ ∧ (1 · (1 · (𝐹‘(𝑋 + 0)))) ∈ ℂ) → Σ𝑘 ∈ (0...0)((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (1 · (1 · (𝐹‘(𝑋 + 0))))) |
45 | 7, 28, 44 | sylancr 578 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...0)((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (1 · (1 · (𝐹‘(𝑋 + 0))))) |
46 | 45, 27 | eqtrd 2814 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...0)((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (𝐹‘𝑋)) |
47 | 20, 46 | eqtrd 2814 |
1
⊢ (𝜑 → ((0
△ |
Colors of variables: wff setvar class |
Syntax hints:
→ wi 4 ∧ wa 387
= wceq 1507 ∈
wcel 2050 ⊆ wss 3829
{csn 4441 ⟶wf 6184
‘cfv 6188 (class class class)co 6976
ℂcc 10333 0cc0 10335
1c1 10336 + caddc 10338 · cmul 10340
− cmin 10670 -cneg 10671
ℕ0cn0 11707
ℤcz 11793 ...cfz 12708
↑cexp 13244 Ccbc 13477
Σcsu 14903
△ |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-pm 8209 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-oi 8769 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-z 11794 df-uz 12059 df-rp 12205 df-fz 12709 df-fzo 12850 df-seq 13185 df-exp 13245 df-fac 13449 df-bc 13478 df-hash 13506 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-clim 14706 df-sum 14904 df-fwddifn 33149 |
This theorem is referenced by: (None) |
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