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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 11307 | . . . 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 3604 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
7 | 0z 11388 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
8 | fzsn 12383 | . . . . . . 7 ⊢ (0 ∈ ℤ → (0...0) = {0}) | |
9 | 7, 8 | ax-mp 5 | . . . . . 6 ⊢ (0...0) = {0} |
10 | 9 | eleq2i 2693 | . . . . 5 ⊢ (𝑘 ∈ (0...0) ↔ 𝑘 ∈ {0}) |
11 | velsn 4193 | . . . . 5 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0) | |
12 | 10, 11 | bitri 264 | . . . 4 ⊢ (𝑘 ∈ (0...0) ↔ 𝑘 = 0) |
13 | oveq2 6658 | . . . . . 6 ⊢ (𝑘 = 0 → (𝑋 + 𝑘) = (𝑋 + 0)) | |
14 | 13 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑋 + 𝑘) = (𝑋 + 0)) |
15 | 6 | addid1d 10236 | . . . . . . 7 ⊢ (𝜑 → (𝑋 + 0) = 𝑋) |
16 | 15, 5 | eqeltrd 2701 | . . . . . 6 ⊢ (𝜑 → (𝑋 + 0) ∈ 𝐴) |
17 | 16 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑋 + 0) ∈ 𝐴) |
18 | 14, 17 | eqeltrd 2701 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑋 + 𝑘) ∈ 𝐴) |
19 | 12, 18 | sylan2b 492 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...0)) → (𝑋 + 𝑘) ∈ 𝐴) |
20 | 2, 3, 4, 6, 19 | fwddifnval 32270 |
. 2
⊢ (𝜑 → ((0
△ |
21 | 15 | fveq2d 6195 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝑋 + 0)) = (𝐹‘𝑋)) |
22 | 21 | oveq2d 6666 | . . . . . . . 8 ⊢ (𝜑 → (1 · (𝐹‘(𝑋 + 0))) = (1 · (𝐹‘𝑋))) |
23 | 4, 5 | ffvelrnd 6360 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℂ) |
24 | 23 | mulid2d 10058 | . . . . . . . 8 ⊢ (𝜑 → (1 · (𝐹‘𝑋)) = (𝐹‘𝑋)) |
25 | 22, 24 | eqtrd 2656 | . . . . . . 7 ⊢ (𝜑 → (1 · (𝐹‘(𝑋 + 0))) = (𝐹‘𝑋)) |
26 | 25 | oveq2d 6666 | . . . . . 6 ⊢ (𝜑 → (1 · (1 · (𝐹‘(𝑋 + 0)))) = (1 · (𝐹‘𝑋))) |
27 | 26, 24 | eqtrd 2656 | . . . . 5 ⊢ (𝜑 → (1 · (1 · (𝐹‘(𝑋 + 0)))) = (𝐹‘𝑋)) |
28 | 27, 23 | eqeltrd 2701 | . . . 4 ⊢ (𝜑 → (1 · (1 · (𝐹‘(𝑋 + 0)))) ∈ ℂ) |
29 | oveq2 6658 | . . . . . . 7 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0)) | |
30 | bcnn 13099 | . . . . . . . 8 ⊢ (0 ∈ ℕ0 → (0C0) = 1) | |
31 | 1, 30 | ax-mp 5 | . . . . . . 7 ⊢ (0C0) = 1 |
32 | 29, 31 | syl6eq 2672 | . . . . . 6 ⊢ (𝑘 = 0 → (0C𝑘) = 1) |
33 | oveq2 6658 | . . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0)) | |
34 | 0m0e0 11130 | . . . . . . . . . 10 ⊢ (0 − 0) = 0 | |
35 | 33, 34 | syl6eq 2672 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0) |
36 | 35 | oveq2d 6666 | . . . . . . . 8 ⊢ (𝑘 = 0 → (-1↑(0 − 𝑘)) = (-1↑0)) |
37 | neg1cn 11124 | . . . . . . . . 9 ⊢ -1 ∈ ℂ | |
38 | exp0 12864 | . . . . . . . . 9 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
39 | 37, 38 | ax-mp 5 | . . . . . . . 8 ⊢ (-1↑0) = 1 |
40 | 36, 39 | syl6eq 2672 | . . . . . . 7 ⊢ (𝑘 = 0 → (-1↑(0 − 𝑘)) = 1) |
41 | 13 | fveq2d 6195 | . . . . . . 7 ⊢ (𝑘 = 0 → (𝐹‘(𝑋 + 𝑘)) = (𝐹‘(𝑋 + 0))) |
42 | 40, 41 | oveq12d 6668 | . . . . . 6 ⊢ (𝑘 = 0 → ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘))) = (1 · (𝐹‘(𝑋 + 0)))) |
43 | 32, 42 | oveq12d 6668 | . . . . 5 ⊢ (𝑘 = 0 → ((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (1 · (1 · (𝐹‘(𝑋 + 0))))) |
44 | 43 | fsum1 14476 | . . . 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 695 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...0)((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (1 · (1 · (𝐹‘(𝑋 + 0))))) |
46 | 45, 27 | eqtrd 2656 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...0)((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (𝐹‘𝑋)) |
47 | 20, 46 | eqtrd 2656 |
1
⊢ (𝜑 → ((0
△ |
Colors of variables: wff setvar class |
Syntax hints:
→ wi 4 ∧ wa 384
= wceq 1483 ∈
wcel 1990 ⊆ wss 3574
{csn 4177 ⟶wf 5884
‘cfv 5888 (class class class)co 6650
ℂcc 9934 0cc0 9936
1c1 9937 + caddc 9939 · cmul 9941
− cmin 10266 -cneg 10267
ℕ0cn0 11292
ℤcz 11377 ...cfz 12326
↑cexp 12860 Ccbc 13089
Σcsu 14416
△ |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-fwddifn 32268 |
This theorem is referenced by: (None) |
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