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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 11913 | . . . 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 3968 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
7 | 0z 11993 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
8 | fzsn 12950 | . . . . . . 7 ⊢ (0 ∈ ℤ → (0...0) = {0}) | |
9 | 7, 8 | ax-mp 5 | . . . . . 6 ⊢ (0...0) = {0} |
10 | 9 | eleq2i 2904 | . . . . 5 ⊢ (𝑘 ∈ (0...0) ↔ 𝑘 ∈ {0}) |
11 | velsn 4583 | . . . . 5 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0) | |
12 | 10, 11 | bitri 277 | . . . 4 ⊢ (𝑘 ∈ (0...0) ↔ 𝑘 = 0) |
13 | oveq2 7164 | . . . . . 6 ⊢ (𝑘 = 0 → (𝑋 + 𝑘) = (𝑋 + 0)) | |
14 | 13 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑋 + 𝑘) = (𝑋 + 0)) |
15 | 6 | addid1d 10840 | . . . . . . 7 ⊢ (𝜑 → (𝑋 + 0) = 𝑋) |
16 | 15, 5 | eqeltrd 2913 | . . . . . 6 ⊢ (𝜑 → (𝑋 + 0) ∈ 𝐴) |
17 | 16 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑋 + 0) ∈ 𝐴) |
18 | 14, 17 | eqeltrd 2913 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑋 + 𝑘) ∈ 𝐴) |
19 | 12, 18 | sylan2b 595 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...0)) → (𝑋 + 𝑘) ∈ 𝐴) |
20 | 2, 3, 4, 6, 19 | fwddifnval 33624 |
. 2
⊢ (𝜑 → ((0
△ |
21 | 15 | fveq2d 6674 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝑋 + 0)) = (𝐹‘𝑋)) |
22 | 21 | oveq2d 7172 | . . . . . . . 8 ⊢ (𝜑 → (1 · (𝐹‘(𝑋 + 0))) = (1 · (𝐹‘𝑋))) |
23 | 4, 5 | ffvelrnd 6852 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℂ) |
24 | 23 | mulid2d 10659 | . . . . . . . 8 ⊢ (𝜑 → (1 · (𝐹‘𝑋)) = (𝐹‘𝑋)) |
25 | 22, 24 | eqtrd 2856 | . . . . . . 7 ⊢ (𝜑 → (1 · (𝐹‘(𝑋 + 0))) = (𝐹‘𝑋)) |
26 | 25 | oveq2d 7172 | . . . . . 6 ⊢ (𝜑 → (1 · (1 · (𝐹‘(𝑋 + 0)))) = (1 · (𝐹‘𝑋))) |
27 | 26, 24 | eqtrd 2856 | . . . . 5 ⊢ (𝜑 → (1 · (1 · (𝐹‘(𝑋 + 0)))) = (𝐹‘𝑋)) |
28 | 27, 23 | eqeltrd 2913 | . . . 4 ⊢ (𝜑 → (1 · (1 · (𝐹‘(𝑋 + 0)))) ∈ ℂ) |
29 | oveq2 7164 | . . . . . . 7 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0)) | |
30 | bcnn 13673 | . . . . . . . 8 ⊢ (0 ∈ ℕ0 → (0C0) = 1) | |
31 | 1, 30 | ax-mp 5 | . . . . . . 7 ⊢ (0C0) = 1 |
32 | 29, 31 | syl6eq 2872 | . . . . . 6 ⊢ (𝑘 = 0 → (0C𝑘) = 1) |
33 | oveq2 7164 | . . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0)) | |
34 | 0m0e0 11758 | . . . . . . . . . 10 ⊢ (0 − 0) = 0 | |
35 | 33, 34 | syl6eq 2872 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0) |
36 | 35 | oveq2d 7172 | . . . . . . . 8 ⊢ (𝑘 = 0 → (-1↑(0 − 𝑘)) = (-1↑0)) |
37 | neg1cn 11752 | . . . . . . . . 9 ⊢ -1 ∈ ℂ | |
38 | exp0 13434 | . . . . . . . . 9 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
39 | 37, 38 | ax-mp 5 | . . . . . . . 8 ⊢ (-1↑0) = 1 |
40 | 36, 39 | syl6eq 2872 | . . . . . . 7 ⊢ (𝑘 = 0 → (-1↑(0 − 𝑘)) = 1) |
41 | 13 | fveq2d 6674 | . . . . . . 7 ⊢ (𝑘 = 0 → (𝐹‘(𝑋 + 𝑘)) = (𝐹‘(𝑋 + 0))) |
42 | 40, 41 | oveq12d 7174 | . . . . . 6 ⊢ (𝑘 = 0 → ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘))) = (1 · (𝐹‘(𝑋 + 0)))) |
43 | 32, 42 | oveq12d 7174 | . . . . 5 ⊢ (𝑘 = 0 → ((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (1 · (1 · (𝐹‘(𝑋 + 0))))) |
44 | 43 | fsum1 15102 | . . . 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 589 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...0)((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (1 · (1 · (𝐹‘(𝑋 + 0))))) |
46 | 45, 27 | eqtrd 2856 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...0)((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (𝐹‘𝑋)) |
47 | 20, 46 | eqtrd 2856 |
1
⊢ (𝜑 → ((0
△ |
Colors of variables: wff setvar class |
Syntax hints:
→ wi 4 ∧ wa 398
= wceq 1537 ∈
wcel 2114 ⊆ wss 3936
{csn 4567 ⟶wf 6351
‘cfv 6355 (class class class)co 7156
ℂcc 10535 0cc0 10537
1c1 10538 + caddc 10540 · cmul 10542
− cmin 10870 -cneg 10871
ℕ0cn0 11898
ℤcz 11982 ...cfz 12893
↑cexp 13430 Ccbc 13663
Σcsu 15042
△ |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-fwddifn 33622 |
This theorem is referenced by: (None) |
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