![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
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 △n 𝐹)‘𝑋) = (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 12539 | . . . 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 3996 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
7 | 0z 12622 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
8 | fzsn 13603 | . . . . . . 7 ⊢ (0 ∈ ℤ → (0...0) = {0}) | |
9 | 7, 8 | ax-mp 5 | . . . . . 6 ⊢ (0...0) = {0} |
10 | 9 | eleq2i 2831 | . . . . 5 ⊢ (𝑘 ∈ (0...0) ↔ 𝑘 ∈ {0}) |
11 | velsn 4647 | . . . . 5 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0) | |
12 | 10, 11 | bitri 275 | . . . 4 ⊢ (𝑘 ∈ (0...0) ↔ 𝑘 = 0) |
13 | oveq2 7439 | . . . . . 6 ⊢ (𝑘 = 0 → (𝑋 + 𝑘) = (𝑋 + 0)) | |
14 | 13 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑋 + 𝑘) = (𝑋 + 0)) |
15 | 6 | addridd 11459 | . . . . . . 7 ⊢ (𝜑 → (𝑋 + 0) = 𝑋) |
16 | 15, 5 | eqeltrd 2839 | . . . . . 6 ⊢ (𝜑 → (𝑋 + 0) ∈ 𝐴) |
17 | 16 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑋 + 0) ∈ 𝐴) |
18 | 14, 17 | eqeltrd 2839 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑋 + 𝑘) ∈ 𝐴) |
19 | 12, 18 | sylan2b 594 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...0)) → (𝑋 + 𝑘) ∈ 𝐴) |
20 | 2, 3, 4, 6, 19 | fwddifnval 36145 | . 2 ⊢ (𝜑 → ((0 △n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘))))) |
21 | 15 | fveq2d 6911 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝑋 + 0)) = (𝐹‘𝑋)) |
22 | 21 | oveq2d 7447 | . . . . . . . 8 ⊢ (𝜑 → (1 · (𝐹‘(𝑋 + 0))) = (1 · (𝐹‘𝑋))) |
23 | 4, 5 | ffvelcdmd 7105 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℂ) |
24 | 23 | mullidd 11277 | . . . . . . . 8 ⊢ (𝜑 → (1 · (𝐹‘𝑋)) = (𝐹‘𝑋)) |
25 | 22, 24 | eqtrd 2775 | . . . . . . 7 ⊢ (𝜑 → (1 · (𝐹‘(𝑋 + 0))) = (𝐹‘𝑋)) |
26 | 25 | oveq2d 7447 | . . . . . 6 ⊢ (𝜑 → (1 · (1 · (𝐹‘(𝑋 + 0)))) = (1 · (𝐹‘𝑋))) |
27 | 26, 24 | eqtrd 2775 | . . . . 5 ⊢ (𝜑 → (1 · (1 · (𝐹‘(𝑋 + 0)))) = (𝐹‘𝑋)) |
28 | 27, 23 | eqeltrd 2839 | . . . 4 ⊢ (𝜑 → (1 · (1 · (𝐹‘(𝑋 + 0)))) ∈ ℂ) |
29 | oveq2 7439 | . . . . . . 7 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0)) | |
30 | bcnn 14348 | . . . . . . . 8 ⊢ (0 ∈ ℕ0 → (0C0) = 1) | |
31 | 1, 30 | ax-mp 5 | . . . . . . 7 ⊢ (0C0) = 1 |
32 | 29, 31 | eqtrdi 2791 | . . . . . 6 ⊢ (𝑘 = 0 → (0C𝑘) = 1) |
33 | oveq2 7439 | . . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0)) | |
34 | 0m0e0 12384 | . . . . . . . . . 10 ⊢ (0 − 0) = 0 | |
35 | 33, 34 | eqtrdi 2791 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0) |
36 | 35 | oveq2d 7447 | . . . . . . . 8 ⊢ (𝑘 = 0 → (-1↑(0 − 𝑘)) = (-1↑0)) |
37 | neg1cn 12378 | . . . . . . . . 9 ⊢ -1 ∈ ℂ | |
38 | exp0 14103 | . . . . . . . . 9 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
39 | 37, 38 | ax-mp 5 | . . . . . . . 8 ⊢ (-1↑0) = 1 |
40 | 36, 39 | eqtrdi 2791 | . . . . . . 7 ⊢ (𝑘 = 0 → (-1↑(0 − 𝑘)) = 1) |
41 | 13 | fveq2d 6911 | . . . . . . 7 ⊢ (𝑘 = 0 → (𝐹‘(𝑋 + 𝑘)) = (𝐹‘(𝑋 + 0))) |
42 | 40, 41 | oveq12d 7449 | . . . . . 6 ⊢ (𝑘 = 0 → ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘))) = (1 · (𝐹‘(𝑋 + 0)))) |
43 | 32, 42 | oveq12d 7449 | . . . . 5 ⊢ (𝑘 = 0 → ((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (1 · (1 · (𝐹‘(𝑋 + 0))))) |
44 | 43 | fsum1 15780 | . . . 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 587 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...0)((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (1 · (1 · (𝐹‘(𝑋 + 0))))) |
46 | 45, 27 | eqtrd 2775 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...0)((0C𝑘) · ((-1↑(0 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) = (𝐹‘𝑋)) |
47 | 20, 46 | eqtrd 2775 | 1 ⊢ (𝜑 → ((0 △n 𝐹)‘𝑋) = (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 {csn 4631 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 -cneg 11491 ℕ0cn0 12524 ℤcz 12611 ...cfz 13544 ↑cexp 14099 Ccbc 14338 Σcsu 15719 △n cfwddifn 36142 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-fwddifn 36143 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |