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Mirrors > Home > MPE Home > Th. List > 0risefac | Structured version Visualization version GIF version |
Description: The value of the zero rising factorial at natural 𝑁. (Contributed by Scott Fenton, 17-Feb-2018.) |
Ref | Expression |
---|---|
0risefac | ⊢ (𝑁 ∈ ℕ → (0 RiseFac 𝑁) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 10429 | . . 3 ⊢ 0 ∈ ℂ | |
2 | nnnn0 11713 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | risefallfac 15236 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (0 RiseFac 𝑁) = ((-1↑𝑁) · (-0 FallFac 𝑁))) | |
4 | 1, 2, 3 | sylancr 579 | . 2 ⊢ (𝑁 ∈ ℕ → (0 RiseFac 𝑁) = ((-1↑𝑁) · (-0 FallFac 𝑁))) |
5 | neg0 10731 | . . . . 5 ⊢ -0 = 0 | |
6 | 5 | oveq1i 6984 | . . . 4 ⊢ (-0 FallFac 𝑁) = (0 FallFac 𝑁) |
7 | 0fallfac 15249 | . . . 4 ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0) | |
8 | 6, 7 | syl5eq 2819 | . . 3 ⊢ (𝑁 ∈ ℕ → (-0 FallFac 𝑁) = 0) |
9 | 8 | oveq2d 6990 | . 2 ⊢ (𝑁 ∈ ℕ → ((-1↑𝑁) · (-0 FallFac 𝑁)) = ((-1↑𝑁) · 0)) |
10 | neg1cn 11559 | . . . 4 ⊢ -1 ∈ ℂ | |
11 | expcl 13260 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (-1↑𝑁) ∈ ℂ) | |
12 | 10, 2, 11 | sylancr 579 | . . 3 ⊢ (𝑁 ∈ ℕ → (-1↑𝑁) ∈ ℂ) |
13 | 12 | mul01d 10637 | . 2 ⊢ (𝑁 ∈ ℕ → ((-1↑𝑁) · 0) = 0) |
14 | 4, 9, 13 | 3eqtrd 2811 | 1 ⊢ (𝑁 ∈ ℕ → (0 RiseFac 𝑁) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 (class class class)co 6974 ℂcc 10331 0cc0 10333 1c1 10334 · cmul 10338 -cneg 10669 ℕcn 11437 ℕ0cn0 11705 ↑cexp 13242 FallFac cfallfac 15216 RiseFac crisefac 15217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-sup 8699 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-n0 11706 df-z 11792 df-uz 12057 df-rp 12203 df-fz 12707 df-fzo 12848 df-seq 13183 df-exp 13243 df-hash 13504 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-clim 14704 df-prod 15118 df-risefac 15218 df-fallfac 15219 |
This theorem is referenced by: (None) |
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