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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 10822 | . . 3 ⊢ 0 ∈ ℂ | |
2 | nnnn0 12094 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | risefallfac 15583 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (0 RiseFac 𝑁) = ((-1↑𝑁) · (-0 FallFac 𝑁))) | |
4 | 1, 2, 3 | sylancr 590 | . 2 ⊢ (𝑁 ∈ ℕ → (0 RiseFac 𝑁) = ((-1↑𝑁) · (-0 FallFac 𝑁))) |
5 | neg0 11121 | . . . . 5 ⊢ -0 = 0 | |
6 | 5 | oveq1i 7220 | . . . 4 ⊢ (-0 FallFac 𝑁) = (0 FallFac 𝑁) |
7 | 0fallfac 15596 | . . . 4 ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0) | |
8 | 6, 7 | syl5eq 2790 | . . 3 ⊢ (𝑁 ∈ ℕ → (-0 FallFac 𝑁) = 0) |
9 | 8 | oveq2d 7226 | . 2 ⊢ (𝑁 ∈ ℕ → ((-1↑𝑁) · (-0 FallFac 𝑁)) = ((-1↑𝑁) · 0)) |
10 | neg1cn 11941 | . . . 4 ⊢ -1 ∈ ℂ | |
11 | expcl 13650 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (-1↑𝑁) ∈ ℂ) | |
12 | 10, 2, 11 | sylancr 590 | . . 3 ⊢ (𝑁 ∈ ℕ → (-1↑𝑁) ∈ ℂ) |
13 | 12 | mul01d 11028 | . 2 ⊢ (𝑁 ∈ ℕ → ((-1↑𝑁) · 0) = 0) |
14 | 4, 9, 13 | 3eqtrd 2781 | 1 ⊢ (𝑁 ∈ ℕ → (0 RiseFac 𝑁) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 (class class class)co 7210 ℂcc 10724 0cc0 10726 1c1 10727 · cmul 10731 -cneg 11060 ℕcn 11827 ℕ0cn0 12087 ↑cexp 13632 FallFac cfallfac 15563 RiseFac crisefac 15564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-inf2 9253 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-se 5507 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-isom 6386 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-sup 9055 df-oi 9123 df-card 9552 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-3 11891 df-n0 12088 df-z 12174 df-uz 12436 df-rp 12584 df-fz 13093 df-fzo 13236 df-seq 13572 df-exp 13633 df-hash 13894 df-cj 14659 df-re 14660 df-im 14661 df-sqrt 14795 df-abs 14796 df-clim 15046 df-prod 15465 df-risefac 15565 df-fallfac 15566 |
This theorem is referenced by: (None) |
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