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Mirrors > Home > MPE Home > Th. List > facnn | Structured version Visualization version GIF version |
Description: Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Ref | Expression |
---|---|
facnn | ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10350 | . . . 4 ⊢ 0 ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 0 ∈ V) |
3 | 1ex 10352 | . . . 4 ⊢ 1 ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 1 ∈ V) |
5 | df-fac 13354 | . . . 4 ⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | |
6 | nnuz 12005 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
7 | dfn2 11633 | . . . . . . . 8 ⊢ ℕ = (ℕ0 ∖ {0}) | |
8 | 6, 7 | eqtr3i 2851 | . . . . . . 7 ⊢ (ℤ≥‘1) = (ℕ0 ∖ {0}) |
9 | 8 | reseq2i 5626 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = (seq1( · , I ) ↾ (ℕ0 ∖ {0})) |
10 | 1z 11735 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
11 | seqfn 13107 | . . . . . . 7 ⊢ (1 ∈ ℤ → seq1( · , I ) Fn (ℤ≥‘1)) | |
12 | fnresdm 6233 | . . . . . . 7 ⊢ (seq1( · , I ) Fn (ℤ≥‘1) → (seq1( · , I ) ↾ (ℤ≥‘1)) = seq1( · , I )) | |
13 | 10, 11, 12 | mp2b 10 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = seq1( · , I ) |
14 | 9, 13 | eqtr3i 2851 | . . . . 5 ⊢ (seq1( · , I ) ↾ (ℕ0 ∖ {0})) = seq1( · , I ) |
15 | 14 | uneq2i 3991 | . . . 4 ⊢ ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) = ({〈0, 1〉} ∪ seq1( · , I )) |
16 | 5, 15 | eqtr4i 2852 | . . 3 ⊢ ! = ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) |
17 | id 22 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 𝑁 ∈ (ℕ0 ∖ {0})) | |
18 | 2, 4, 16, 17 | fvsnun2 6703 | . 2 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
19 | 18, 7 | eleq2s 2924 | 1 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∖ cdif 3795 ∪ cun 3796 {csn 4397 〈cop 4403 I cid 5249 ↾ cres 5344 Fn wfn 6118 ‘cfv 6123 0cc0 10252 1c1 10253 · cmul 10257 ℕcn 11350 ℕ0cn0 11618 ℤcz 11704 ℤ≥cuz 11968 seqcseq 13095 !cfa 13353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-seq 13096 df-fac 13354 |
This theorem is referenced by: fac1 13357 facp1 13358 bcval5 13398 fprodfac 15076 logfac 24746 wilthlem3 25209 |
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