Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10629 | . . . 4 ⊢ 0 ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 0 ∈ V) |
3 | 1ex 10631 | . . . 4 ⊢ 1 ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 1 ∈ V) |
5 | df-fac 13628 | . . . 4 ⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | |
6 | nnuz 12275 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
7 | dfn2 11904 | . . . . . . . 8 ⊢ ℕ = (ℕ0 ∖ {0}) | |
8 | 6, 7 | eqtr3i 2846 | . . . . . . 7 ⊢ (ℤ≥‘1) = (ℕ0 ∖ {0}) |
9 | 8 | reseq2i 5844 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = (seq1( · , I ) ↾ (ℕ0 ∖ {0})) |
10 | 1z 12006 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
11 | seqfn 13375 | . . . . . . 7 ⊢ (1 ∈ ℤ → seq1( · , I ) Fn (ℤ≥‘1)) | |
12 | fnresdm 6460 | . . . . . . 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 2846 | . . . . 5 ⊢ (seq1( · , I ) ↾ (ℕ0 ∖ {0})) = seq1( · , I ) |
15 | 14 | uneq2i 4135 | . . . 4 ⊢ ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) = ({〈0, 1〉} ∪ seq1( · , I )) |
16 | 5, 15 | eqtr4i 2847 | . . 3 ⊢ ! = ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) |
17 | id 22 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 𝑁 ∈ (ℕ0 ∖ {0})) | |
18 | 2, 4, 16, 17 | fvsnun2 6939 | . 2 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
19 | 18, 7 | eleq2s 2931 | 1 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 ∪ cun 3933 {csn 4560 〈cop 4566 I cid 5453 ↾ cres 5551 Fn wfn 6344 ‘cfv 6349 0cc0 10531 1c1 10532 · cmul 10536 ℕcn 11632 ℕ0cn0 11891 ℤcz 11975 ℤ≥cuz 12237 seqcseq 13363 !cfa 13627 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13364 df-fac 13628 |
This theorem is referenced by: fac1 13631 facp1 13632 bcval5 13672 fprodfac 15321 logfac 25178 wilthlem3 25641 |
Copyright terms: Public domain | W3C validator |