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Mirrors > Home > MPE Home > Th. List > fac0 | Structured version Visualization version GIF version |
Description: The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Ref | Expression |
---|---|
fac0 | ⊢ (!‘0) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10624 | . . . 4 ⊢ 0 ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → 0 ∈ V) |
3 | 1ex 10626 | . . . 4 ⊢ 1 ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → 1 ∈ V) |
5 | df-fac 13624 | . . . 4 ⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | |
6 | nnuz 12270 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
7 | dfn2 11899 | . . . . . . . 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 12001 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
11 | seqfn 13371 | . . . . . . 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 | 2, 4, 16 | fvsnun1 6937 | . 2 ⊢ (⊤ → (!‘0) = 1) |
18 | 17 | mptru 1535 | 1 ⊢ (!‘0) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 Vcvv 3495 ∖ cdif 3932 ∪ cun 3933 {csn 4559 〈cop 4565 I cid 5453 ↾ cres 5551 Fn wfn 6344 ‘cfv 6349 0cc0 10526 1c1 10527 · cmul 10531 ℕcn 11627 ℕ0cn0 11886 ℤcz 11970 ℤ≥cuz 12232 seqcseq 13359 !cfa 13623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 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 3497 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 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-n0 11887 df-z 11971 df-uz 12233 df-seq 13360 df-fac 13624 |
This theorem is referenced by: facp1 13628 faccl 13633 facwordi 13639 faclbnd 13640 faclbnd4lem3 13645 facubnd 13650 bcn0 13660 bcval5 13668 hashf1 13805 fprodfac 15317 fallfacfac 15389 ef0lem 15422 ege2le3 15433 eft0val 15455 prmfac1 16053 pcfac 16225 tayl0 24879 logfac 25111 advlogexp 25165 facgam 25571 logexprlim 25729 subfacval2 32332 faclim 32876 bccn0 40555 mccl 41759 dvnxpaek 42107 dvnprodlem3 42113 etransclem14 42414 etransclem24 42424 etransclem25 42425 etransclem35 42435 |
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