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Theorem fac0 13630

Description: The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

**Assertion**
Ref	Expression
fac0	⊢ (!‘0) = 1

**Proof of Theorem fac0**
Step	Hyp	Ref	Expression
1		c0ex 10629	. . . 4 ⊢ 0 ∈ V
2	1	a1i 11	. . 3 ⊢ (⊤ → 0 ∈ V)
3		1ex 10631	. . . 4 ⊢ 1 ∈ V
4	3	a1i 11	. . 3 ⊢ (⊤ → 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	2, 4, 16	fvsnun1 6938	. 2 ⊢ (⊤ → (!‘0) = 1)
18	17	mptru 1540	1 ⊢ (!‘0) = 1

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ 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: facp1 13632 faccl 13637 facwordi 13643 faclbnd 13644 faclbnd4lem3 13649 facubnd 13654 bcn0 13664 bcval5 13672 hashf1 13809 fprodfac 15321 fallfacfac 15393 ef0lem 15426 ege2le3 15437 eft0val 15459 prmfac1 16057 pcfac 16229 tayl0 24944 logfac 25178 advlogexp 25232 facgam 25637 logexprlim 25795 subfacval2 32429 faclim 32973 bccn0 40668 mccl 41872 dvnxpaek 42220 dvnprodlem3 42226 etransclem14 42527 etransclem24 42537 etransclem25 42538 etransclem35 42548

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