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Theorem fac0 13249

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 10218	. 2 ⊢ 0 ∈ V
2		1ex 10219	. 2 ⊢ 1 ∈ V
3		df-fac 13247	. . 3 ⊢ ! = ({⟨0, 1⟩} ∪ seq1( · , I ))
4		nnuz 11908	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
5		dfn2 11489	. . . . . . 7 ⊢ ℕ = (ℕ₀ ∖ {0})
6	4, 5	eqtr3i 2776	. . . . . 6 ⊢ (ℤ_≥‘1) = (ℕ₀ ∖ {0})
7	6	reseq2i 5540	. . . . 5 ⊢ (seq1( · , I ) ↾ (ℤ_≥‘1)) = (seq1( · , I ) ↾ (ℕ₀ ∖ {0}))
8		1z 11591	. . . . . 6 ⊢ 1 ∈ ℤ
9		seqfn 12999	. . . . . 6 ⊢ (1 ∈ ℤ → seq1( · , I ) Fn (ℤ_≥‘1))
10		fnresdm 6153	. . . . . 6 ⊢ (seq1( · , I ) Fn (ℤ_≥‘1) → (seq1( · , I ) ↾ (ℤ_≥‘1)) = seq1( · , I ))
11	8, 9, 10	mp2b 10	. . . . 5 ⊢ (seq1( · , I ) ↾ (ℤ_≥‘1)) = seq1( · , I )
12	7, 11	eqtr3i 2776	. . . 4 ⊢ (seq1( · , I ) ↾ (ℕ₀ ∖ {0})) = seq1( · , I )
13	12	uneq2i 3899	. . 3 ⊢ ({⟨0, 1⟩} ∪ (seq1( · , I ) ↾ (ℕ₀ ∖ {0}))) = ({⟨0, 1⟩} ∪ seq1( · , I ))
14	3, 13	eqtr4i 2777	. 2 ⊢ ! = ({⟨0, 1⟩} ∪ (seq1( · , I ) ↾ (ℕ₀ ∖ {0})))
15	1, 2, 14	fvsnun1 6604	1 ⊢ (!‘0) = 1

Colors of variables: wff setvar class

Syntax hints: = wceq 1624 ∈ wcel 2131 ∖ cdif 3704 ∪ cun 3705 {csn 4313 ⟨cop 4319 I cid 5165 ↾ cres 5260 Fn wfn 6036 ‘cfv 6041 0cc0 10120 1c1 10121 · cmul 10125 ℕcn 11204 ℕ₀cn0 11476 ℤcz 11561 ℤ_≥cuz 11871 seqcseq 12987 !cfa 13246

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-nn 11205 df-n0 11477 df-z 11562 df-uz 11872 df-seq 12988 df-fac 13247

This theorem is referenced by: facp1 13251 faccl 13256 facwordi 13262 faclbnd 13263 faclbnd4lem3 13268 facubnd 13273 bcn0 13283 bcval5 13291 hashf1 13425 fprodfac 14894 fallfacfac 14967 ef0lem 15000 ege2le3 15011 eft0val 15033 prmfac1 15625 pcfac 15797 tayl0 24307 logfac 24538 advlogexp 24592 facgam 24983 logexprlim 25141 subfacval2 31468 faclim 31931 bccn0 39036 mccl 40325 dvnxpaek 40652 dvnprodlem3 40658 etransclem14 40960 etransclem24 40970 etransclem25 40971 etransclem35 40981

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