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Theorem fac0 12925

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 9913	. 2 ⊢ 0 ∈ V
2		1ex 9914	. 2 ⊢ 1 ∈ V
3		df-fac 12923	. . 3 ⊢ ! = ({⟨0, 1⟩} ∪ seq1( · , I ))
4		nnuz 11599	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
5		dfn2 11182	. . . . . . 7 ⊢ ℕ = (ℕ₀ ∖ {0})
6	4, 5	eqtr3i 2634	. . . . . 6 ⊢ (ℤ_≥‘1) = (ℕ₀ ∖ {0})
7	6	reseq2i 5314	. . . . 5 ⊢ (seq1( · , I ) ↾ (ℤ_≥‘1)) = (seq1( · , I ) ↾ (ℕ₀ ∖ {0}))
8		1z 11284	. . . . . 6 ⊢ 1 ∈ ℤ
9		seqfn 12675	. . . . . 6 ⊢ (1 ∈ ℤ → seq1( · , I ) Fn (ℤ_≥‘1))
10		fnresdm 5914	. . . . . 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 2634	. . . 4 ⊢ (seq1( · , I ) ↾ (ℕ₀ ∖ {0})) = seq1( · , I )
13	12	uneq2i 3726	. . 3 ⊢ ({⟨0, 1⟩} ∪ (seq1( · , I ) ↾ (ℕ₀ ∖ {0}))) = ({⟨0, 1⟩} ∪ seq1( · , I ))
14	3, 13	eqtr4i 2635	. 2 ⊢ ! = ({⟨0, 1⟩} ∪ (seq1( · , I ) ↾ (ℕ₀ ∖ {0})))
15	1, 2, 14	fvsnun1 6353	1 ⊢ (!‘0) = 1

Colors of variables: wff setvar class

Syntax hints: = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 {csn 4125 ⟨cop 4131 I cid 4948 ↾ cres 5040 Fn wfn 5799 ‘cfv 5804 0cc0 9815 1c1 9816 · cmul 9820 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 seqcseq 12663 !cfa 12922

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-fac 12923

This theorem is referenced by: facp1 12927 faccl 12932 facwordi 12938 faclbnd 12939 faclbnd4lem3 12944 facubnd 12949 bcn0 12959 bcval5 12967 hashf1 13098 fprodfac 14542 fallfacfac 14615 ef0lem 14648 ege2le3 14659 eft0val 14681 prmfac1 15269 pcfac 15441 tayl0 23920 logfac 24151 advlogexp 24201 facgam 24592 logexprlim 24750 subfacval2 30423 faclim 30885 bccn0 37564 mccl 38665 dvnxpaek 38832 dvnprodlem3 38838 etransclem14 39141 etransclem24 39151 etransclem25 39152 etransclem35 39162

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