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Theorem fac0 14261

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 11232	. . . 4 ⊢ 0 ∈ V
2	1	a1i 11	. . 3 ⊢ (⊤ → 0 ∈ V)
3		1ex 11234	. . . 4 ⊢ 1 ∈ V
4	3	a1i 11	. . 3 ⊢ (⊤ → 1 ∈ V)
5		df-fac 14259	. . . 4 ⊢ ! = ({⟨0, 1⟩} ∪ seq1( · , I ))
6		nnuz 12889	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
7		dfn2 12509	. . . . . . . 8 ⊢ ℕ = (ℕ₀ ∖ {0})
8	6, 7	eqtr3i 2758	. . . . . . 7 ⊢ (ℤ_≥‘1) = (ℕ₀ ∖ {0})
9	8	reseq2i 5976	. . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ_≥‘1)) = (seq1( · , I ) ↾ (ℕ₀ ∖ {0}))
10		1z 12616	. . . . . . 7 ⊢ 1 ∈ ℤ
11		seqfn 14004	. . . . . . 7 ⊢ (1 ∈ ℤ → seq1( · , I ) Fn (ℤ_≥‘1))
12		fnresdm 6668	. . . . . . 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 2758	. . . . 5 ⊢ (seq1( · , I ) ↾ (ℕ₀ ∖ {0})) = seq1( · , I )
15	14	uneq2i 4156	. . . 4 ⊢ ({⟨0, 1⟩} ∪ (seq1( · , I ) ↾ (ℕ₀ ∖ {0}))) = ({⟨0, 1⟩} ∪ seq1( · , I ))
16	5, 15	eqtr4i 2759	. . 3 ⊢ ! = ({⟨0, 1⟩} ∪ (seq1( · , I ) ↾ (ℕ₀ ∖ {0})))
17	2, 4, 16	fvsnun1 7185	. 2 ⊢ (⊤ → (!‘0) = 1)
18	17	mptru 1541	1 ⊢ (!‘0) = 1

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 Vcvv 3470 ∖ cdif 3942 ∪ cun 3943 {csn 4624 ⟨cop 4630 I cid 5569 ↾ cres 5674 Fn wfn 6537 ‘cfv 6542 0cc0 11132 1c1 11133 · cmul 11137 ℕcn 12236 ℕ₀cn0 12496 ℤcz 12582 ℤ_≥cuz 12846 seqcseq 13992 !cfa 14258

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-z 12583 df-uz 12847 df-seq 13993 df-fac 14259

This theorem is referenced by: facp1 14263 faccl 14268 facwordi 14274 faclbnd 14275 faclbnd4lem3 14280 facubnd 14285 bcn0 14295 bcval5 14303 hashf1 14444 fprodfac 15943 fallfacfac 16015 ef0lem 16048 ege2le3 16060 eft0val 16082 prmfac1 16685 pcfac 16861 tayl0 26289 logfac 26528 advlogexp 26582 facgam 26991 logexprlim 27151 subfacval2 34791 faclim 35334 bccn0 43774 mccl 44980 dvnxpaek 45324 dvnprodlem3 45330 etransclem14 45630 etransclem24 45640 etransclem25 45641 etransclem35 45651

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