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Theorem facnnt 4870

Description: Value of the factorial function for positive integers.

**Assertion**
Ref	Expression
facnnt	⊢ (A ∈ ℕ → (! ‘A) = (( · seq(I ↾ ℕ)) ‘A))

**Proof of Theorem facnnt**
Step	Hyp	Ref	Expression
1		df-fac 4869	. . . . . 6 ⊢ ! = (( · seq(I ↾ ℕ)) ∪ {⟨0, 1⟩})
2		reseq1 2575	. . . . . 6 ⊢ (! = (( · seq(I ↾ ℕ)) ∪ {⟨0, 1⟩}) → (! ↾ ℕ) = ((( · seq(I ↾ ℕ)) ∪ {⟨0, 1⟩}) ↾ ℕ))
3	1, 2	ax-mp 6	. . . . 5 ⊢ (! ↾ ℕ) = ((( · seq(I ↾ ℕ)) ∪ {⟨0, 1⟩}) ↾ ℕ)
4		resundir 2583	. . . . . 6 ⊢ ((( · seq(I ↾ ℕ)) ∪ {⟨0, 1⟩}) ↾ ℕ) = ((( · seq(I ↾ ℕ)) ↾ ℕ) ∪ ({⟨0, 1⟩} ↾ ℕ))
5		incom 1636	. . . . . . . . 9 ⊢ ({0} ∩ ℕ) = (ℕ ∩ {0})
6		0nnn 4443	. . . . . . . . . 10 ⊢ ¬ 0 ∈ ℕ
7		disjsn 1836	. . . . . . . . . 10 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
8	6, 7	mpbir 165	. . . . . . . . 9 ⊢ (ℕ ∩ {0}) = ∅
9	5, 8	eqtr 1119	. . . . . . . 8 ⊢ ({0} ∩ ℕ) = ∅
10		0nn0 4546	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
11	10	elisseti 1355	. . . . . . . . . . 11 ⊢ 0 ∈ V
12		1nn 4432	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
13	12	elisseti 1355	. . . . . . . . . . 11 ⊢ 1 ∈ V
14	11, 13	f1osn 2827	. . . . . . . . . 10 ⊢ {⟨0, 1⟩}:{0}–1-1-onto→{1}
15		f1ofn 2801	. . . . . . . . . 10 ⊢ ({⟨0, 1⟩}:{0}–1-1-onto→{1} → {⟨0, 1⟩} Fn {0})
16	14, 15	ax-mp 6	. . . . . . . . 9 ⊢ {⟨0, 1⟩} Fn {0}
17		fnresdisj 2732	. . . . . . . . 9 ⊢ ({⟨0, 1⟩} Fn {0} → (({0} ∩ ℕ) = ∅ ↔ ({⟨0, 1⟩} ↾ ℕ) = ∅))
18	16, 17	ax-mp 6	. . . . . . . 8 ⊢ (({0} ∩ ℕ) = ∅ ↔ ({⟨0, 1⟩} ↾ ℕ) = ∅)
19	9, 18	mpbi 164	. . . . . . 7 ⊢ ({⟨0, 1⟩} ↾ ℕ) = ∅
20	19	uneq2i 1608	. . . . . 6 ⊢ ((( · seq(I ↾ ℕ)) ↾ ℕ) ∪ ({⟨0, 1⟩} ↾ ℕ)) = ((( · seq(I ↾ ℕ)) ↾ ℕ) ∪ ∅)
21	4, 20	eqtr 1119	. . . . 5 ⊢ ((( · seq(I ↾ ℕ)) ∪ {⟨0, 1⟩}) ↾ ℕ) = ((( · seq(I ↾ ℕ)) ↾ ℕ) ∪ ∅)
22		un0 1721	. . . . 5 ⊢ ((( · seq(I ↾ ℕ)) ↾ ℕ) ∪ ∅) = (( · seq(I ↾ ℕ)) ↾ ℕ)
23	3, 21, 22	3eqtr 1123	. . . 4 ⊢ (! ↾ ℕ) = (( · seq(I ↾ ℕ)) ↾ ℕ)
24	23	fveq1i 2833	. . 3 ⊢ ((! ↾ ℕ) ‘A) = ((( · seq(I ↾ ℕ)) ↾ ℕ) ‘A)
25	24	a1i 7	. 2 ⊢ (A ∈ ℕ → ((! ↾ ℕ) ‘A) = ((( · seq(I ↾ ℕ)) ↾ ℕ) ‘A))
26		fvres 2840	. 2 ⊢ (A ∈ ℕ → ((! ↾ ℕ) ‘A) = (! ‘A))
27		fvres 2840	. 2 ⊢ (A ∈ ℕ → ((( · seq(I ↾ ℕ)) ↾ ℕ) ‘A) = (( · seq(I ↾ ℕ)) ‘A))
28	25, 26, 27	3eqtr3d 1133	1 ⊢ (A ∈ ℕ → (! ‘A) = (( · seq(I ↾ ℕ)) ‘A))

Colors of variables: wff set class

Syntax hints: ¬ wn 1 → wi 2 ↔ wb 127 = wceq 1091 ∈ wcel 1092 ∪ cun 1485 ∩ cin 1486 ∅c0 1707 {csn 1808 ⟨cop 1810 Icid 2057 ↾ cres 2412 Fn wfn 2417 –1-1-onto→wf1o 2421 ‘cfv 2422 (class class class)co 3001 0cc0 4028 1c1 4029 · cmulc 4032 ℕcn 4093 ℕ₀cn0 4094 seqcseq 4660 !cfa 4868

This theorem is referenced by: fac1 4872 facp1t 4873

This theorem was proved from axioms: ax-1 3 ax-2 4 ax-3 5 ax-mp 6 ax-4 673 ax-5 674 ax-6 675 ax-7 676 ax-gen 677 ax-8 798 ax-9 799 ax-10 800 ax-11 801 ax-12 802 ax-13 804 ax-14 805 ax-16 922 ax-17 925 ax-ext 1074 ax-rep 1075 ax-un 1076 ax-pow 1077 ax-reg 1078 ax-inf 1079

This theorem depends on definitions: df-bi 128 df-or 197 df-an 198 df-3or 582 df-3an 583 df-ex 679 df-sb 853 df-eu 1009 df-mo 1010 df-clab 1093 df-cleq 1097 df-clel 1099 df-ne 1192 df-ral 1205 df-rex 1206 df-reu 1207 df-rab 1208 df-v 1349 df-sbc 1441 df-dif 1489 df-un 1490 df-in 1491 df-ss 1492 df-pss 1494 df-nul 1708 df-if 1777 df-pw 1799 df-sn 1811 df-pr 1812 df-tp 1814 df-op 1815 df-uni 1920 df-int 1966 df-iun 1996 df-tr 2042 df-br 2063 df-opab 2098 df-eprel 2122 df-id 2125 df-po 2128 df-so 2138 df-fr 2169 df-we 2186 df-ord 2202 df-on 2203 df-lim 2204 df-suc 2205 df-om 2373 df-xp 2424 df-rel 2425 df-cnv 2426 df-co 2427 df-dm 2428 df-rn 2429 df-res 2430 df-ima 2431 df-fun 2432 df-fn 2433 df-f 2434 df-f1 2435 df-fo 2436 df-f1o 2437 df-fv 2438 df-rdg 2970 df-opr 3003 df-oprab 3004 df-1o 3104 df-oadd 3106 df-omul 3107 df-er 3200 df-ec 3202 df-qs 3205 df-ni 3794 df-pli 3795 &nsp;df-mi 3796 df-lti 3797 df-plpq 3829 df-mpq 3830 df-enq 3831 df-nq 3832 df-plq 3833 df-mq 3834 df-rq 3835 df-ltq 3836 df-1q 3837 df-np 3880 df-1p 3881 df-plp 3882 df-mp 3883 df-ltp 3884 df-plpr 3958 df-mpr 3959 df-enr 3960 df-nr 3961 df-plr 3962 df-mr 3963 df-ltr 3964 df-0r 3965 df-1r 3966 df-m1r 3967 df-c 4034 df-0 4035 df-1 4036 df-r 4038 df-plus 4039 df-mul 4040 df-lt 4041 df-sub 4133 df-neg 4135 df-le 4277 df-n 4423 df-n0 4535 df-fac 4869

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