facnn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > facnn		Structured version Visualization version GIF version

Theorem facnn 14234

Description: Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

**Assertion**
Ref	Expression
facnn	⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁))

**Proof of Theorem facnn**
Step	Hyp	Ref	Expression
1		c0ex 11207	. . . 4 ⊢ 0 ∈ V
2	1	a1i 11	. . 3 ⊢ (𝑁 ∈ (ℕ₀ ∖ {0}) → 0 ∈ V)
3		1ex 11209	. . . 4 ⊢ 1 ∈ V
4	3	a1i 11	. . 3 ⊢ (𝑁 ∈ (ℕ₀ ∖ {0}) → 1 ∈ V)
5		df-fac 14233	. . . 4 ⊢ ! = ({⟨0, 1⟩} ∪ seq1( · , I ))
6		nnuz 12864	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
7		dfn2 12484	. . . . . . . 8 ⊢ ℕ = (ℕ₀ ∖ {0})
8	6, 7	eqtr3i 2762	. . . . . . 7 ⊢ (ℤ_≥‘1) = (ℕ₀ ∖ {0})
9	8	reseq2i 5978	. . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ_≥‘1)) = (seq1( · , I ) ↾ (ℕ₀ ∖ {0}))
10		1z 12591	. . . . . . 7 ⊢ 1 ∈ ℤ
11		seqfn 13977	. . . . . . 7 ⊢ (1 ∈ ℤ → seq1( · , I ) Fn (ℤ_≥‘1))
12		fnresdm 6669	. . . . . . 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 2762	. . . . 5 ⊢ (seq1( · , I ) ↾ (ℕ₀ ∖ {0})) = seq1( · , I )
15	14	uneq2i 4160	. . . 4 ⊢ ({⟨0, 1⟩} ∪ (seq1( · , I ) ↾ (ℕ₀ ∖ {0}))) = ({⟨0, 1⟩} ∪ seq1( · , I ))
16	5, 15	eqtr4i 2763	. . 3 ⊢ ! = ({⟨0, 1⟩} ∪ (seq1( · , I ) ↾ (ℕ₀ ∖ {0})))
17		id 22	. . 3 ⊢ (𝑁 ∈ (ℕ₀ ∖ {0}) → 𝑁 ∈ (ℕ₀ ∖ {0}))
18	2, 4, 16, 17	fvsnun2 7180	. 2 ⊢ (𝑁 ∈ (ℕ₀ ∖ {0}) → (!‘𝑁) = (seq1( · , I )‘𝑁))
19	18, 7	eleq2s 2851	1 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∖ cdif 3945 ∪ cun 3946 {csn 4628 ⟨cop 4634 I cid 5573 ↾ cres 5678 Fn wfn 6538 ‘cfv 6543 0cc0 11109 1c1 11110 · cmul 11114 ℕcn 12211 ℕ₀cn0 12471 ℤcz 12557 ℤ_≥cuz 12821 seqcseq 13965 !cfa 14232

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-seq 13966 df-fac 14233

This theorem is referenced by: fac1 14236 facp1 14237 bcval5 14277 fprodfac 15916 logfac 26108 wilthlem3 26571

W3C validator