facnn2 - Metamath Proof Explorer

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Theorem facnn2 14195

Description: Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)

**Assertion**
Ref	Expression
facnn2	⊢ (𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁))

**Proof of Theorem facnn2**
Step	Hyp	Ref	Expression
1		elnnnn0 12430	. 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀))
2		facp1 14191	. . . 4 ⊢ ((𝑁 − 1) ∈ ℕ₀ → (!‘((𝑁 − 1) + 1)) = ((!‘(𝑁 − 1)) · ((𝑁 − 1) + 1)))
3	2	adantl 481	. . 3 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (!‘((𝑁 − 1) + 1)) = ((!‘(𝑁 − 1)) · ((𝑁 − 1) + 1)))
4		npcan1 11548	. . . . 5 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
5	4	fveq2d 6832	. . . 4 ⊢ (𝑁 ∈ ℂ → (!‘((𝑁 − 1) + 1)) = (!‘𝑁))
6	5	adantr 480	. . 3 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (!‘((𝑁 − 1) + 1)) = (!‘𝑁))
7	4	oveq2d 7368	. . . 4 ⊢ (𝑁 ∈ ℂ → ((!‘(𝑁 − 1)) · ((𝑁 − 1) + 1)) = ((!‘(𝑁 − 1)) · 𝑁))
8	7	adantr 480	. . 3 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → ((!‘(𝑁 − 1)) · ((𝑁 − 1) + 1)) = ((!‘(𝑁 − 1)) · 𝑁))
9	3, 6, 8	3eqtr3d 2774	. 2 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁))
10	1, 9	sylbi 217	1 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ‘cfv 6487 (class class class)co 7352 ℂcc 11010 1c1 11013 + caddc 11015 · cmul 11017 − cmin 11350 ℕcn 12131 ℕ₀cn0 12387 !cfa 14186

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-nn 12132 df-n0 12388 df-z 12475 df-uz 12739 df-seq 13915 df-fac 14187

This theorem is referenced by: faclbnd4lem1 14206 bcn1 14226 bcm1k 14228 hashf1 14370 dvdsfac 16243 chtublem 27155 bcmono 27221 dvnxpaek 46045

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