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Theorem faclbnd 14004

Description: A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)

**Assertion**
Ref	Expression
faclbnd	⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))

Proof of Theorem faclbnd Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn0 12235	. 2 ⊢ (𝑀 ∈ ℕ₀ ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
2		oveq1 7282	. . . . . . . 8 ⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
3	2	oveq2d 7291	. . . . . . 7 ⊢ (𝑗 = 0 → (𝑀↑(𝑗 + 1)) = (𝑀↑(0 + 1)))
4		fveq2 6774	. . . . . . . 8 ⊢ (𝑗 = 0 → (!‘𝑗) = (!‘0))
5	4	oveq2d 7291	. . . . . . 7 ⊢ (𝑗 = 0 → ((𝑀↑𝑀) · (!‘𝑗)) = ((𝑀↑𝑀) · (!‘0)))
6	3, 5	breq12d 5087	. . . . . 6 ⊢ (𝑗 = 0 → ((𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗)) ↔ (𝑀↑(0 + 1)) ≤ ((𝑀↑𝑀) · (!‘0))))
7	6	imbi2d 341	. . . . 5 ⊢ (𝑗 = 0 → ((𝑀 ∈ ℕ → (𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗))) ↔ (𝑀 ∈ ℕ → (𝑀↑(0 + 1)) ≤ ((𝑀↑𝑀) · (!‘0)))))
8		oveq1 7282	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
9	8	oveq2d 7291	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑀↑(𝑗 + 1)) = (𝑀↑(𝑘 + 1)))
10		fveq2 6774	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (!‘𝑗) = (!‘𝑘))
11	10	oveq2d 7291	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝑀↑𝑀) · (!‘𝑗)) = ((𝑀↑𝑀) · (!‘𝑘)))
12	9, 11	breq12d 5087	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗)) ↔ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))))
13	12	imbi2d 341	. . . . 5 ⊢ (𝑗 = 𝑘 → ((𝑀 ∈ ℕ → (𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗))) ↔ (𝑀 ∈ ℕ → (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)))))
14		oveq1 7282	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 + 1) = ((𝑘 + 1) + 1))
15	14	oveq2d 7291	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → (𝑀↑(𝑗 + 1)) = (𝑀↑((𝑘 + 1) + 1)))
16		fveq2 6774	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (!‘𝑗) = (!‘(𝑘 + 1)))
17	16	oveq2d 7291	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → ((𝑀↑𝑀) · (!‘𝑗)) = ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
18	15, 17	breq12d 5087	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → ((𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗)) ↔ (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1)))))
19	18	imbi2d 341	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ((𝑀 ∈ ℕ → (𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗))) ↔ (𝑀 ∈ ℕ → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))))
20		oveq1 7282	. . . . . . . 8 ⊢ (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
21	20	oveq2d 7291	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝑀↑(𝑗 + 1)) = (𝑀↑(𝑁 + 1)))
22		fveq2 6774	. . . . . . . 8 ⊢ (𝑗 = 𝑁 → (!‘𝑗) = (!‘𝑁))
23	22	oveq2d 7291	. . . . . . 7 ⊢ (𝑗 = 𝑁 → ((𝑀↑𝑀) · (!‘𝑗)) = ((𝑀↑𝑀) · (!‘𝑁)))
24	21, 23	breq12d 5087	. . . . . 6 ⊢ (𝑗 = 𝑁 → ((𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗)) ↔ (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁))))
25	24	imbi2d 341	. . . . 5 ⊢ (𝑗 = 𝑁 → ((𝑀 ∈ ℕ → (𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗))) ↔ (𝑀 ∈ ℕ → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))))
26		nnre 11980	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
27		nnge1 12001	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
28		elnnuz 12622	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
29	28	biimpi 215	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ_≥‘1))
30	26, 27, 29	leexp2ad 13971	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀↑1) ≤ (𝑀↑𝑀))
31		0p1e1 12095	. . . . . . . 8 ⊢ (0 + 1) = 1
32	31	oveq2i 7286	. . . . . . 7 ⊢ (𝑀↑(0 + 1)) = (𝑀↑1)
33	32	a1i 11	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀↑(0 + 1)) = (𝑀↑1))
34		fac0 13990	. . . . . . . 8 ⊢ (!‘0) = 1
35	34	oveq2i 7286	. . . . . . 7 ⊢ ((𝑀↑𝑀) · (!‘0)) = ((𝑀↑𝑀) · 1)
36		nnnn0 12240	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
37	26, 36	reexpcld 13881	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑀↑𝑀) ∈ ℝ)
38	37	recnd 11003	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀↑𝑀) ∈ ℂ)
39	38	mulid1d 10992	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → ((𝑀↑𝑀) · 1) = (𝑀↑𝑀))
40	35, 39	eqtrid 2790	. . . . . 6 ⊢ (𝑀 ∈ ℕ → ((𝑀↑𝑀) · (!‘0)) = (𝑀↑𝑀))
41	30, 33, 40	3brtr4d 5106	. . . . 5 ⊢ (𝑀 ∈ ℕ → (𝑀↑(0 + 1)) ≤ ((𝑀↑𝑀) · (!‘0)))
42	26	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 𝑀 ∈ ℝ)
43		simpllr 773	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 𝑘 ∈ ℕ₀)
44		peano2nn0 12273	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 + 1) ∈ ℕ₀)
45	43, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑘 + 1) ∈ ℕ₀)
46	42, 45	reexpcld 13881	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑀↑(𝑘 + 1)) ∈ ℝ)
47	36	ad3antrrr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 𝑀 ∈ ℕ₀)
48	42, 47	reexpcld 13881	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑀↑𝑀) ∈ ℝ)
49	43	faccld 13998	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (!‘𝑘) ∈ ℕ)
50	49	nnred 11988	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (!‘𝑘) ∈ ℝ)
51	48, 50	remulcld 11005	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → ((𝑀↑𝑀) · (!‘𝑘)) ∈ ℝ)
52		nn0re 12242	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
53		peano2re 11148	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
54	43, 52, 53	3syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ)
55		nngt0 12004	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀)
56	55	ad3antrrr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 0 < 𝑀)
57		0re 10977	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
58		ltle 11063	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (0 < 𝑀 → 0 ≤ 𝑀))
59	57, 58	mpan 687	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℝ → (0 < 𝑀 → 0 ≤ 𝑀))
60	42, 56, 59	sylc 65	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 0 ≤ 𝑀)
61	42, 45, 60	expge0d 13882	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 0 ≤ (𝑀↑(𝑘 + 1)))
62		simplr 766	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)))
63		simprr 770	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 𝑀 ≤ (𝑘 + 1))
64	46, 51, 42, 54, 61, 60, 62, 63	lemul12ad 11917	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → ((𝑀↑(𝑘 + 1)) · 𝑀) ≤ (((𝑀↑𝑀) · (!‘𝑘)) · (𝑘 + 1)))
65	64	anandis 675	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) ∧ 𝑀 ≤ (𝑘 + 1))) → ((𝑀↑(𝑘 + 1)) · 𝑀) ≤ (((𝑀↑𝑀) · (!‘𝑘)) · (𝑘 + 1)))
66		nncn 11981	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
67		expp1 13789	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℂ ∧ (𝑘 + 1) ∈ ℕ₀) → (𝑀↑((𝑘 + 1) + 1)) = ((𝑀↑(𝑘 + 1)) · 𝑀))
68	66, 44, 67	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀↑((𝑘 + 1) + 1)) = ((𝑀↑(𝑘 + 1)) · 𝑀))
69	68	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑀↑((𝑘 + 1) + 1)) = ((𝑀↑(𝑘 + 1)) · 𝑀))
70		facp1 13992	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (!‘(𝑘 + 1)) = ((!‘𝑘) · (𝑘 + 1)))
71	70	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (!‘(𝑘 + 1)) = ((!‘𝑘) · (𝑘 + 1)))
72	71	oveq2d 7291	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) = ((𝑀↑𝑀) · ((!‘𝑘) · (𝑘 + 1))))
73	38	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀↑𝑀) ∈ ℂ)
74		faccl 13997	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ∈ ℕ)
75	74	nncnd 11989	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ∈ ℂ)
76	75	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (!‘𝑘) ∈ ℂ)
77		nn0cn 12243	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
78		peano2cn 11147	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℂ → (𝑘 + 1) ∈ ℂ)
79	77, 78	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 + 1) ∈ ℂ)
80	79	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℂ)
81	73, 76, 80	mulassd 10998	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (((𝑀↑𝑀) · (!‘𝑘)) · (𝑘 + 1)) = ((𝑀↑𝑀) · ((!‘𝑘) · (𝑘 + 1))))
82	72, 81	eqtr4d 2781	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) = (((𝑀↑𝑀) · (!‘𝑘)) · (𝑘 + 1)))
83	82	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) ∧ 𝑀 ≤ (𝑘 + 1))) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) = (((𝑀↑𝑀) · (!‘𝑘)) · (𝑘 + 1)))
84	65, 69, 83	3brtr4d 5106	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
85	84	exp32 421	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) → (𝑀 ≤ (𝑘 + 1) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))))
86	85	com23 86	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀 ≤ (𝑘 + 1) → ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))))
87		nn0ltp1le 12378	. . . . . . . . . . 11 ⊢ (((𝑘 + 1) ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) → ((𝑘 + 1) < 𝑀 ↔ ((𝑘 + 1) + 1) ≤ 𝑀))
88	44, 36, 87	syl2anr 597	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) < 𝑀 ↔ ((𝑘 + 1) + 1) ≤ 𝑀))
89		peano2nn0 12273	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ₀ → ((𝑘 + 1) + 1) ∈ ℕ₀)
90	44, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → ((𝑘 + 1) + 1) ∈ ℕ₀)
91		reexpcl 13799	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℝ ∧ ((𝑘 + 1) + 1) ∈ ℕ₀) → (𝑀↑((𝑘 + 1) + 1)) ∈ ℝ)
92	26, 90, 91	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀↑((𝑘 + 1) + 1)) ∈ ℝ)
93	92	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑((𝑘 + 1) + 1)) ∈ ℝ)
94	37	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑𝑀) ∈ ℝ)
95	44	faccld 13998	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (!‘(𝑘 + 1)) ∈ ℕ)
96	95	nnred 11988	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → (!‘(𝑘 + 1)) ∈ ℝ)
97		remulcl 10956	. . . . . . . . . . . . . 14 ⊢ (((𝑀↑𝑀) ∈ ℝ ∧ (!‘(𝑘 + 1)) ∈ ℝ) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) ∈ ℝ)
98	37, 96, 97	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) ∈ ℝ)
99	98	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) ∈ ℝ)
100	26	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → 𝑀 ∈ ℝ)
101	27	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → 1 ≤ 𝑀)
102		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → ((𝑘 + 1) + 1) ≤ 𝑀)
103	90	ad2antlr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → ((𝑘 + 1) + 1) ∈ ℕ₀)
104	103	nn0zd 12424	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → ((𝑘 + 1) + 1) ∈ ℤ)
105		nnz 12342	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
106	105	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → 𝑀 ∈ ℤ)
107		eluz 12596	. . . . . . . . . . . . . . 15 ⊢ ((((𝑘 + 1) + 1) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (ℤ_≥‘((𝑘 + 1) + 1)) ↔ ((𝑘 + 1) + 1) ≤ 𝑀))
108	104, 106, 107	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀 ∈ (ℤ_≥‘((𝑘 + 1) + 1)) ↔ ((𝑘 + 1) + 1) ≤ 𝑀))
109	102, 108	mpbird 256	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → 𝑀 ∈ (ℤ_≥‘((𝑘 + 1) + 1)))
110	100, 101, 109	leexp2ad 13971	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑((𝑘 + 1) + 1)) ≤ (𝑀↑𝑀))
111	37, 96	anim12i 613	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑𝑀) ∈ ℝ ∧ (!‘(𝑘 + 1)) ∈ ℝ))
112		nn0re 12242	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
113		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℕ₀)
114		nn0ge0 12258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ₀ → 0 ≤ 𝑀)
115	112, 113, 114	expge0d 13882	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ₀ → 0 ≤ (𝑀↑𝑀))
116	36, 115	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 ≤ (𝑀↑𝑀))
117	95	nnge1d 12021	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → 1 ≤ (!‘(𝑘 + 1)))
118	116, 117	anim12i 613	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (0 ≤ (𝑀↑𝑀) ∧ 1 ≤ (!‘(𝑘 + 1))))
119		lemulge11 11837	. . . . . . . . . . . . . 14 ⊢ ((((𝑀↑𝑀) ∈ ℝ ∧ (!‘(𝑘 + 1)) ∈ ℝ) ∧ (0 ≤ (𝑀↑𝑀) ∧ 1 ≤ (!‘(𝑘 + 1)))) → (𝑀↑𝑀) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
120	111, 118, 119	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀↑𝑀) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
121	120	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑𝑀) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
122	93, 94, 99, 110, 121	letrd 11132	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
123	122	ex 413	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) + 1) ≤ 𝑀 → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1)))))
124	88, 123	sylbid 239	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) < 𝑀 → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1)))))
125	124	a1dd 50	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) < 𝑀 → ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))))
126	52, 53	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 + 1) ∈ ℝ)
127		lelttric 11082	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → (𝑀 ≤ (𝑘 + 1) ∨ (𝑘 + 1) < 𝑀))
128	26, 126, 127	syl2an 596	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀 ≤ (𝑘 + 1) ∨ (𝑘 + 1) < 𝑀))
129	86, 125, 128	mpjaod 857	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1)))))
130	129	expcom 414	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ → (𝑀 ∈ ℕ → ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))))
131	130	a2d 29	. . . . 5 ⊢ (𝑘 ∈ ℕ₀ → ((𝑀 ∈ ℕ → (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) → (𝑀 ∈ ℕ → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))))
132	7, 13, 19, 25, 41, 131	nn0ind 12415	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝑀 ∈ ℕ → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁))))
133	132	impcom 408	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))
134		faccl 13997	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (!‘𝑁) ∈ ℕ)
135	134	nnnn0d 12293	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (!‘𝑁) ∈ ℕ₀)
136	135	nn0ge0d 12296	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ (!‘𝑁))
137		nn0p1nn 12272	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
138	137	0expd 13857	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (0↑(𝑁 + 1)) = 0)
139		0exp0e1 13787	. . . . . . . 8 ⊢ (0↑0) = 1
140	139	oveq1i 7285	. . . . . . 7 ⊢ ((0↑0) · (!‘𝑁)) = (1 · (!‘𝑁))
141	134	nncnd 11989	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (!‘𝑁) ∈ ℂ)
142	141	mulid2d 10993	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (1 · (!‘𝑁)) = (!‘𝑁))
143	140, 142	eqtrid 2790	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → ((0↑0) · (!‘𝑁)) = (!‘𝑁))
144	136, 138, 143	3brtr4d 5106	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (0↑(𝑁 + 1)) ≤ ((0↑0) · (!‘𝑁)))
145		oveq1 7282	. . . . . 6 ⊢ (𝑀 = 0 → (𝑀↑(𝑁 + 1)) = (0↑(𝑁 + 1)))
146		oveq12 7284	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝑀 = 0) → (𝑀↑𝑀) = (0↑0))
147	146	anidms 567	. . . . . . 7 ⊢ (𝑀 = 0 → (𝑀↑𝑀) = (0↑0))
148	147	oveq1d 7290	. . . . . 6 ⊢ (𝑀 = 0 → ((𝑀↑𝑀) · (!‘𝑁)) = ((0↑0) · (!‘𝑁)))
149	145, 148	breq12d 5087	. . . . 5 ⊢ (𝑀 = 0 → ((𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)) ↔ (0↑(𝑁 + 1)) ≤ ((0↑0) · (!‘𝑁))))
150	144, 149	syl5ibr 245	. . . 4 ⊢ (𝑀 = 0 → (𝑁 ∈ ℕ₀ → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁))))
151	150	imp 407	. . 3 ⊢ ((𝑀 = 0 ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))
152	133, 151	jaoian 954	. 2 ⊢ (((𝑀 ∈ ℕ ∨ 𝑀 = 0) ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))
153	1, 152	sylanb 581	1 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ↑cexp 13782 !cfa 13987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-seq 13722 df-exp 13783 df-fac 13988

This theorem is referenced by: faclbnd2 14005 faclbnd3 14006

W3C validator