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Theorem faclbnd 12939

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 11171	. 2 ⊢ (𝑀 ∈ ℕ₀ ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
2		oveq1 6556	. . . . . . . 8 ⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
3	2	oveq2d 6565	. . . . . . 7 ⊢ (𝑗 = 0 → (𝑀↑(𝑗 + 1)) = (𝑀↑(0 + 1)))
4		fveq2 6103	. . . . . . . 8 ⊢ (𝑗 = 0 → (!‘𝑗) = (!‘0))
5	4	oveq2d 6565	. . . . . . 7 ⊢ (𝑗 = 0 → ((𝑀↑𝑀) · (!‘𝑗)) = ((𝑀↑𝑀) · (!‘0)))
6	3, 5	breq12d 4596	. . . . . 6 ⊢ (𝑗 = 0 → ((𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗)) ↔ (𝑀↑(0 + 1)) ≤ ((𝑀↑𝑀) · (!‘0))))
7	6	imbi2d 329	. . . . 5 ⊢ (𝑗 = 0 → ((𝑀 ∈ ℕ → (𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗))) ↔ (𝑀 ∈ ℕ → (𝑀↑(0 + 1)) ≤ ((𝑀↑𝑀) · (!‘0)))))
8		oveq1 6556	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
9	8	oveq2d 6565	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑀↑(𝑗 + 1)) = (𝑀↑(𝑘 + 1)))
10		fveq2 6103	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (!‘𝑗) = (!‘𝑘))
11	10	oveq2d 6565	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝑀↑𝑀) · (!‘𝑗)) = ((𝑀↑𝑀) · (!‘𝑘)))
12	9, 11	breq12d 4596	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗)) ↔ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))))
13	12	imbi2d 329	. . . . 5 ⊢ (𝑗 = 𝑘 → ((𝑀 ∈ ℕ → (𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗))) ↔ (𝑀 ∈ ℕ → (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)))))
14		oveq1 6556	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 + 1) = ((𝑘 + 1) + 1))
15	14	oveq2d 6565	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → (𝑀↑(𝑗 + 1)) = (𝑀↑((𝑘 + 1) + 1)))
16		fveq2 6103	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (!‘𝑗) = (!‘(𝑘 + 1)))
17	16	oveq2d 6565	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → ((𝑀↑𝑀) · (!‘𝑗)) = ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
18	15, 17	breq12d 4596	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → ((𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗)) ↔ (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1)))))
19	18	imbi2d 329	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ((𝑀 ∈ ℕ → (𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗))) ↔ (𝑀 ∈ ℕ → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))))
20		oveq1 6556	. . . . . . . 8 ⊢ (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
21	20	oveq2d 6565	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝑀↑(𝑗 + 1)) = (𝑀↑(𝑁 + 1)))
22		fveq2 6103	. . . . . . . 8 ⊢ (𝑗 = 𝑁 → (!‘𝑗) = (!‘𝑁))
23	22	oveq2d 6565	. . . . . . 7 ⊢ (𝑗 = 𝑁 → ((𝑀↑𝑀) · (!‘𝑗)) = ((𝑀↑𝑀) · (!‘𝑁)))
24	21, 23	breq12d 4596	. . . . . 6 ⊢ (𝑗 = 𝑁 → ((𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗)) ↔ (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁))))
25	24	imbi2d 329	. . . . 5 ⊢ (𝑗 = 𝑁 → ((𝑀 ∈ ℕ → (𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗))) ↔ (𝑀 ∈ ℕ → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))))
26		nnre 10904	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
27		nnge1 10923	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
28		elnnuz 11600	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
29	28	biimpi 205	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ_≥‘1))
30	26, 27, 29	leexp2ad 12903	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀↑1) ≤ (𝑀↑𝑀))
31		0p1e1 11009	. . . . . . . 8 ⊢ (0 + 1) = 1
32	31	oveq2i 6560	. . . . . . 7 ⊢ (𝑀↑(0 + 1)) = (𝑀↑1)
33	32	a1i 11	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀↑(0 + 1)) = (𝑀↑1))
34		fac0 12925	. . . . . . . 8 ⊢ (!‘0) = 1
35	34	oveq2i 6560	. . . . . . 7 ⊢ ((𝑀↑𝑀) · (!‘0)) = ((𝑀↑𝑀) · 1)
36		nnnn0 11176	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
37	26, 36	reexpcld 12887	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑀↑𝑀) ∈ ℝ)
38	37	recnd 9947	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀↑𝑀) ∈ ℂ)
39	38	mulid1d 9936	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → ((𝑀↑𝑀) · 1) = (𝑀↑𝑀))
40	35, 39	syl5eq 2656	. . . . . 6 ⊢ (𝑀 ∈ ℕ → ((𝑀↑𝑀) · (!‘0)) = (𝑀↑𝑀))
41	30, 33, 40	3brtr4d 4615	. . . . 5 ⊢ (𝑀 ∈ ℕ → (𝑀↑(0 + 1)) ≤ ((𝑀↑𝑀) · (!‘0)))
42	26	ad3antrrr 762	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 𝑀 ∈ ℝ)
43		simpllr 795	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 𝑘 ∈ ℕ₀)
44		peano2nn0 11210	. . . . . . . . . . . . . . 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 12887	. . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 𝑀 ∈ ℕ₀)
48	42, 47	reexpcld 12887	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑀↑𝑀) ∈ ℝ)
49		faccl 12932	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ∈ ℕ)
50	43, 49	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (!‘𝑘) ∈ ℕ)
51	50	nnred 10912	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (!‘𝑘) ∈ ℝ)
52	48, 51	remulcld 9949	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → ((𝑀↑𝑀) · (!‘𝑘)) ∈ ℝ)
53		nn0re 11178	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
54		peano2re 10088	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
55	43, 53, 54	3syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ)
56		nngt0 10926	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀)
57	56	ad3antrrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 0 < 𝑀)
58		0re 9919	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
59		ltle 10005	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (0 < 𝑀 → 0 ≤ 𝑀))
60	58, 59	mpan 702	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℝ → (0 < 𝑀 → 0 ≤ 𝑀))
61	42, 57, 60	sylc 63	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 0 ≤ 𝑀)
62	42, 45, 61	expge0d 12888	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 0 ≤ (𝑀↑(𝑘 + 1)))
63		simplr 788	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)))
64		simprr 792	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 𝑀 ≤ (𝑘 + 1))
65	46, 52, 42, 55, 62, 61, 63, 64	lemul12ad 10845	. . . . . . . . . . . 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)))
66	65	anandis 869	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) ∧ 𝑀 ≤ (𝑘 + 1))) → ((𝑀↑(𝑘 + 1)) · 𝑀) ≤ (((𝑀↑𝑀) · (!‘𝑘)) · (𝑘 + 1)))
67		nncn 10905	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
68		expp1 12729	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℂ ∧ (𝑘 + 1) ∈ ℕ₀) → (𝑀↑((𝑘 + 1) + 1)) = ((𝑀↑(𝑘 + 1)) · 𝑀))
69	67, 44, 68	syl2an 493	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀↑((𝑘 + 1) + 1)) = ((𝑀↑(𝑘 + 1)) · 𝑀))
70	69	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑀↑((𝑘 + 1) + 1)) = ((𝑀↑(𝑘 + 1)) · 𝑀))
71		facp1 12927	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (!‘(𝑘 + 1)) = ((!‘𝑘) · (𝑘 + 1)))
72	71	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (!‘(𝑘 + 1)) = ((!‘𝑘) · (𝑘 + 1)))
73	72	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) = ((𝑀↑𝑀) · ((!‘𝑘) · (𝑘 + 1))))
74	38	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀↑𝑀) ∈ ℂ)
75	49	nncnd 10913	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ∈ ℂ)
76	75	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (!‘𝑘) ∈ ℂ)
77		nn0cn 11179	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
78		peano2cn 10087	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℂ → (𝑘 + 1) ∈ ℂ)
79	77, 78	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 + 1) ∈ ℂ)
80	79	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℂ)
81	74, 76, 80	mulassd 9942	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (((𝑀↑𝑀) · (!‘𝑘)) · (𝑘 + 1)) = ((𝑀↑𝑀) · ((!‘𝑘) · (𝑘 + 1))))
82	73, 81	eqtr4d 2647	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) = (((𝑀↑𝑀) · (!‘𝑘)) · (𝑘 + 1)))
83	82	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) ∧ 𝑀 ≤ (𝑘 + 1))) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) = (((𝑀↑𝑀) · (!‘𝑘)) · (𝑘 + 1)))
84	66, 70, 83	3brtr4d 4615	. . . . . . . . . 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 629	. . . . . . . . 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 84	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀 ≤ (𝑘 + 1) → ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))))
87		nn0ltp1le 11312	. . . . . . . . . . 11 ⊢ (((𝑘 + 1) ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) → ((𝑘 + 1) < 𝑀 ↔ ((𝑘 + 1) + 1) ≤ 𝑀))
88	44, 36, 87	syl2anr 494	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) < 𝑀 ↔ ((𝑘 + 1) + 1) ≤ 𝑀))
89		peano2nn0 11210	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ₀ → ((𝑘 + 1) + 1) ∈ ℕ₀)
90	44, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → ((𝑘 + 1) + 1) ∈ ℕ₀)
91		reexpcl 12739	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℝ ∧ ((𝑘 + 1) + 1) ∈ ℕ₀) → (𝑀↑((𝑘 + 1) + 1)) ∈ ℝ)
92	26, 90, 91	syl2an 493	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀↑((𝑘 + 1) + 1)) ∈ ℝ)
93	92	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑((𝑘 + 1) + 1)) ∈ ℝ)
94	37	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑𝑀) ∈ ℝ)
95		faccl 12932	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 + 1) ∈ ℕ₀ → (!‘(𝑘 + 1)) ∈ ℕ)
96	44, 95	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (!‘(𝑘 + 1)) ∈ ℕ)
97	96	nnred 10912	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → (!‘(𝑘 + 1)) ∈ ℝ)
98		remulcl 9900	. . . . . . . . . . . . . 14 ⊢ (((𝑀↑𝑀) ∈ ℝ ∧ (!‘(𝑘 + 1)) ∈ ℝ) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) ∈ ℝ)
99	37, 97, 98	syl2an 493	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) ∈ ℝ)
100	99	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) ∈ ℝ)
101	26	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → 𝑀 ∈ ℝ)
102	27	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → 1 ≤ 𝑀)
103		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → ((𝑘 + 1) + 1) ≤ 𝑀)
104	90	ad2antlr 759	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → ((𝑘 + 1) + 1) ∈ ℕ₀)
105	104	nn0zd 11356	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → ((𝑘 + 1) + 1) ∈ ℤ)
106		nnz 11276	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
107	106	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → 𝑀 ∈ ℤ)
108		eluz 11577	. . . . . . . . . . . . . . 15 ⊢ ((((𝑘 + 1) + 1) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (ℤ_≥‘((𝑘 + 1) + 1)) ↔ ((𝑘 + 1) + 1) ≤ 𝑀))
109	105, 107, 108	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀 ∈ (ℤ_≥‘((𝑘 + 1) + 1)) ↔ ((𝑘 + 1) + 1) ≤ 𝑀))
110	103, 109	mpbird 246	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → 𝑀 ∈ (ℤ_≥‘((𝑘 + 1) + 1)))
111	101, 102, 110	leexp2ad 12903	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑((𝑘 + 1) + 1)) ≤ (𝑀↑𝑀))
112	37, 97	anim12i 588	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑𝑀) ∈ ℝ ∧ (!‘(𝑘 + 1)) ∈ ℝ))
113		nn0re 11178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
114		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℕ₀)
115		nn0ge0 11195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ₀ → 0 ≤ 𝑀)
116	113, 114, 115	expge0d 12888	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ₀ → 0 ≤ (𝑀↑𝑀))
117	36, 116	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 ≤ (𝑀↑𝑀))
118	96	nnge1d 10940	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → 1 ≤ (!‘(𝑘 + 1)))
119	117, 118	anim12i 588	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (0 ≤ (𝑀↑𝑀) ∧ 1 ≤ (!‘(𝑘 + 1))))
120		lemulge11 10764	. . . . . . . . . . . . . 14 ⊢ ((((𝑀↑𝑀) ∈ ℝ ∧ (!‘(𝑘 + 1)) ∈ ℝ) ∧ (0 ≤ (𝑀↑𝑀) ∧ 1 ≤ (!‘(𝑘 + 1)))) → (𝑀↑𝑀) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
121	112, 119, 120	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀↑𝑀) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
122	121	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑𝑀) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
123	93, 94, 100, 111, 122	letrd 10073	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
124	123	ex 449	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) + 1) ≤ 𝑀 → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1)))))
125	88, 124	sylbid 229	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) < 𝑀 → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1)))))
126	125	a1dd 48	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) < 𝑀 → ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))))
127	53, 54	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 + 1) ∈ ℝ)
128		lelttric 10023	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → (𝑀 ≤ (𝑘 + 1) ∨ (𝑘 + 1) < 𝑀))
129	26, 127, 128	syl2an 493	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀 ≤ (𝑘 + 1) ∨ (𝑘 + 1) < 𝑀))
130	86, 126, 129	mpjaod 395	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1)))))
131	130	expcom 450	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ → (𝑀 ∈ ℕ → ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))))
132	131	a2d 29	. . . . 5 ⊢ (𝑘 ∈ ℕ₀ → ((𝑀 ∈ ℕ → (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) → (𝑀 ∈ ℕ → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))))
133	7, 13, 19, 25, 41, 132	nn0ind 11348	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝑀 ∈ ℕ → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁))))
134	133	impcom 445	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))
135		faccl 12932	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (!‘𝑁) ∈ ℕ)
136	135	nnnn0d 11228	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (!‘𝑁) ∈ ℕ₀)
137	136	nn0ge0d 11231	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ (!‘𝑁))
138		nn0p1nn 11209	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
139	138	0expd 12886	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (0↑(𝑁 + 1)) = 0)
140		0exp0e1 12727	. . . . . . . 8 ⊢ (0↑0) = 1
141	140	oveq1i 6559	. . . . . . 7 ⊢ ((0↑0) · (!‘𝑁)) = (1 · (!‘𝑁))
142	135	nncnd 10913	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (!‘𝑁) ∈ ℂ)
143	142	mulid2d 9937	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (1 · (!‘𝑁)) = (!‘𝑁))
144	141, 143	syl5eq 2656	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → ((0↑0) · (!‘𝑁)) = (!‘𝑁))
145	137, 139, 144	3brtr4d 4615	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (0↑(𝑁 + 1)) ≤ ((0↑0) · (!‘𝑁)))
146		oveq1 6556	. . . . . 6 ⊢ (𝑀 = 0 → (𝑀↑(𝑁 + 1)) = (0↑(𝑁 + 1)))
147		oveq12 6558	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝑀 = 0) → (𝑀↑𝑀) = (0↑0))
148	147	anidms 675	. . . . . . 7 ⊢ (𝑀 = 0 → (𝑀↑𝑀) = (0↑0))
149	148	oveq1d 6564	. . . . . 6 ⊢ (𝑀 = 0 → ((𝑀↑𝑀) · (!‘𝑁)) = ((0↑0) · (!‘𝑁)))
150	146, 149	breq12d 4596	. . . . 5 ⊢ (𝑀 = 0 → ((𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)) ↔ (0↑(𝑁 + 1)) ≤ ((0↑0) · (!‘𝑁))))
151	145, 150	syl5ibr 235	. . . 4 ⊢ (𝑀 = 0 → (𝑁 ∈ ℕ₀ → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁))))
152	151	imp 444	. . 3 ⊢ ((𝑀 = 0 ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))
153	134, 152	jaoian 820	. 2 ⊢ (((𝑀 ∈ ℕ ∨ 𝑀 = 0) ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))
154	1, 153	sylanb 488	1 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ↑cexp 12722 !cfa 12922

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-fac 12923

This theorem is referenced by: faclbnd2 12940 faclbnd3 12941

W3C validator