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Theorem faclbnd 13644

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 11893	. 2 ⊢ (𝑀 ∈ ℕ₀ ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
2		oveq1 7157	. . . . . . . 8 ⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
3	2	oveq2d 7166	. . . . . . 7 ⊢ (𝑗 = 0 → (𝑀↑(𝑗 + 1)) = (𝑀↑(0 + 1)))
4		fveq2 6665	. . . . . . . 8 ⊢ (𝑗 = 0 → (!‘𝑗) = (!‘0))
5	4	oveq2d 7166	. . . . . . 7 ⊢ (𝑗 = 0 → ((𝑀↑𝑀) · (!‘𝑗)) = ((𝑀↑𝑀) · (!‘0)))
6	3, 5	breq12d 5072	. . . . . 6 ⊢ (𝑗 = 0 → ((𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗)) ↔ (𝑀↑(0 + 1)) ≤ ((𝑀↑𝑀) · (!‘0))))
7	6	imbi2d 343	. . . . 5 ⊢ (𝑗 = 0 → ((𝑀 ∈ ℕ → (𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗))) ↔ (𝑀 ∈ ℕ → (𝑀↑(0 + 1)) ≤ ((𝑀↑𝑀) · (!‘0)))))
8		oveq1 7157	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
9	8	oveq2d 7166	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑀↑(𝑗 + 1)) = (𝑀↑(𝑘 + 1)))
10		fveq2 6665	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (!‘𝑗) = (!‘𝑘))
11	10	oveq2d 7166	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝑀↑𝑀) · (!‘𝑗)) = ((𝑀↑𝑀) · (!‘𝑘)))
12	9, 11	breq12d 5072	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗)) ↔ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))))
13	12	imbi2d 343	. . . . 5 ⊢ (𝑗 = 𝑘 → ((𝑀 ∈ ℕ → (𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗))) ↔ (𝑀 ∈ ℕ → (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)))))
14		oveq1 7157	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 + 1) = ((𝑘 + 1) + 1))
15	14	oveq2d 7166	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → (𝑀↑(𝑗 + 1)) = (𝑀↑((𝑘 + 1) + 1)))
16		fveq2 6665	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (!‘𝑗) = (!‘(𝑘 + 1)))
17	16	oveq2d 7166	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → ((𝑀↑𝑀) · (!‘𝑗)) = ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
18	15, 17	breq12d 5072	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → ((𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗)) ↔ (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1)))))
19	18	imbi2d 343	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ((𝑀 ∈ ℕ → (𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗))) ↔ (𝑀 ∈ ℕ → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))))
20		oveq1 7157	. . . . . . . 8 ⊢ (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
21	20	oveq2d 7166	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝑀↑(𝑗 + 1)) = (𝑀↑(𝑁 + 1)))
22		fveq2 6665	. . . . . . . 8 ⊢ (𝑗 = 𝑁 → (!‘𝑗) = (!‘𝑁))
23	22	oveq2d 7166	. . . . . . 7 ⊢ (𝑗 = 𝑁 → ((𝑀↑𝑀) · (!‘𝑗)) = ((𝑀↑𝑀) · (!‘𝑁)))
24	21, 23	breq12d 5072	. . . . . 6 ⊢ (𝑗 = 𝑁 → ((𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗)) ↔ (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁))))
25	24	imbi2d 343	. . . . 5 ⊢ (𝑗 = 𝑁 → ((𝑀 ∈ ℕ → (𝑀↑(𝑗 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑗))) ↔ (𝑀 ∈ ℕ → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))))
26		nnre 11639	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
27		nnge1 11659	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
28		elnnuz 12276	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
29	28	biimpi 218	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ_≥‘1))
30	26, 27, 29	leexp2ad 13611	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀↑1) ≤ (𝑀↑𝑀))
31		0p1e1 11753	. . . . . . . 8 ⊢ (0 + 1) = 1
32	31	oveq2i 7161	. . . . . . 7 ⊢ (𝑀↑(0 + 1)) = (𝑀↑1)
33	32	a1i 11	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀↑(0 + 1)) = (𝑀↑1))
34		fac0 13630	. . . . . . . 8 ⊢ (!‘0) = 1
35	34	oveq2i 7161	. . . . . . 7 ⊢ ((𝑀↑𝑀) · (!‘0)) = ((𝑀↑𝑀) · 1)
36		nnnn0 11898	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
37	26, 36	reexpcld 13521	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑀↑𝑀) ∈ ℝ)
38	37	recnd 10663	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀↑𝑀) ∈ ℂ)
39	38	mulid1d 10652	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → ((𝑀↑𝑀) · 1) = (𝑀↑𝑀))
40	35, 39	syl5eq 2868	. . . . . 6 ⊢ (𝑀 ∈ ℕ → ((𝑀↑𝑀) · (!‘0)) = (𝑀↑𝑀))
41	30, 33, 40	3brtr4d 5091	. . . . 5 ⊢ (𝑀 ∈ ℕ → (𝑀↑(0 + 1)) ≤ ((𝑀↑𝑀) · (!‘0)))
42	26	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 𝑀 ∈ ℝ)
43		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 𝑘 ∈ ℕ₀)
44		peano2nn0 11931	. . . . . . . . . . . . . . 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 13521	. . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 𝑀 ∈ ℕ₀)
48	42, 47	reexpcld 13521	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑀↑𝑀) ∈ ℝ)
49	43	faccld 13638	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (!‘𝑘) ∈ ℕ)
50	49	nnred 11647	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → (!‘𝑘) ∈ ℝ)
51	48, 50	remulcld 10665	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → ((𝑀↑𝑀) · (!‘𝑘)) ∈ ℝ)
52		nn0re 11900	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
53		peano2re 10807	. . . . . . . . . . . . . 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 11662	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀)
56	55	ad3antrrr 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 0 < 𝑀)
57		0re 10637	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
58		ltle 10723	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (0 < 𝑀 → 0 ≤ 𝑀))
59	57, 58	mpan 688	. . . . . . . . . . . . . . 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 13522	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ (𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘))) ∧ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑀 ≤ (𝑘 + 1))) → 0 ≤ (𝑀↑(𝑘 + 1)))
62		simplr 767	. . . . . . . . . . . . 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 771	. . . . . . . . . . . . 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 11576	. . . . . . . . . . . 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 676	. . . . . . . . . . 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 11640	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
67		expp1 13430	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℂ ∧ (𝑘 + 1) ∈ ℕ₀) → (𝑀↑((𝑘 + 1) + 1)) = ((𝑀↑(𝑘 + 1)) · 𝑀))
68	66, 44, 67	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀↑((𝑘 + 1) + 1)) = ((𝑀↑(𝑘 + 1)) · 𝑀))
69	68	adantr 483	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) ∧ 𝑀 ≤ (𝑘 + 1))) → (𝑀↑((𝑘 + 1) + 1)) = ((𝑀↑(𝑘 + 1)) · 𝑀))
70		facp1 13632	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (!‘(𝑘 + 1)) = ((!‘𝑘) · (𝑘 + 1)))
71	70	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (!‘(𝑘 + 1)) = ((!‘𝑘) · (𝑘 + 1)))
72	71	oveq2d 7166	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) = ((𝑀↑𝑀) · ((!‘𝑘) · (𝑘 + 1))))
73	38	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀↑𝑀) ∈ ℂ)
74		faccl 13637	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ∈ ℕ)
75	74	nncnd 11648	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ∈ ℂ)
76	75	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (!‘𝑘) ∈ ℂ)
77		nn0cn 11901	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
78		peano2cn 10806	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℂ → (𝑘 + 1) ∈ ℂ)
79	77, 78	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 + 1) ∈ ℂ)
80	79	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℂ)
81	73, 76, 80	mulassd 10658	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (((𝑀↑𝑀) · (!‘𝑘)) · (𝑘 + 1)) = ((𝑀↑𝑀) · ((!‘𝑘) · (𝑘 + 1))))
82	72, 81	eqtr4d 2859	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) = (((𝑀↑𝑀) · (!‘𝑘)) · (𝑘 + 1)))
83	82	adantr 483	. . . . . . . . . . 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 5091	. . . . . . . . . 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 423	. . . . . . . . 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 12034	. . . . . . . . . . 11 ⊢ (((𝑘 + 1) ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) → ((𝑘 + 1) < 𝑀 ↔ ((𝑘 + 1) + 1) ≤ 𝑀))
88	44, 36, 87	syl2anr 598	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑘 + 1) < 𝑀 ↔ ((𝑘 + 1) + 1) ≤ 𝑀))
89		peano2nn0 11931	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ₀ → ((𝑘 + 1) + 1) ∈ ℕ₀)
90	44, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → ((𝑘 + 1) + 1) ∈ ℕ₀)
91		reexpcl 13440	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℝ ∧ ((𝑘 + 1) + 1) ∈ ℕ₀) → (𝑀↑((𝑘 + 1) + 1)) ∈ ℝ)
92	26, 90, 91	syl2an 597	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀↑((𝑘 + 1) + 1)) ∈ ℝ)
93	92	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑((𝑘 + 1) + 1)) ∈ ℝ)
94	37	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑𝑀) ∈ ℝ)
95	44	faccld 13638	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (!‘(𝑘 + 1)) ∈ ℕ)
96	95	nnred 11647	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → (!‘(𝑘 + 1)) ∈ ℝ)
97		remulcl 10616	. . . . . . . . . . . . . 14 ⊢ (((𝑀↑𝑀) ∈ ℝ ∧ (!‘(𝑘 + 1)) ∈ ℝ) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) ∈ ℝ)
98	37, 96, 97	syl2an 597	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) ∈ ℝ)
99	98	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → ((𝑀↑𝑀) · (!‘(𝑘 + 1))) ∈ ℝ)
100	26	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → 𝑀 ∈ ℝ)
101	27	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → 1 ≤ 𝑀)
102		simpr 487	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → ((𝑘 + 1) + 1) ≤ 𝑀)
103	90	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → ((𝑘 + 1) + 1) ∈ ℕ₀)
104	103	nn0zd 12079	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → ((𝑘 + 1) + 1) ∈ ℤ)
105		nnz 11998	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
106	105	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → 𝑀 ∈ ℤ)
107		eluz 12251	. . . . . . . . . . . . . . 15 ⊢ ((((𝑘 + 1) + 1) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (ℤ_≥‘((𝑘 + 1) + 1)) ↔ ((𝑘 + 1) + 1) ≤ 𝑀))
108	104, 106, 107	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀 ∈ (ℤ_≥‘((𝑘 + 1) + 1)) ↔ ((𝑘 + 1) + 1) ≤ 𝑀))
109	102, 108	mpbird 259	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → 𝑀 ∈ (ℤ_≥‘((𝑘 + 1) + 1)))
110	100, 101, 109	leexp2ad 13611	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑((𝑘 + 1) + 1)) ≤ (𝑀↑𝑀))
111	37, 96	anim12i 614	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑𝑀) ∈ ℝ ∧ (!‘(𝑘 + 1)) ∈ ℝ))
112		nn0re 11900	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
113		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℕ₀)
114		nn0ge0 11916	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ₀ → 0 ≤ 𝑀)
115	112, 113, 114	expge0d 13522	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ₀ → 0 ≤ (𝑀↑𝑀))
116	36, 115	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 ≤ (𝑀↑𝑀))
117	95	nnge1d 11679	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → 1 ≤ (!‘(𝑘 + 1)))
118	116, 117	anim12i 614	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (0 ≤ (𝑀↑𝑀) ∧ 1 ≤ (!‘(𝑘 + 1))))
119		lemulge11 11496	. . . . . . . . . . . . . 14 ⊢ ((((𝑀↑𝑀) ∈ ℝ ∧ (!‘(𝑘 + 1)) ∈ ℝ) ∧ (0 ≤ (𝑀↑𝑀) ∧ 1 ≤ (!‘(𝑘 + 1)))) → (𝑀↑𝑀) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
120	111, 118, 119	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀↑𝑀) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
121	120	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑𝑀) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
122	93, 94, 99, 110, 121	letrd 10791	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) ∧ ((𝑘 + 1) + 1) ≤ 𝑀) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1))))
123	122	ex 415	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (((𝑘 + 1) + 1) ≤ 𝑀 → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1)))))
124	88, 123	sylbid 242	. . . . . . . . 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 10741	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → (𝑀 ≤ (𝑘 + 1) ∨ (𝑘 + 1) < 𝑀))
128	26, 126, 127	syl2an 597	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑀 ≤ (𝑘 + 1) ∨ (𝑘 + 1) < 𝑀))
129	86, 125, 128	mpjaod 856	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → ((𝑀↑(𝑘 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑘)) → (𝑀↑((𝑘 + 1) + 1)) ≤ ((𝑀↑𝑀) · (!‘(𝑘 + 1)))))
130	129	expcom 416	. . . . . 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 12071	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝑀 ∈ ℕ → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁))))
133	132	impcom 410	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))
134		faccl 13637	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (!‘𝑁) ∈ ℕ)
135	134	nnnn0d 11949	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (!‘𝑁) ∈ ℕ₀)
136	135	nn0ge0d 11952	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ (!‘𝑁))
137		nn0p1nn 11930	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
138	137	0expd 13497	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (0↑(𝑁 + 1)) = 0)
139		0exp0e1 13428	. . . . . . . 8 ⊢ (0↑0) = 1
140	139	oveq1i 7160	. . . . . . 7 ⊢ ((0↑0) · (!‘𝑁)) = (1 · (!‘𝑁))
141	134	nncnd 11648	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (!‘𝑁) ∈ ℂ)
142	141	mulid2d 10653	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (1 · (!‘𝑁)) = (!‘𝑁))
143	140, 142	syl5eq 2868	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → ((0↑0) · (!‘𝑁)) = (!‘𝑁))
144	136, 138, 143	3brtr4d 5091	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (0↑(𝑁 + 1)) ≤ ((0↑0) · (!‘𝑁)))
145		oveq1 7157	. . . . . 6 ⊢ (𝑀 = 0 → (𝑀↑(𝑁 + 1)) = (0↑(𝑁 + 1)))
146		oveq12 7159	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝑀 = 0) → (𝑀↑𝑀) = (0↑0))
147	146	anidms 569	. . . . . . 7 ⊢ (𝑀 = 0 → (𝑀↑𝑀) = (0↑0))
148	147	oveq1d 7165	. . . . . 6 ⊢ (𝑀 = 0 → ((𝑀↑𝑀) · (!‘𝑁)) = ((0↑0) · (!‘𝑁)))
149	145, 148	breq12d 5072	. . . . 5 ⊢ (𝑀 = 0 → ((𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)) ↔ (0↑(𝑁 + 1)) ≤ ((0↑0) · (!‘𝑁))))
150	144, 149	syl5ibr 248	. . . 4 ⊢ (𝑀 = 0 → (𝑁 ∈ ℕ₀ → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁))))
151	150	imp 409	. . 3 ⊢ ((𝑀 = 0 ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))
152	133, 151	jaoian 953	. 2 ⊢ (((𝑀 ∈ ℕ ∨ 𝑀 = 0) ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))
153	1, 152	sylanb 583	1 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 < clt 10669 ≤ cle 10670 ℕcn 11632 ℕ₀cn0 11891 ℤcz 11975 ℤ_≥cuz 12237 ↑cexp 13423 !cfa 13627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-fac 13628

This theorem is referenced by: faclbnd2 13645 faclbnd3 13646

W3C validator