stoweidlem3 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem3		Structured version Visualization version GIF version

Theorem stoweidlem3 42295

Description: Lemma for stoweid 42355: if 𝐴 is positive and all 𝑀 terms of a finite product are larger than 𝐴, then the finite product is larger than A^M. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
stoweidlem3.1	⊢ Ⅎ𝑖𝐹
stoweidlem3.2	⊢ Ⅎ𝑖𝜑
stoweidlem3.3	⊢ 𝑋 = seq1( · , 𝐹)
stoweidlem3.4	⊢ (𝜑 → 𝑀 ∈ ℕ)
stoweidlem3.5	⊢ (𝜑 → 𝐹:(1...𝑀)⟶ℝ)
stoweidlem3.6	⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐴 < (𝐹‘𝑖))
stoweidlem3.7	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)

**Assertion**
Ref	Expression
stoweidlem3	⊢ (𝜑 → (𝐴↑𝑀) < (𝑋‘𝑀))

Distinct variable groups: 𝐴,𝑖 𝑖,𝑀 Allowed substitution hints: 𝜑(𝑖) 𝐹(𝑖) 𝑋(𝑖)

Proof of Theorem stoweidlem3 Dummy variables 𝑎 𝑚 𝑗 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem3.4	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2		elnnuz 12285	. . . 4 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
3	1, 2	sylib 220	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
4		eluzfz2 12918	. . 3 ⊢ (𝑀 ∈ (ℤ_≥‘1) → 𝑀 ∈ (1...𝑀))
5	3, 4	syl 17	. 2 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
6		oveq2 7166	. . . . 5 ⊢ (𝑛 = 1 → (𝐴↑𝑛) = (𝐴↑1))
7		fveq2 6672	. . . . 5 ⊢ (𝑛 = 1 → (𝑋‘𝑛) = (𝑋‘1))
8	6, 7	breq12d 5081	. . . 4 ⊢ (𝑛 = 1 → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑1) < (𝑋‘1)))
9	8	imbi2d 343	. . 3 ⊢ (𝑛 = 1 → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑1) < (𝑋‘1))))
10		oveq2 7166	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴↑𝑛) = (𝐴↑𝑚))
11		fveq2 6672	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑋‘𝑛) = (𝑋‘𝑚))
12	10, 11	breq12d 5081	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑𝑚) < (𝑋‘𝑚)))
13	12	imbi2d 343	. . 3 ⊢ (𝑛 = 𝑚 → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚))))
14		oveq2 7166	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝐴↑𝑛) = (𝐴↑(𝑚 + 1)))
15		fveq2 6672	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑋‘𝑛) = (𝑋‘(𝑚 + 1)))
16	14, 15	breq12d 5081	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1))))
17	16	imbi2d 343	. . 3 ⊢ (𝑛 = (𝑚 + 1) → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1)))))
18		oveq2 7166	. . . . 5 ⊢ (𝑛 = 𝑀 → (𝐴↑𝑛) = (𝐴↑𝑀))
19		fveq2 6672	. . . . 5 ⊢ (𝑛 = 𝑀 → (𝑋‘𝑛) = (𝑋‘𝑀))
20	18, 19	breq12d 5081	. . . 4 ⊢ (𝑛 = 𝑀 → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑𝑀) < (𝑋‘𝑀)))
21	20	imbi2d 343	. . 3 ⊢ (𝑛 = 𝑀 → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑𝑀) < (𝑋‘𝑀))))
22		1zzd 12016	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
23	1	nnzd 12089	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
24	22, 23, 22	3jca 1124	. . . . . . . 8 ⊢ (𝜑 → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 1 ∈ ℤ))
25		1le1 11270	. . . . . . . . 9 ⊢ 1 ≤ 1
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ 1)
27	1	nnge1d 11688	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝑀)
28	24, 26, 27	jca32 518	. . . . . . 7 ⊢ (𝜑 → ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (1 ≤ 1 ∧ 1 ≤ 𝑀)))
29		elfz2 12902	. . . . . . 7 ⊢ (1 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (1 ≤ 1 ∧ 1 ≤ 𝑀)))
30	28, 29	sylibr 236	. . . . . 6 ⊢ (𝜑 → 1 ∈ (1...𝑀))
31	30	ancli 551	. . . . . 6 ⊢ (𝜑 → (𝜑 ∧ 1 ∈ (1...𝑀)))
32		stoweidlem3.2	. . . . . . . . 9 ⊢ Ⅎ𝑖𝜑
33		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑖1 ∈ (1...𝑀)
34	32, 33	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ 1 ∈ (1...𝑀))
35		nfcv 2979	. . . . . . . . 9 ⊢ Ⅎ𝑖𝐴
36		nfcv 2979	. . . . . . . . 9 ⊢ Ⅎ𝑖 <
37		stoweidlem3.1	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝐹
38		nfcv 2979	. . . . . . . . . 10 ⊢ Ⅎ𝑖1
39	37, 38	nffv 6682	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐹‘1)
40	35, 36, 39	nfbr 5115	. . . . . . . 8 ⊢ Ⅎ𝑖 𝐴 < (𝐹‘1)
41	34, 40	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑖((𝜑 ∧ 1 ∈ (1...𝑀)) → 𝐴 < (𝐹‘1))
42		eleq1 2902	. . . . . . . . 9 ⊢ (𝑖 = 1 → (𝑖 ∈ (1...𝑀) ↔ 1 ∈ (1...𝑀)))
43	42	anbi2d 630	. . . . . . . 8 ⊢ (𝑖 = 1 → ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ↔ (𝜑 ∧ 1 ∈ (1...𝑀))))
44		fveq2 6672	. . . . . . . . 9 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = (𝐹‘1))
45	44	breq2d 5080	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐴 < (𝐹‘𝑖) ↔ 𝐴 < (𝐹‘1)))
46	43, 45	imbi12d 347	. . . . . . 7 ⊢ (𝑖 = 1 → (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐴 < (𝐹‘𝑖)) ↔ ((𝜑 ∧ 1 ∈ (1...𝑀)) → 𝐴 < (𝐹‘1))))
47		stoweidlem3.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐴 < (𝐹‘𝑖))
48	41, 46, 47	vtoclg1f 3568	. . . . . 6 ⊢ (1 ∈ (1...𝑀) → ((𝜑 ∧ 1 ∈ (1...𝑀)) → 𝐴 < (𝐹‘1)))
49	30, 31, 48	sylc 65	. . . . 5 ⊢ (𝜑 → 𝐴 < (𝐹‘1))
50		stoweidlem3.7	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
51	50	rpcnd 12436	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
52	51	exp1d 13508	. . . . 5 ⊢ (𝜑 → (𝐴↑1) = 𝐴)
53		stoweidlem3.3	. . . . . . . 8 ⊢ 𝑋 = seq1( · , 𝐹)
54	53	fveq1i 6673	. . . . . . 7 ⊢ (𝑋‘1) = (seq1( · , 𝐹)‘1)
55		1z 12015	. . . . . . . 8 ⊢ 1 ∈ ℤ
56		seq1 13385	. . . . . . . 8 ⊢ (1 ∈ ℤ → (seq1( · , 𝐹)‘1) = (𝐹‘1))
57	55, 56	ax-mp 5	. . . . . . 7 ⊢ (seq1( · , 𝐹)‘1) = (𝐹‘1)
58	54, 57	eqtri 2846	. . . . . 6 ⊢ (𝑋‘1) = (𝐹‘1)
59	58	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑋‘1) = (𝐹‘1))
60	49, 52, 59	3brtr4d 5100	. . . 4 ⊢ (𝜑 → (𝐴↑1) < (𝑋‘1))
61	60	a1i 11	. . 3 ⊢ (𝑀 ∈ (ℤ_≥‘1) → (𝜑 → (𝐴↑1) < (𝑋‘1)))
62	50	3ad2ant3 1131	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 ∈ ℝ⁺)
63	62	rpred 12434	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 ∈ ℝ)
64		elfzouz 13045	. . . . . . . . 9 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ (ℤ_≥‘1))
65		elnnuz 12285	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ_≥‘1))
66		nnnn0 11907	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
67	65, 66	sylbir 237	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ_≥‘1) → 𝑚 ∈ ℕ₀)
68	64, 67	syl 17	. . . . . . . 8 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ ℕ₀)
69	68	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝑚 ∈ ℕ₀)
70	63, 69	reexpcld 13530	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑𝑚) ∈ ℝ)
71	53	fveq1i 6673	. . . . . . . 8 ⊢ (𝑋‘𝑚) = (seq1( · , 𝐹)‘𝑚)
72	64	adantr 483	. . . . . . . . 9 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → 𝑚 ∈ (ℤ_≥‘1))
73		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑚 ∈ (1..^𝑀)
74	73, 32	nfan 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑚 ∈ (1..^𝑀) ∧ 𝜑)
75		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖 𝑎 ∈ (1...𝑚)
76	74, 75	nfan 1900	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚))
77		nfcv 2979	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖𝑎
78	37, 77	nffv 6682	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐹‘𝑎)
79	78	nfel1 2996	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝐹‘𝑎) ∈ ℝ
80	76, 79	nfim 1897	. . . . . . . . . 10 ⊢ Ⅎ𝑖(((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚)) → (𝐹‘𝑎) ∈ ℝ)
81		eleq1 2902	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (1...𝑚) ↔ 𝑎 ∈ (1...𝑚)))
82	81	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑎 → (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) ↔ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚))))
83		fveq2 6672	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑎 → (𝐹‘𝑖) = (𝐹‘𝑎))
84	83	eleq1d 2899	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑎 → ((𝐹‘𝑖) ∈ ℝ ↔ (𝐹‘𝑎) ∈ ℝ))
85	82, 84	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → ((((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → (𝐹‘𝑖) ∈ ℝ) ↔ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚)) → (𝐹‘𝑎) ∈ ℝ)))
86		stoweidlem3.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(1...𝑀)⟶ℝ)
87	86	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝐹:(1...𝑀)⟶ℝ)
88		1zzd 12016	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 1 ∈ ℤ)
89	23	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑀 ∈ ℤ)
90		elfzelz 12911	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℤ)
91	90	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℤ)
92	88, 89, 91	3jca 1124	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ))
93		elfzle1 12913	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑚) → 1 ≤ 𝑖)
94	93	adantl 484	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 1 ≤ 𝑖)
95	90	zred 12090	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℝ)
96	95	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℝ)
97		elfzoelz 13041	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ ℤ)
98	97	zred 12090	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ ℝ)
99	98	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑚 ∈ ℝ)
100	1	nnred 11655	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
101	100	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑀 ∈ ℝ)
102		elfzle2 12914	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ≤ 𝑚)
103	102	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ≤ 𝑚)
104		elfzoel2 13040	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑀 ∈ ℤ)
105	104	zred 12090	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑀 ∈ ℝ)
106		elfzolt2 13050	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 < 𝑀)
107	98, 105, 106	ltled 10790	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ≤ 𝑀)
108	107	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑚 ≤ 𝑀)
109	96, 99, 101, 103, 108	letrd 10799	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ≤ 𝑀)
110	92, 94, 109	jca32 518	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)))
111		elfz2 12902	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)))
112	110, 111	sylibr 236	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ (1...𝑀))
113	87, 112	ffvelrnd 6854	. . . . . . . . . 10 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → (𝐹‘𝑖) ∈ ℝ)
114	80, 85, 113	chvarfv 2242	. . . . . . . . 9 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚)) → (𝐹‘𝑎) ∈ ℝ)
115		remulcl 10624	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑎 · 𝑗) ∈ ℝ)
116	115	adantl 484	. . . . . . . . 9 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ)
117	72, 114, 116	seqcl 13393	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (seq1( · , 𝐹)‘𝑚) ∈ ℝ)
118	71, 117	eqeltrid 2919	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (𝑋‘𝑚) ∈ ℝ)
119	118	3adant2 1127	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑋‘𝑚) ∈ ℝ)
120	86	3ad2ant3 1131	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐹:(1...𝑀)⟶ℝ)
121		fzofzp1 13137	. . . . . . . 8 ⊢ (𝑚 ∈ (1..^𝑀) → (𝑚 + 1) ∈ (1...𝑀))
122	121	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑚 + 1) ∈ (1...𝑀))
123	120, 122	ffvelrnd 6854	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐹‘(𝑚 + 1)) ∈ ℝ)
124	50	rpge0d 12438	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝐴)
125	124	3ad2ant3 1131	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 0 ≤ 𝐴)
126	63, 69, 125	expge0d 13531	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 0 ≤ (𝐴↑𝑚))
127		simp3 1134	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝜑)
128		simp2 1133	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)))
129	127, 128	mpd 15	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑𝑚) < (𝑋‘𝑚))
130	121	adantr 483	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (𝑚 + 1) ∈ (1...𝑀))
131		simpr 487	. . . . . . . . 9 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → 𝜑)
132	131, 130	jca 514	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀)))
133		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝑚 + 1) ∈ (1...𝑀)
134	32, 133	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀))
135		nfcv 2979	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑚 + 1)
136	37, 135	nffv 6682	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝐹‘(𝑚 + 1))
137	35, 36, 136	nfbr 5115	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐴 < (𝐹‘(𝑚 + 1))
138	134, 137	nfim 1897	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀)) → 𝐴 < (𝐹‘(𝑚 + 1)))
139		eleq1 2902	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑚 + 1) → (𝑖 ∈ (1...𝑀) ↔ (𝑚 + 1) ∈ (1...𝑀)))
140	139	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑖 = (𝑚 + 1) → ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ↔ (𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀))))
141		fveq2 6672	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑚 + 1) → (𝐹‘𝑖) = (𝐹‘(𝑚 + 1)))
142	141	breq2d 5080	. . . . . . . . . 10 ⊢ (𝑖 = (𝑚 + 1) → (𝐴 < (𝐹‘𝑖) ↔ 𝐴 < (𝐹‘(𝑚 + 1))))
143	140, 142	imbi12d 347	. . . . . . . . 9 ⊢ (𝑖 = (𝑚 + 1) → (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐴 < (𝐹‘𝑖)) ↔ ((𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀)) → 𝐴 < (𝐹‘(𝑚 + 1)))))
144	138, 143, 47	vtoclg1f 3568	. . . . . . . 8 ⊢ ((𝑚 + 1) ∈ (1...𝑀) → ((𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀)) → 𝐴 < (𝐹‘(𝑚 + 1))))
145	130, 132, 144	sylc 65	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → 𝐴 < (𝐹‘(𝑚 + 1)))
146	145	3adant2 1127	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 < (𝐹‘(𝑚 + 1)))
147	70, 119, 63, 123, 126, 129, 125, 146	ltmul12ad 11583	. . . . 5 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → ((𝐴↑𝑚) · 𝐴) < ((𝑋‘𝑚) · (𝐹‘(𝑚 + 1))))
148	51	3ad2ant3 1131	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 ∈ ℂ)
149	148, 69	expp1d 13514	. . . . 5 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑(𝑚 + 1)) = ((𝐴↑𝑚) · 𝐴))
150	53	fveq1i 6673	. . . . . . 7 ⊢ (𝑋‘(𝑚 + 1)) = (seq1( · , 𝐹)‘(𝑚 + 1))
151	150	a1i 11	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑋‘(𝑚 + 1)) = (seq1( · , 𝐹)‘(𝑚 + 1)))
152	64	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝑚 ∈ (ℤ_≥‘1))
153		seqp1 13387	. . . . . . 7 ⊢ (𝑚 ∈ (ℤ_≥‘1) → (seq1( · , 𝐹)‘(𝑚 + 1)) = ((seq1( · , 𝐹)‘𝑚) · (𝐹‘(𝑚 + 1))))
154	152, 153	syl 17	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (seq1( · , 𝐹)‘(𝑚 + 1)) = ((seq1( · , 𝐹)‘𝑚) · (𝐹‘(𝑚 + 1))))
155	71	a1i 11	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑋‘𝑚) = (seq1( · , 𝐹)‘𝑚))
156	155	eqcomd 2829	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (seq1( · , 𝐹)‘𝑚) = (𝑋‘𝑚))
157	156	oveq1d 7173	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → ((seq1( · , 𝐹)‘𝑚) · (𝐹‘(𝑚 + 1))) = ((𝑋‘𝑚) · (𝐹‘(𝑚 + 1))))
158	151, 154, 157	3eqtrd 2862	. . . . 5 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑋‘(𝑚 + 1)) = ((𝑋‘𝑚) · (𝐹‘(𝑚 + 1))))
159	147, 149, 158	3brtr4d 5100	. . . 4 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1)))
160	159	3exp 1115	. . 3 ⊢ (𝑚 ∈ (1..^𝑀) → ((𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) → (𝜑 → (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1)))))
161	9, 13, 17, 21, 61, 160	fzind2 13158	. 2 ⊢ (𝑀 ∈ (1...𝑀) → (𝜑 → (𝐴↑𝑀) < (𝑋‘𝑀)))
162	5, 161	mpcom 38	1 ⊢ (𝜑 → (𝐴↑𝑀) < (𝑋‘𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 < clt 10677 ≤ cle 10678 ℕcn 11640 ℕ₀cn0 11900 ℤcz 11984 ℤ_≥cuz 12246 ℝ⁺crp 12392 ...cfz 12895 ..^cfzo 13036 seqcseq 13372 ↑cexp 13432

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433

This theorem is referenced by: stoweidlem42 42334

W3C validator