stoweidlem3 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem3 42645

Description: Lemma for stoweid 42705: 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 12270	. . . 4 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
3	1, 2	sylib 221	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
4		eluzfz2 12910	. . 3 ⊢ (𝑀 ∈ (ℤ_≥‘1) → 𝑀 ∈ (1...𝑀))
5	3, 4	syl 17	. 2 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
6		oveq2 7143	. . . . 5 ⊢ (𝑛 = 1 → (𝐴↑𝑛) = (𝐴↑1))
7		fveq2 6645	. . . . 5 ⊢ (𝑛 = 1 → (𝑋‘𝑛) = (𝑋‘1))
8	6, 7	breq12d 5043	. . . 4 ⊢ (𝑛 = 1 → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑1) < (𝑋‘1)))
9	8	imbi2d 344	. . 3 ⊢ (𝑛 = 1 → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑1) < (𝑋‘1))))
10		oveq2 7143	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴↑𝑛) = (𝐴↑𝑚))
11		fveq2 6645	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑋‘𝑛) = (𝑋‘𝑚))
12	10, 11	breq12d 5043	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑𝑚) < (𝑋‘𝑚)))
13	12	imbi2d 344	. . 3 ⊢ (𝑛 = 𝑚 → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚))))
14		oveq2 7143	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝐴↑𝑛) = (𝐴↑(𝑚 + 1)))
15		fveq2 6645	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑋‘𝑛) = (𝑋‘(𝑚 + 1)))
16	14, 15	breq12d 5043	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1))))
17	16	imbi2d 344	. . 3 ⊢ (𝑛 = (𝑚 + 1) → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1)))))
18		oveq2 7143	. . . . 5 ⊢ (𝑛 = 𝑀 → (𝐴↑𝑛) = (𝐴↑𝑀))
19		fveq2 6645	. . . . 5 ⊢ (𝑛 = 𝑀 → (𝑋‘𝑛) = (𝑋‘𝑀))
20	18, 19	breq12d 5043	. . . 4 ⊢ (𝑛 = 𝑀 → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑𝑀) < (𝑋‘𝑀)))
21	20	imbi2d 344	. . 3 ⊢ (𝑛 = 𝑀 → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑𝑀) < (𝑋‘𝑀))))
22		1zzd 12001	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
23	1	nnzd 12074	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
24	22, 23, 22	3jca 1125	. . . . . . . 8 ⊢ (𝜑 → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 1 ∈ ℤ))
25		1le1 11257	. . . . . . . . 9 ⊢ 1 ≤ 1
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ 1)
27	1	nnge1d 11673	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝑀)
28	24, 26, 27	jca32 519	. . . . . . 7 ⊢ (𝜑 → ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (1 ≤ 1 ∧ 1 ≤ 𝑀)))
29		elfz2 12892	. . . . . . 7 ⊢ (1 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (1 ≤ 1 ∧ 1 ≤ 𝑀)))
30	28, 29	sylibr 237	. . . . . 6 ⊢ (𝜑 → 1 ∈ (1...𝑀))
31	30	ancli 552	. . . . . 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 2955	. . . . . . . . 9 ⊢ Ⅎ𝑖𝐴
36		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎ𝑖 <
37		stoweidlem3.1	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝐹
38		nfcv 2955	. . . . . . . . . 10 ⊢ Ⅎ𝑖1
39	37, 38	nffv 6655	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐹‘1)
40	35, 36, 39	nfbr 5077	. . . . . . . 8 ⊢ Ⅎ𝑖 𝐴 < (𝐹‘1)
41	34, 40	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑖((𝜑 ∧ 1 ∈ (1...𝑀)) → 𝐴 < (𝐹‘1))
42		eleq1 2877	. . . . . . . . 9 ⊢ (𝑖 = 1 → (𝑖 ∈ (1...𝑀) ↔ 1 ∈ (1...𝑀)))
43	42	anbi2d 631	. . . . . . . 8 ⊢ (𝑖 = 1 → ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ↔ (𝜑 ∧ 1 ∈ (1...𝑀))))
44		fveq2 6645	. . . . . . . . 9 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = (𝐹‘1))
45	44	breq2d 5042	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐴 < (𝐹‘𝑖) ↔ 𝐴 < (𝐹‘1)))
46	43, 45	imbi12d 348	. . . . . . 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 3514	. . . . . 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 12421	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
52	51	exp1d 13501	. . . . 5 ⊢ (𝜑 → (𝐴↑1) = 𝐴)
53		stoweidlem3.3	. . . . . . . 8 ⊢ 𝑋 = seq1( · , 𝐹)
54	53	fveq1i 6646	. . . . . . 7 ⊢ (𝑋‘1) = (seq1( · , 𝐹)‘1)
55		1z 12000	. . . . . . . 8 ⊢ 1 ∈ ℤ
56		seq1 13377	. . . . . . . 8 ⊢ (1 ∈ ℤ → (seq1( · , 𝐹)‘1) = (𝐹‘1))
57	55, 56	ax-mp 5	. . . . . . 7 ⊢ (seq1( · , 𝐹)‘1) = (𝐹‘1)
58	54, 57	eqtri 2821	. . . . . 6 ⊢ (𝑋‘1) = (𝐹‘1)
59	58	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑋‘1) = (𝐹‘1))
60	49, 52, 59	3brtr4d 5062	. . . 4 ⊢ (𝜑 → (𝐴↑1) < (𝑋‘1))
61	60	a1i 11	. . 3 ⊢ (𝑀 ∈ (ℤ_≥‘1) → (𝜑 → (𝐴↑1) < (𝑋‘1)))
62	50	3ad2ant3 1132	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 ∈ ℝ⁺)
63	62	rpred 12419	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 ∈ ℝ)
64		elfzouz 13037	. . . . . . . . 9 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ (ℤ_≥‘1))
65		elnnuz 12270	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ_≥‘1))
66		nnnn0 11892	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
67	65, 66	sylbir 238	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ_≥‘1) → 𝑚 ∈ ℕ₀)
68	64, 67	syl 17	. . . . . . . 8 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ ℕ₀)
69	68	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝑚 ∈ ℕ₀)
70	63, 69	reexpcld 13523	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑𝑚) ∈ ℝ)
71	53	fveq1i 6646	. . . . . . . 8 ⊢ (𝑋‘𝑚) = (seq1( · , 𝐹)‘𝑚)
72	64	adantr 484	. . . . . . . . 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 2955	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖𝑎
78	37, 77	nffv 6655	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐹‘𝑎)
79	78	nfel1 2971	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝐹‘𝑎) ∈ ℝ
80	76, 79	nfim 1897	. . . . . . . . . 10 ⊢ Ⅎ𝑖(((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚)) → (𝐹‘𝑎) ∈ ℝ)
81		eleq1 2877	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (1...𝑚) ↔ 𝑎 ∈ (1...𝑚)))
82	81	anbi2d 631	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑎 → (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) ↔ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚))))
83		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑎 → (𝐹‘𝑖) = (𝐹‘𝑎))
84	83	eleq1d 2874	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑎 → ((𝐹‘𝑖) ∈ ℝ ↔ (𝐹‘𝑎) ∈ ℝ))
85	82, 84	imbi12d 348	. . . . . . . . . 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 726	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝐹:(1...𝑀)⟶ℝ)
88		1zzd 12001	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 1 ∈ ℤ)
89	23	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑀 ∈ ℤ)
90		elfzelz 12902	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℤ)
91	90	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℤ)
92	88, 89, 91	3jca 1125	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ))
93		elfzle1 12905	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑚) → 1 ≤ 𝑖)
94	93	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 1 ≤ 𝑖)
95	90	zred 12075	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℝ)
96	95	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℝ)
97		elfzoelz 13033	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ ℤ)
98	97	zred 12075	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ ℝ)
99	98	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑚 ∈ ℝ)
100	1	nnred 11640	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
101	100	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑀 ∈ ℝ)
102		elfzle2 12906	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ≤ 𝑚)
103	102	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ≤ 𝑚)
104		elfzoel2 13032	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑀 ∈ ℤ)
105	104	zred 12075	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑀 ∈ ℝ)
106		elfzolt2 13042	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 < 𝑀)
107	98, 105, 106	ltled 10777	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ≤ 𝑀)
108	107	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑚 ≤ 𝑀)
109	96, 99, 101, 103, 108	letrd 10786	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ≤ 𝑀)
110	92, 94, 109	jca32 519	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)))
111		elfz2 12892	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)))
112	110, 111	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ (1...𝑀))
113	87, 112	ffvelrnd 6829	. . . . . . . . . 10 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → (𝐹‘𝑖) ∈ ℝ)
114	80, 85, 113	chvarfv 2240	. . . . . . . . 9 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚)) → (𝐹‘𝑎) ∈ ℝ)
115		remulcl 10611	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑎 · 𝑗) ∈ ℝ)
116	115	adantl 485	. . . . . . . . 9 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ)
117	72, 114, 116	seqcl 13386	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (seq1( · , 𝐹)‘𝑚) ∈ ℝ)
118	71, 117	eqeltrid 2894	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (𝑋‘𝑚) ∈ ℝ)
119	118	3adant2 1128	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑋‘𝑚) ∈ ℝ)
120	86	3ad2ant3 1132	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐹:(1...𝑀)⟶ℝ)
121		fzofzp1 13129	. . . . . . . 8 ⊢ (𝑚 ∈ (1..^𝑀) → (𝑚 + 1) ∈ (1...𝑀))
122	121	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑚 + 1) ∈ (1...𝑀))
123	120, 122	ffvelrnd 6829	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐹‘(𝑚 + 1)) ∈ ℝ)
124	50	rpge0d 12423	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝐴)
125	124	3ad2ant3 1132	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 0 ≤ 𝐴)
126	63, 69, 125	expge0d 13524	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 0 ≤ (𝐴↑𝑚))
127		simp3 1135	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝜑)
128		simp2 1134	. . . . . . 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 484	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (𝑚 + 1) ∈ (1...𝑀))
131		simpr 488	. . . . . . . . 9 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → 𝜑)
132	131, 130	jca 515	. . . . . . . 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 2955	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑚 + 1)
136	37, 135	nffv 6655	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝐹‘(𝑚 + 1))
137	35, 36, 136	nfbr 5077	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐴 < (𝐹‘(𝑚 + 1))
138	134, 137	nfim 1897	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀)) → 𝐴 < (𝐹‘(𝑚 + 1)))
139		eleq1 2877	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑚 + 1) → (𝑖 ∈ (1...𝑀) ↔ (𝑚 + 1) ∈ (1...𝑀)))
140	139	anbi2d 631	. . . . . . . . . 10 ⊢ (𝑖 = (𝑚 + 1) → ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ↔ (𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀))))
141		fveq2 6645	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑚 + 1) → (𝐹‘𝑖) = (𝐹‘(𝑚 + 1)))
142	141	breq2d 5042	. . . . . . . . . 10 ⊢ (𝑖 = (𝑚 + 1) → (𝐴 < (𝐹‘𝑖) ↔ 𝐴 < (𝐹‘(𝑚 + 1))))
143	140, 142	imbi12d 348	. . . . . . . . 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 3514	. . . . . . . 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 1128	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 < (𝐹‘(𝑚 + 1)))
147	70, 119, 63, 123, 126, 129, 125, 146	ltmul12ad 11570	. . . . 5 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → ((𝐴↑𝑚) · 𝐴) < ((𝑋‘𝑚) · (𝐹‘(𝑚 + 1))))
148	51	3ad2ant3 1132	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 ∈ ℂ)
149	148, 69	expp1d 13507	. . . . 5 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑(𝑚 + 1)) = ((𝐴↑𝑚) · 𝐴))
150	53	fveq1i 6646	. . . . . . 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 1130	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝑚 ∈ (ℤ_≥‘1))
153		seqp1 13379	. . . . . . 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 2804	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (seq1( · , 𝐹)‘𝑚) = (𝑋‘𝑚))
157	156	oveq1d 7150	. . . . . 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 2837	. . . . 5 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑋‘(𝑚 + 1)) = ((𝑋‘𝑚) · (𝐹‘(𝑚 + 1))))
159	147, 149, 158	3brtr4d 5062	. . . 4 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1)))
160	159	3exp 1116	. . 3 ⊢ (𝑚 ∈ (1..^𝑀) → ((𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) → (𝜑 → (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1)))))
161	9, 13, 17, 21, 61, 160	fzind2 13150	. 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 399 ∧ w3a 1084 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 ℕcn 11625 ℕ₀cn0 11885 ℤcz 11969 ℤ_≥cuz 12231 ℝ⁺crp 12377 ...cfz 12885 ..^cfzo 13028 seqcseq 13364 ↑cexp 13425

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426

This theorem is referenced by: stoweidlem42 42684

W3C validator