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 46577

Description: Lemma for stoweid 46637: 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 12879	. . . 4 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
3	1, 2	sylib 220	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
4		eluzfz2 13537	. . 3 ⊢ (𝑀 ∈ (ℤ_≥‘1) → 𝑀 ∈ (1...𝑀))
5	3, 4	syl 17	. 2 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
6		oveq2 7404	. . . . 5 ⊢ (𝑛 = 1 → (𝐴↑𝑛) = (𝐴↑1))
7		fveq2 6867	. . . . 5 ⊢ (𝑛 = 1 → (𝑋‘𝑛) = (𝑋‘1))
8	6, 7	breq12d 5113	. . . 4 ⊢ (𝑛 = 1 → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑1) < (𝑋‘1)))
9	8	imbi2d 342	. . 3 ⊢ (𝑛 = 1 → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑1) < (𝑋‘1))))
10		oveq2 7404	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴↑𝑛) = (𝐴↑𝑚))
11		fveq2 6867	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑋‘𝑛) = (𝑋‘𝑚))
12	10, 11	breq12d 5113	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑𝑚) < (𝑋‘𝑚)))
13	12	imbi2d 342	. . 3 ⊢ (𝑛 = 𝑚 → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚))))
14		oveq2 7404	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝐴↑𝑛) = (𝐴↑(𝑚 + 1)))
15		fveq2 6867	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑋‘𝑛) = (𝑋‘(𝑚 + 1)))
16	14, 15	breq12d 5113	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1))))
17	16	imbi2d 342	. . 3 ⊢ (𝑛 = (𝑚 + 1) → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1)))))
18		oveq2 7404	. . . . 5 ⊢ (𝑛 = 𝑀 → (𝐴↑𝑛) = (𝐴↑𝑀))
19		fveq2 6867	. . . . 5 ⊢ (𝑛 = 𝑀 → (𝑋‘𝑛) = (𝑋‘𝑀))
20	18, 19	breq12d 5113	. . . 4 ⊢ (𝑛 = 𝑀 → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑𝑀) < (𝑋‘𝑀)))
21	20	imbi2d 342	. . 3 ⊢ (𝑛 = 𝑀 → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑𝑀) < (𝑋‘𝑀))))
22		1zzd 12602	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
23	1	nnzd 12594	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
24		1le1 11815	. . . . . . . 8 ⊢ 1 ≤ 1
25	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ≤ 1)
26	1	nnge1d 12261	. . . . . . 7 ⊢ (𝜑 → 1 ≤ 𝑀)
27	22, 23, 22, 25, 26	elfzd 13520	. . . . . 6 ⊢ (𝜑 → 1 ∈ (1...𝑀))
28	27	ancli 556	. . . . . 6 ⊢ (𝜑 → (𝜑 ∧ 1 ∈ (1...𝑀)))
29		stoweidlem3.2	. . . . . . . . 9 ⊢ Ⅎ𝑖𝜑
30		nfv 1934	. . . . . . . . 9 ⊢ Ⅎ𝑖1 ∈ (1...𝑀)
31	29, 30	nfan 1919	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ 1 ∈ (1...𝑀))
32		nfcv 2924	. . . . . . . . 9 ⊢ Ⅎ𝑖𝐴
33		nfcv 2924	. . . . . . . . 9 ⊢ Ⅎ𝑖 <
34		stoweidlem3.1	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝐹
35		nfcv 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑖1
36	34, 35	nffv 6877	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐹‘1)
37	32, 33, 36	nfbr 5147	. . . . . . . 8 ⊢ Ⅎ𝑖 𝐴 < (𝐹‘1)
38	31, 37	nfim 1916	. . . . . . 7 ⊢ Ⅎ𝑖((𝜑 ∧ 1 ∈ (1...𝑀)) → 𝐴 < (𝐹‘1))
39		eleq1 2850	. . . . . . . . 9 ⊢ (𝑖 = 1 → (𝑖 ∈ (1...𝑀) ↔ 1 ∈ (1...𝑀)))
40	39	anbi2d 639	. . . . . . . 8 ⊢ (𝑖 = 1 → ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ↔ (𝜑 ∧ 1 ∈ (1...𝑀))))
41		fveq2 6867	. . . . . . . . 9 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = (𝐹‘1))
42	41	breq2d 5112	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐴 < (𝐹‘𝑖) ↔ 𝐴 < (𝐹‘1)))
43	40, 42	imbi12d 346	. . . . . . 7 ⊢ (𝑖 = 1 → (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐴 < (𝐹‘𝑖)) ↔ ((𝜑 ∧ 1 ∈ (1...𝑀)) → 𝐴 < (𝐹‘1))))
44		stoweidlem3.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐴 < (𝐹‘𝑖))
45	38, 43, 44	vtoclg1f 3535	. . . . . 6 ⊢ (1 ∈ (1...𝑀) → ((𝜑 ∧ 1 ∈ (1...𝑀)) → 𝐴 < (𝐹‘1)))
46	27, 28, 45	sylc 65	. . . . 5 ⊢ (𝜑 → 𝐴 < (𝐹‘1))
47		stoweidlem3.7	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
48	47	rpcnd 13039	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
49	48	exp1d 14154	. . . . 5 ⊢ (𝜑 → (𝐴↑1) = 𝐴)
50		stoweidlem3.3	. . . . . . . 8 ⊢ 𝑋 = seq1( · , 𝐹)
51	50	fveq1i 6868	. . . . . . 7 ⊢ (𝑋‘1) = (seq1( · , 𝐹)‘1)
52		1z 12601	. . . . . . . 8 ⊢ 1 ∈ ℤ
53		seq1 14027	. . . . . . . 8 ⊢ (1 ∈ ℤ → (seq1( · , 𝐹)‘1) = (𝐹‘1))
54	52, 53	ax-mp 5	. . . . . . 7 ⊢ (seq1( · , 𝐹)‘1) = (𝐹‘1)
55	51, 54	eqtri 2785	. . . . . 6 ⊢ (𝑋‘1) = (𝐹‘1)
56	55	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑋‘1) = (𝐹‘1))
57	46, 49, 56	3brtr4d 5132	. . . 4 ⊢ (𝜑 → (𝐴↑1) < (𝑋‘1))
58	57	a1i 11	. . 3 ⊢ (𝑀 ∈ (ℤ_≥‘1) → (𝜑 → (𝐴↑1) < (𝑋‘1)))
59	47	3ad2ant3 1148	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 ∈ ℝ⁺)
60	59	rpred 13037	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 ∈ ℝ)
61		elfzouz 13669	. . . . . . . . 9 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ (ℤ_≥‘1))
62		elnnuz 12879	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ_≥‘1))
63		nnnn0 12488	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
64	62, 63	sylbir 237	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ_≥‘1) → 𝑚 ∈ ℕ₀)
65	61, 64	syl 17	. . . . . . . 8 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ ℕ₀)
66	65	3ad2ant1 1146	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝑚 ∈ ℕ₀)
67	60, 66	reexpcld 14176	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑𝑚) ∈ ℝ)
68	50	fveq1i 6868	. . . . . . . 8 ⊢ (𝑋‘𝑚) = (seq1( · , 𝐹)‘𝑚)
69	61	adantr 484	. . . . . . . . 9 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → 𝑚 ∈ (ℤ_≥‘1))
70		nfv 1934	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑚 ∈ (1..^𝑀)
71	70, 29	nfan 1919	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑚 ∈ (1..^𝑀) ∧ 𝜑)
72		nfv 1934	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖 𝑎 ∈ (1...𝑚)
73	71, 72	nfan 1919	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚))
74		nfcv 2924	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖𝑎
75	34, 74	nffv 6877	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐹‘𝑎)
76	75	nfel1 2940	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝐹‘𝑎) ∈ ℝ
77	73, 76	nfim 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑖(((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚)) → (𝐹‘𝑎) ∈ ℝ)
78		eleq1 2850	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (1...𝑚) ↔ 𝑎 ∈ (1...𝑚)))
79	78	anbi2d 639	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑎 → (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) ↔ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚))))
80		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑎 → (𝐹‘𝑖) = (𝐹‘𝑎))
81	80	eleq1d 2847	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑎 → ((𝐹‘𝑖) ∈ ℝ ↔ (𝐹‘𝑎) ∈ ℝ))
82	79, 81	imbi12d 346	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → ((((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → (𝐹‘𝑖) ∈ ℝ) ↔ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚)) → (𝐹‘𝑎) ∈ ℝ)))
83		stoweidlem3.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(1...𝑀)⟶ℝ)
84	83	ad2antlr 737	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝐹:(1...𝑀)⟶ℝ)
85		1zzd 12602	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 1 ∈ ℤ)
86	23	ad2antlr 737	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑀 ∈ ℤ)
87		elfzelz 13529	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℤ)
88	87	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℤ)
89		elfzle1 13532	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑚) → 1 ≤ 𝑖)
90	89	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 1 ≤ 𝑖)
91	87	zred 12677	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℝ)
92	91	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℝ)
93		elfzoelz 13664	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ ℤ)
94	93	zred 12677	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ ℝ)
95	94	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑚 ∈ ℝ)
96	1	nnred 12225	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
97	96	ad2antlr 737	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑀 ∈ ℝ)
98		elfzle2 13533	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ≤ 𝑚)
99	98	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ≤ 𝑚)
100		elfzoel2 13663	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑀 ∈ ℤ)
101	100	zred 12677	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑀 ∈ ℝ)
102		elfzolt2 13674	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 < 𝑀)
103	94, 101, 102	ltled 11331	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ≤ 𝑀)
104	103	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑚 ≤ 𝑀)
105	92, 95, 97, 99, 104	letrd 11340	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ≤ 𝑀)
106	85, 86, 88, 90, 105	elfzd 13520	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ (1...𝑀))
107	84, 106	ffvelcdmd 7066	. . . . . . . . . 10 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → (𝐹‘𝑖) ∈ ℝ)
108	77, 82, 107	chvarfv 2275	. . . . . . . . 9 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚)) → (𝐹‘𝑎) ∈ ℝ)
109		remulcl 11158	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑎 · 𝑗) ∈ ℝ)
110	109	adantl 485	. . . . . . . . 9 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ)
111	69, 108, 110	seqcl 14035	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (seq1( · , 𝐹)‘𝑚) ∈ ℝ)
112	68, 111	eqeltrid 2866	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (𝑋‘𝑚) ∈ ℝ)
113	112	3adant2 1144	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑋‘𝑚) ∈ ℝ)
114	83	3ad2ant3 1148	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐹:(1...𝑀)⟶ℝ)
115		fzofzp1 13770	. . . . . . . 8 ⊢ (𝑚 ∈ (1..^𝑀) → (𝑚 + 1) ∈ (1...𝑀))
116	115	3ad2ant1 1146	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑚 + 1) ∈ (1...𝑀))
117	114, 116	ffvelcdmd 7066	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐹‘(𝑚 + 1)) ∈ ℝ)
118	47	rpge0d 13041	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝐴)
119	118	3ad2ant3 1148	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 0 ≤ 𝐴)
120	60, 66, 119	expge0d 14177	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 0 ≤ (𝐴↑𝑚))
121		simp3 1151	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝜑)
122		simp2 1150	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)))
123	121, 122	mpd 15	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑𝑚) < (𝑋‘𝑚))
124	115	adantr 484	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (𝑚 + 1) ∈ (1...𝑀))
125		simpr 488	. . . . . . . . 9 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → 𝜑)
126	125, 124	jca 519	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀)))
127		nfv 1934	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝑚 + 1) ∈ (1...𝑀)
128	29, 127	nfan 1919	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀))
129		nfcv 2924	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑚 + 1)
130	34, 129	nffv 6877	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝐹‘(𝑚 + 1))
131	32, 33, 130	nfbr 5147	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐴 < (𝐹‘(𝑚 + 1))
132	128, 131	nfim 1916	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀)) → 𝐴 < (𝐹‘(𝑚 + 1)))
133		eleq1 2850	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑚 + 1) → (𝑖 ∈ (1...𝑀) ↔ (𝑚 + 1) ∈ (1...𝑀)))
134	133	anbi2d 639	. . . . . . . . . 10 ⊢ (𝑖 = (𝑚 + 1) → ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ↔ (𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀))))
135		fveq2 6867	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑚 + 1) → (𝐹‘𝑖) = (𝐹‘(𝑚 + 1)))
136	135	breq2d 5112	. . . . . . . . . 10 ⊢ (𝑖 = (𝑚 + 1) → (𝐴 < (𝐹‘𝑖) ↔ 𝐴 < (𝐹‘(𝑚 + 1))))
137	134, 136	imbi12d 346	. . . . . . . . 9 ⊢ (𝑖 = (𝑚 + 1) → (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐴 < (𝐹‘𝑖)) ↔ ((𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀)) → 𝐴 < (𝐹‘(𝑚 + 1)))))
138	132, 137, 44	vtoclg1f 3535	. . . . . . . 8 ⊢ ((𝑚 + 1) ∈ (1...𝑀) → ((𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀)) → 𝐴 < (𝐹‘(𝑚 + 1))))
139	124, 126, 138	sylc 65	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → 𝐴 < (𝐹‘(𝑚 + 1)))
140	139	3adant2 1144	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 < (𝐹‘(𝑚 + 1)))
141	67, 113, 60, 117, 120, 123, 119, 140	ltmul12ad 12133	. . . . 5 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → ((𝐴↑𝑚) · 𝐴) < ((𝑋‘𝑚) · (𝐹‘(𝑚 + 1))))
142	48	3ad2ant3 1148	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 ∈ ℂ)
143	142, 66	expp1d 14160	. . . . 5 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑(𝑚 + 1)) = ((𝐴↑𝑚) · 𝐴))
144	50	fveq1i 6868	. . . . . . 7 ⊢ (𝑋‘(𝑚 + 1)) = (seq1( · , 𝐹)‘(𝑚 + 1))
145	144	a1i 11	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑋‘(𝑚 + 1)) = (seq1( · , 𝐹)‘(𝑚 + 1)))
146	61	3ad2ant1 1146	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝑚 ∈ (ℤ_≥‘1))
147		seqp1 14029	. . . . . . 7 ⊢ (𝑚 ∈ (ℤ_≥‘1) → (seq1( · , 𝐹)‘(𝑚 + 1)) = ((seq1( · , 𝐹)‘𝑚) · (𝐹‘(𝑚 + 1))))
148	146, 147	syl 17	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (seq1( · , 𝐹)‘(𝑚 + 1)) = ((seq1( · , 𝐹)‘𝑚) · (𝐹‘(𝑚 + 1))))
149	68	a1i 11	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑋‘𝑚) = (seq1( · , 𝐹)‘𝑚))
150	149	eqcomd 2768	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (seq1( · , 𝐹)‘𝑚) = (𝑋‘𝑚))
151	150	oveq1d 7411	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → ((seq1( · , 𝐹)‘𝑚) · (𝐹‘(𝑚 + 1))) = ((𝑋‘𝑚) · (𝐹‘(𝑚 + 1))))
152	145, 148, 151	3eqtrd 2801	. . . . 5 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑋‘(𝑚 + 1)) = ((𝑋‘𝑚) · (𝐹‘(𝑚 + 1))))
153	141, 143, 152	3brtr4d 5132	. . . 4 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1)))
154	153	3exp 1132	. . 3 ⊢ (𝑚 ∈ (1..^𝑀) → ((𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) → (𝜑 → (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1)))))
155	9, 13, 17, 21, 58, 154	fzind2 13794	. 2 ⊢ (𝑀 ∈ (1...𝑀) → (𝜑 → (𝐴↑𝑀) < (𝑋‘𝑀)))
156	5, 155	mpcom 38	1 ⊢ (𝜑 → (𝐴↑𝑀) < (𝑋‘𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1098 = wceq 1560 Ⅎwnf 1803 ∈ wcel 2142 Ⅎwnfc 2909 class class class wbr 5100 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ℂcc 11071 ℝcr 11072 0cc0 11073 1c1 11074 + caddc 11076 · cmul 11078 < clt 11216 ≤ cle 11217 ℕcn 12210 ℕ₀cn0 12481 ℤcz 12568 ℤ_≥cuz 12839 ℝ⁺crp 12993 ...cfz 13512 ..^cfzo 13659 seqcseq 14014 ↑cexp 14074

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-n0 12482 df-z 12569 df-uz 12840 df-rp 12994 df-fz 13513 df-fzo 13660 df-seq 14015 df-exp 14075

This theorem is referenced by: stoweidlem42 46616

W3C validator