stoweidlem3 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem3 43544

Description: Lemma for stoweid 43604: 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 12622	. . . 4 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
3	1, 2	sylib 217	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
4		eluzfz2 13264	. . 3 ⊢ (𝑀 ∈ (ℤ_≥‘1) → 𝑀 ∈ (1...𝑀))
5	3, 4	syl 17	. 2 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
6		oveq2 7283	. . . . 5 ⊢ (𝑛 = 1 → (𝐴↑𝑛) = (𝐴↑1))
7		fveq2 6774	. . . . 5 ⊢ (𝑛 = 1 → (𝑋‘𝑛) = (𝑋‘1))
8	6, 7	breq12d 5087	. . . 4 ⊢ (𝑛 = 1 → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑1) < (𝑋‘1)))
9	8	imbi2d 341	. . 3 ⊢ (𝑛 = 1 → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑1) < (𝑋‘1))))
10		oveq2 7283	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴↑𝑛) = (𝐴↑𝑚))
11		fveq2 6774	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑋‘𝑛) = (𝑋‘𝑚))
12	10, 11	breq12d 5087	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑𝑚) < (𝑋‘𝑚)))
13	12	imbi2d 341	. . 3 ⊢ (𝑛 = 𝑚 → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚))))
14		oveq2 7283	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝐴↑𝑛) = (𝐴↑(𝑚 + 1)))
15		fveq2 6774	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑋‘𝑛) = (𝑋‘(𝑚 + 1)))
16	14, 15	breq12d 5087	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1))))
17	16	imbi2d 341	. . 3 ⊢ (𝑛 = (𝑚 + 1) → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1)))))
18		oveq2 7283	. . . . 5 ⊢ (𝑛 = 𝑀 → (𝐴↑𝑛) = (𝐴↑𝑀))
19		fveq2 6774	. . . . 5 ⊢ (𝑛 = 𝑀 → (𝑋‘𝑛) = (𝑋‘𝑀))
20	18, 19	breq12d 5087	. . . 4 ⊢ (𝑛 = 𝑀 → ((𝐴↑𝑛) < (𝑋‘𝑛) ↔ (𝐴↑𝑀) < (𝑋‘𝑀)))
21	20	imbi2d 341	. . 3 ⊢ (𝑛 = 𝑀 → ((𝜑 → (𝐴↑𝑛) < (𝑋‘𝑛)) ↔ (𝜑 → (𝐴↑𝑀) < (𝑋‘𝑀))))
22		1zzd 12351	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
23	1	nnzd 12425	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
24		1le1 11603	. . . . . . . 8 ⊢ 1 ≤ 1
25	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ≤ 1)
26	1	nnge1d 12021	. . . . . . 7 ⊢ (𝜑 → 1 ≤ 𝑀)
27	22, 23, 22, 25, 26	elfzd 13247	. . . . . 6 ⊢ (𝜑 → 1 ∈ (1...𝑀))
28	27	ancli 549	. . . . . 6 ⊢ (𝜑 → (𝜑 ∧ 1 ∈ (1...𝑀)))
29		stoweidlem3.2	. . . . . . . . 9 ⊢ Ⅎ𝑖𝜑
30		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑖1 ∈ (1...𝑀)
31	29, 30	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ 1 ∈ (1...𝑀))
32		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑖𝐴
33		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑖 <
34		stoweidlem3.1	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝐹
35		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑖1
36	34, 35	nffv 6784	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐹‘1)
37	32, 33, 36	nfbr 5121	. . . . . . . 8 ⊢ Ⅎ𝑖 𝐴 < (𝐹‘1)
38	31, 37	nfim 1899	. . . . . . 7 ⊢ Ⅎ𝑖((𝜑 ∧ 1 ∈ (1...𝑀)) → 𝐴 < (𝐹‘1))
39		eleq1 2826	. . . . . . . . 9 ⊢ (𝑖 = 1 → (𝑖 ∈ (1...𝑀) ↔ 1 ∈ (1...𝑀)))
40	39	anbi2d 629	. . . . . . . 8 ⊢ (𝑖 = 1 → ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ↔ (𝜑 ∧ 1 ∈ (1...𝑀))))
41		fveq2 6774	. . . . . . . . 9 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = (𝐹‘1))
42	41	breq2d 5086	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐴 < (𝐹‘𝑖) ↔ 𝐴 < (𝐹‘1)))
43	40, 42	imbi12d 345	. . . . . . 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 3504	. . . . . 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 12774	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
49	48	exp1d 13859	. . . . 5 ⊢ (𝜑 → (𝐴↑1) = 𝐴)
50		stoweidlem3.3	. . . . . . . 8 ⊢ 𝑋 = seq1( · , 𝐹)
51	50	fveq1i 6775	. . . . . . 7 ⊢ (𝑋‘1) = (seq1( · , 𝐹)‘1)
52		1z 12350	. . . . . . . 8 ⊢ 1 ∈ ℤ
53		seq1 13734	. . . . . . . 8 ⊢ (1 ∈ ℤ → (seq1( · , 𝐹)‘1) = (𝐹‘1))
54	52, 53	ax-mp 5	. . . . . . 7 ⊢ (seq1( · , 𝐹)‘1) = (𝐹‘1)
55	51, 54	eqtri 2766	. . . . . 6 ⊢ (𝑋‘1) = (𝐹‘1)
56	55	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑋‘1) = (𝐹‘1))
57	46, 49, 56	3brtr4d 5106	. . . 4 ⊢ (𝜑 → (𝐴↑1) < (𝑋‘1))
58	57	a1i 11	. . 3 ⊢ (𝑀 ∈ (ℤ_≥‘1) → (𝜑 → (𝐴↑1) < (𝑋‘1)))
59	47	3ad2ant3 1134	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 ∈ ℝ⁺)
60	59	rpred 12772	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 ∈ ℝ)
61		elfzouz 13391	. . . . . . . . 9 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ (ℤ_≥‘1))
62		elnnuz 12622	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ_≥‘1))
63		nnnn0 12240	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
64	62, 63	sylbir 234	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ_≥‘1) → 𝑚 ∈ ℕ₀)
65	61, 64	syl 17	. . . . . . . 8 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ ℕ₀)
66	65	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝑚 ∈ ℕ₀)
67	60, 66	reexpcld 13881	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑𝑚) ∈ ℝ)
68	50	fveq1i 6775	. . . . . . . 8 ⊢ (𝑋‘𝑚) = (seq1( · , 𝐹)‘𝑚)
69	61	adantr 481	. . . . . . . . 9 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → 𝑚 ∈ (ℤ_≥‘1))
70		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑚 ∈ (1..^𝑀)
71	70, 29	nfan 1902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑚 ∈ (1..^𝑀) ∧ 𝜑)
72		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖 𝑎 ∈ (1...𝑚)
73	71, 72	nfan 1902	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚))
74		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖𝑎
75	34, 74	nffv 6784	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐹‘𝑎)
76	75	nfel1 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝐹‘𝑎) ∈ ℝ
77	73, 76	nfim 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑖(((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚)) → (𝐹‘𝑎) ∈ ℝ)
78		eleq1 2826	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (1...𝑚) ↔ 𝑎 ∈ (1...𝑚)))
79	78	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑎 → (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) ↔ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚))))
80		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑎 → (𝐹‘𝑖) = (𝐹‘𝑎))
81	80	eleq1d 2823	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑎 → ((𝐹‘𝑖) ∈ ℝ ↔ (𝐹‘𝑎) ∈ ℝ))
82	79, 81	imbi12d 345	. . . . . . . . . 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 724	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝐹:(1...𝑀)⟶ℝ)
85		1zzd 12351	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 1 ∈ ℤ)
86	23	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑀 ∈ ℤ)
87		elfzelz 13256	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℤ)
88	87	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℤ)
89		elfzle1 13259	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑚) → 1 ≤ 𝑖)
90	89	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 1 ≤ 𝑖)
91	87	zred 12426	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℝ)
92	91	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℝ)
93		elfzoelz 13387	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ ℤ)
94	93	zred 12426	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ∈ ℝ)
95	94	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑚 ∈ ℝ)
96	1	nnred 11988	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
97	96	ad2antlr 724	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑀 ∈ ℝ)
98		elfzle2 13260	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ≤ 𝑚)
99	98	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ≤ 𝑚)
100		elfzoel2 13386	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑀 ∈ ℤ)
101	100	zred 12426	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑀 ∈ ℝ)
102		elfzolt2 13396	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 < 𝑀)
103	94, 101, 102	ltled 11123	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1..^𝑀) → 𝑚 ≤ 𝑀)
104	103	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑚 ≤ 𝑀)
105	92, 95, 97, 99, 104	letrd 11132	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ≤ 𝑀)
106	85, 86, 88, 90, 105	elfzd 13247	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ (1...𝑀))
107	84, 106	ffvelrnd 6962	. . . . . . . . . 10 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑖 ∈ (1...𝑚)) → (𝐹‘𝑖) ∈ ℝ)
108	77, 82, 107	chvarfv 2233	. . . . . . . . 9 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ 𝑎 ∈ (1...𝑚)) → (𝐹‘𝑎) ∈ ℝ)
109		remulcl 10956	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑎 · 𝑗) ∈ ℝ)
110	109	adantl 482	. . . . . . . . 9 ⊢ (((𝑚 ∈ (1..^𝑀) ∧ 𝜑) ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ)
111	69, 108, 110	seqcl 13743	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (seq1( · , 𝐹)‘𝑚) ∈ ℝ)
112	68, 111	eqeltrid 2843	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (𝑋‘𝑚) ∈ ℝ)
113	112	3adant2 1130	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑋‘𝑚) ∈ ℝ)
114	83	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐹:(1...𝑀)⟶ℝ)
115		fzofzp1 13484	. . . . . . . 8 ⊢ (𝑚 ∈ (1..^𝑀) → (𝑚 + 1) ∈ (1...𝑀))
116	115	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑚 + 1) ∈ (1...𝑀))
117	114, 116	ffvelrnd 6962	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐹‘(𝑚 + 1)) ∈ ℝ)
118	47	rpge0d 12776	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝐴)
119	118	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 0 ≤ 𝐴)
120	60, 66, 119	expge0d 13882	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 0 ≤ (𝐴↑𝑚))
121		simp3 1137	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝜑)
122		simp2 1136	. . . . . . 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 481	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (𝑚 + 1) ∈ (1...𝑀))
125		simpr 485	. . . . . . . . 9 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → 𝜑)
126	125, 124	jca 512	. . . . . . . 8 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ 𝜑) → (𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀)))
127		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝑚 + 1) ∈ (1...𝑀)
128	29, 127	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀))
129		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑚 + 1)
130	34, 129	nffv 6784	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝐹‘(𝑚 + 1))
131	32, 33, 130	nfbr 5121	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐴 < (𝐹‘(𝑚 + 1))
132	128, 131	nfim 1899	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀)) → 𝐴 < (𝐹‘(𝑚 + 1)))
133		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑚 + 1) → (𝑖 ∈ (1...𝑀) ↔ (𝑚 + 1) ∈ (1...𝑀)))
134	133	anbi2d 629	. . . . . . . . . 10 ⊢ (𝑖 = (𝑚 + 1) → ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ↔ (𝜑 ∧ (𝑚 + 1) ∈ (1...𝑀))))
135		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑚 + 1) → (𝐹‘𝑖) = (𝐹‘(𝑚 + 1)))
136	135	breq2d 5086	. . . . . . . . . 10 ⊢ (𝑖 = (𝑚 + 1) → (𝐴 < (𝐹‘𝑖) ↔ 𝐴 < (𝐹‘(𝑚 + 1))))
137	134, 136	imbi12d 345	. . . . . . . . 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 3504	. . . . . . . 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 1130	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 < (𝐹‘(𝑚 + 1)))
141	67, 113, 60, 117, 120, 123, 119, 140	ltmul12ad 11916	. . . . 5 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → ((𝐴↑𝑚) · 𝐴) < ((𝑋‘𝑚) · (𝐹‘(𝑚 + 1))))
142	48	3ad2ant3 1134	. . . . . 6 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝐴 ∈ ℂ)
143	142, 66	expp1d 13865	. . . . 5 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑(𝑚 + 1)) = ((𝐴↑𝑚) · 𝐴))
144	50	fveq1i 6775	. . . . . . 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 1132	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → 𝑚 ∈ (ℤ_≥‘1))
147		seqp1 13736	. . . . . . 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 2744	. . . . . . 7 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (seq1( · , 𝐹)‘𝑚) = (𝑋‘𝑚))
151	150	oveq1d 7290	. . . . . 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 2782	. . . . 5 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝑋‘(𝑚 + 1)) = ((𝑋‘𝑚) · (𝐹‘(𝑚 + 1))))
153	141, 143, 152	3brtr4d 5106	. . . 4 ⊢ ((𝑚 ∈ (1..^𝑀) ∧ (𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) ∧ 𝜑) → (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1)))
154	153	3exp 1118	. . 3 ⊢ (𝑚 ∈ (1..^𝑀) → ((𝜑 → (𝐴↑𝑚) < (𝑋‘𝑚)) → (𝜑 → (𝐴↑(𝑚 + 1)) < (𝑋‘(𝑚 + 1)))))
155	9, 13, 17, 21, 58, 154	fzind2 13505	. 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 396 ∧ w3a 1086 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2106 Ⅎwnfc 2887 class class class wbr 5074 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ℝ⁺crp 12730 ...cfz 13239 ..^cfzo 13382 seqcseq 13721 ↑cexp 13782

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783

This theorem is referenced by: stoweidlem42 43583

W3C validator