fmul01 - Mathbox for Glauco Siliprandi

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Theorem fmul01 42222

Description: Multiplying a finite number of values in [ 0 , 1 ] , gives the final product itself a number in [ 0 , 1 ]. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
fmul01.1	⊢ Ⅎ𝑖𝐵
fmul01.2	⊢ Ⅎ𝑖𝜑
fmul01.3	⊢ 𝐴 = seq𝐿( · , 𝐵)
fmul01.4	⊢ (𝜑 → 𝐿 ∈ ℤ)
fmul01.5	⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐿))
fmul01.6	⊢ (𝜑 → 𝐾 ∈ (𝐿...𝑀))
fmul01.7	⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
fmul01.8	⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
fmul01.9	⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)

**Assertion**
Ref	Expression
fmul01	⊢ (𝜑 → (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1))

Distinct variable groups: 𝑖,𝐿 𝑖,𝑀 Allowed substitution hints: 𝜑(𝑖) 𝐴(𝑖) 𝐵(𝑖) 𝐾(𝑖)

Proof of Theorem fmul01 Dummy variables 𝑗 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmul01.6	. 2 ⊢ (𝜑 → 𝐾 ∈ (𝐿...𝑀))
2		fveq2 6645	. . . . . 6 ⊢ (𝑘 = 𝐿 → (𝐴‘𝑘) = (𝐴‘𝐿))
3	2	breq2d 5042	. . . . 5 ⊢ (𝑘 = 𝐿 → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘𝐿)))
4	2	breq1d 5040	. . . . 5 ⊢ (𝑘 = 𝐿 → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘𝐿) ≤ 1))
5	3, 4	anbi12d 633	. . . 4 ⊢ (𝑘 = 𝐿 → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘𝐿) ∧ (𝐴‘𝐿) ≤ 1)))
6	5	imbi2d 344	. . 3 ⊢ (𝑘 = 𝐿 → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘𝐿) ∧ (𝐴‘𝐿) ≤ 1))))
7		fveq2 6645	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
8	7	breq2d 5042	. . . . 5 ⊢ (𝑘 = 𝑗 → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘𝑗)))
9	7	breq1d 5040	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘𝑗) ≤ 1))
10	8, 9	anbi12d 633	. . . 4 ⊢ (𝑘 = 𝑗 → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)))
11	10	imbi2d 344	. . 3 ⊢ (𝑘 = 𝑗 → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))))
12		fveq2 6645	. . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑗 + 1)))
13	12	breq2d 5042	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘(𝑗 + 1))))
14	12	breq1d 5040	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘(𝑗 + 1)) ≤ 1))
15	13, 14	anbi12d 633	. . . 4 ⊢ (𝑘 = (𝑗 + 1) → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1)))
16	15	imbi2d 344	. . 3 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))))
17		fveq2 6645	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐴‘𝑘) = (𝐴‘𝐾))
18	17	breq2d 5042	. . . . 5 ⊢ (𝑘 = 𝐾 → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘𝐾)))
19	17	breq1d 5040	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘𝐾) ≤ 1))
20	18, 19	anbi12d 633	. . . 4 ⊢ (𝑘 = 𝐾 → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1)))
21	20	imbi2d 344	. . 3 ⊢ (𝑘 = 𝐾 → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1))))
22		fmul01.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ ℤ)
23	22	zred 12075	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ℝ)
24	23	leidd 11195	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ≤ 𝐿)
25		fmul01.5	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐿))
26		eluzelz 12241	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝑀 ∈ ℤ)
27	25, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
28		eluz 12245	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (ℤ_≥‘𝐿) ↔ 𝐿 ≤ 𝑀))
29	22, 27, 28	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ (ℤ_≥‘𝐿) ↔ 𝐿 ≤ 𝑀))
30	25, 29	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ≤ 𝑀)
31		elfz 12891	. . . . . . . . 9 ⊢ ((𝐿 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ∈ (𝐿...𝑀) ↔ (𝐿 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀)))
32	22, 22, 27, 31	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ∈ (𝐿...𝑀) ↔ (𝐿 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀)))
33	24, 30, 32	mpbir2and 712	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (𝐿...𝑀))
34	33	ancli 552	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)))
35		fmul01.2	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝜑
36		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐿 ∈ (𝐿...𝑀)
37	35, 36	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))
38		nfcv 2955	. . . . . . . . . 10 ⊢ Ⅎ𝑖0
39		nfcv 2955	. . . . . . . . . 10 ⊢ Ⅎ𝑖 ≤
40		fmul01.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝐵
41		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝐿
42	40, 41	nffv 6655	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝐵‘𝐿)
43	38, 39, 42	nfbr 5077	. . . . . . . . 9 ⊢ Ⅎ𝑖0 ≤ (𝐵‘𝐿)
44	37, 43	nfim 1897	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿))
45		eleq1 2877	. . . . . . . . . 10 ⊢ (𝑖 = 𝐿 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝐿 ∈ (𝐿...𝑀)))
46	45	anbi2d 631	. . . . . . . . 9 ⊢ (𝑖 = 𝐿 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))))
47		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑖 = 𝐿 → (𝐵‘𝑖) = (𝐵‘𝐿))
48	47	breq2d 5042	. . . . . . . . 9 ⊢ (𝑖 = 𝐿 → (0 ≤ (𝐵‘𝑖) ↔ 0 ≤ (𝐵‘𝐿)))
49	46, 48	imbi12d 348	. . . . . . . 8 ⊢ (𝑖 = 𝐿 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) ↔ ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿))))
50		fmul01.8	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
51	44, 49, 50	vtoclg1f 3514	. . . . . . 7 ⊢ (𝐿 ∈ (𝐿...𝑀) → ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿)))
52	33, 34, 51	sylc 65	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐵‘𝐿))
53		fmul01.3	. . . . . . . 8 ⊢ 𝐴 = seq𝐿( · , 𝐵)
54	53	fveq1i 6646	. . . . . . 7 ⊢ (𝐴‘𝐿) = (seq𝐿( · , 𝐵)‘𝐿)
55		seq1 13377	. . . . . . . 8 ⊢ (𝐿 ∈ ℤ → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
56	22, 55	syl 17	. . . . . . 7 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
57	54, 56	syl5eq 2845	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝐿) = (𝐵‘𝐿))
58	52, 57	breqtrrd 5058	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐴‘𝐿))
59		nfcv 2955	. . . . . . . . . 10 ⊢ Ⅎ𝑖1
60	42, 39, 59	nfbr 5077	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐵‘𝐿) ≤ 1
61	37, 60	nfim 1897	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ≤ 1)
62	47	breq1d 5040	. . . . . . . . 9 ⊢ (𝑖 = 𝐿 → ((𝐵‘𝑖) ≤ 1 ↔ (𝐵‘𝐿) ≤ 1))
63	46, 62	imbi12d 348	. . . . . . . 8 ⊢ (𝑖 = 𝐿 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) ↔ ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ≤ 1)))
64		fmul01.9	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)
65	61, 63, 64	vtoclg1f 3514	. . . . . . 7 ⊢ (𝐿 ∈ (𝐿...𝑀) → ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ≤ 1))
66	33, 34, 65	sylc 65	. . . . . 6 ⊢ (𝜑 → (𝐵‘𝐿) ≤ 1)
67	57, 66	eqbrtrd 5052	. . . . 5 ⊢ (𝜑 → (𝐴‘𝐿) ≤ 1)
68	58, 67	jca 515	. . . 4 ⊢ (𝜑 → (0 ≤ (𝐴‘𝐿) ∧ (𝐴‘𝐿) ≤ 1))
69	68	a1i 11	. . 3 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → (𝜑 → (0 ≤ (𝐴‘𝐿) ∧ (𝐴‘𝐿) ≤ 1)))
70		elfzouz 13037	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝐿..^𝑀) → 𝑗 ∈ (ℤ_≥‘𝐿))
71	70	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (ℤ_≥‘𝐿))
72		simpl3 1190	. . . . . . . . . 10 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝜑)
73		elfzouz2 13047	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝐿..^𝑀) → 𝑀 ∈ (ℤ_≥‘𝑗))
74		fzss2 12942	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (ℤ_≥‘𝑗) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
75	73, 74	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝐿..^𝑀) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
76	75	3ad2ant1 1130	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
77	76	sselda 3915	. . . . . . . . . 10 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝑘 ∈ (𝐿...𝑀))
78		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑘 ∈ (𝐿...𝑀)
79	35, 78	nfan 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑘 ∈ (𝐿...𝑀))
80		nfcv 2955	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝑘
81	40, 80	nffv 6655	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘𝑘)
82	81	nfel1 2971	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐵‘𝑘) ∈ ℝ
83	79, 82	nfim 1897	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑘 ∈ (𝐿...𝑀)) → (𝐵‘𝑘) ∈ ℝ)
84		eleq1 2877	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑘 ∈ (𝐿...𝑀)))
85	84	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝑘 ∈ (𝐿...𝑀))))
86		fveq2 6645	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝐵‘𝑖) = (𝐵‘𝑘))
87	86	eleq1d 2874	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝑘) ∈ ℝ))
88	85, 87	imbi12d 348	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝑘 ∈ (𝐿...𝑀)) → (𝐵‘𝑘) ∈ ℝ)))
89		fmul01.7	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
90	83, 88, 89	chvarfv 2240	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐿...𝑀)) → (𝐵‘𝑘) ∈ ℝ)
91	72, 77, 90	syl2anc 587	. . . . . . . . 9 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → (𝐵‘𝑘) ∈ ℝ)
92		remulcl 10611	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑘 · 𝑙) ∈ ℝ)
93	92	adantl 485	. . . . . . . . 9 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ (𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ)) → (𝑘 · 𝑙) ∈ ℝ)
94	71, 91, 93	seqcl 13386	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℝ)
95		simp3 1135	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 𝜑)
96		fzofzp1 13129	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝐿..^𝑀) → (𝑗 + 1) ∈ (𝐿...𝑀))
97	96	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝑗 + 1) ∈ (𝐿...𝑀))
98		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑗 + 1) ∈ (𝐿...𝑀)
99	35, 98	nfan 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))
100		nfcv 2955	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(𝑗 + 1)
101	40, 100	nffv 6655	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘(𝑗 + 1))
102	101	nfel1 2971	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐵‘(𝑗 + 1)) ∈ ℝ
103	99, 102	nfim 1897	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
104		eleq1 2877	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 + 1) → (𝑖 ∈ (𝐿...𝑀) ↔ (𝑗 + 1) ∈ (𝐿...𝑀)))
105	104	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))))
106		fveq2 6645	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 + 1) → (𝐵‘𝑖) = (𝐵‘(𝑗 + 1)))
107	106	eleq1d 2874	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘(𝑗 + 1)) ∈ ℝ))
108	105, 107	imbi12d 348	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 + 1) → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)))
109	103, 108, 89	vtoclg1f 3514	. . . . . . . . . 10 ⊢ ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ))
110	109	anabsi7 670	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
111	95, 97, 110	syl2anc 587	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
112		pm3.35 802	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))) → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))
113	112	ancoms 462	. . . . . . . . . . 11 ⊢ (((𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))
114		simpl 486	. . . . . . . . . . 11 ⊢ ((0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1) → 0 ≤ (𝐴‘𝑗))
115	113, 114	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘𝑗))
116	115	3adant1 1127	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘𝑗))
117	53	fveq1i 6646	. . . . . . . . 9 ⊢ (𝐴‘𝑗) = (seq𝐿( · , 𝐵)‘𝑗)
118	116, 117	breqtrdi 5071	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘𝑗))
119		simp1 1133	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (𝐿..^𝑀))
120	96	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → (𝑗 + 1) ∈ (𝐿...𝑀))
121		simpl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → 𝜑)
122	121, 120	jca 515	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
123	38, 39, 101	nfbr 5077	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖0 ≤ (𝐵‘(𝑗 + 1))
124	99, 123	nfim 1897	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))
125	106	breq2d 5042	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → (0 ≤ (𝐵‘𝑖) ↔ 0 ≤ (𝐵‘(𝑗 + 1))))
126	105, 125	imbi12d 348	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 + 1) → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))))
127	124, 126, 50	vtoclg1f 3514	. . . . . . . . . 10 ⊢ ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1))))
128	120, 122, 127	sylc 65	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))
129	95, 119, 128	syl2anc 587	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐵‘(𝑗 + 1)))
130	94, 111, 118, 129	mulge0d 11206	. . . . . . 7 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
131		seqp1 13379	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ_≥‘𝐿) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) = ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
132	71, 131	syl 17	. . . . . . 7 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) = ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
133	130, 132	breqtrrd 5058	. . . . . 6 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘(𝑗 + 1)))
134	53	fveq1i 6646	. . . . . 6 ⊢ (𝐴‘(𝑗 + 1)) = (seq𝐿( · , 𝐵)‘(𝑗 + 1))
135	133, 134	breqtrrdi 5072	. . . . 5 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘(𝑗 + 1)))
136	94, 111	remulcld 10660	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ∈ ℝ)
137		1red 10631	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 1 ∈ ℝ)
138	95, 97	jca 515	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
139	101, 39, 59	nfbr 5077	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘(𝑗 + 1)) ≤ 1
140	99, 139	nfim 1897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1)
141	106	breq1d 5040	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 + 1) → ((𝐵‘𝑖) ≤ 1 ↔ (𝐵‘(𝑗 + 1)) ≤ 1))
142	105, 141	imbi12d 348	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1)))
143	140, 142, 64	vtoclg1f 3514	. . . . . . . . . . 11 ⊢ ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1))
144	97, 138, 143	sylc 65	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐵‘(𝑗 + 1)) ≤ 1)
145	111, 137, 94, 118, 144	lemul2ad 11569	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ ((seq𝐿( · , 𝐵)‘𝑗) · 1))
146	94	recnd 10658	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℂ)
147	146	mulid1d 10647	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · 1) = (seq𝐿( · , 𝐵)‘𝑗))
148	145, 147	breqtrd 5056	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ (seq𝐿( · , 𝐵)‘𝑗))
149		simp2 1134	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)))
150	112	simprd 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))) → (𝐴‘𝑗) ≤ 1)
151	95, 149, 150	syl2anc 587	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐴‘𝑗) ≤ 1)
152	117, 151	eqbrtrrid 5066	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ≤ 1)
153	136, 94, 137, 148, 152	letrd 10786	. . . . . . 7 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ 1)
154	132, 153	eqbrtrd 5052	. . . . . 6 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) ≤ 1)
155	134, 154	eqbrtrid 5065	. . . . 5 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐴‘(𝑗 + 1)) ≤ 1)
156	135, 155	jca 515	. . . 4 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))
157	156	3exp 1116	. . 3 ⊢ (𝑗 ∈ (𝐿..^𝑀) → ((𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) → (𝜑 → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))))
158	6, 11, 16, 21, 69, 157	fzind2 13150	. 2 ⊢ (𝐾 ∈ (𝐿...𝑀) → (𝜑 → (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1)))
159	1, 158	mpcom 38	1 ⊢ (𝜑 → (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 ⊆ wss 3881 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ≤ cle 10665 ℤcz 11969 ℤ_≥cuz 12231 ...cfz 12885 ..^cfzo 13028 seqcseq 13364

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-fz 12886 df-fzo 13029 df-seq 13365

This theorem is referenced by: fmul01lt1lem1 42226 fmul01lt1lem2 42227

W3C validator