fmul01 - Mathbox for Glauco Siliprandi

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Theorem fmul01 46117

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 6862	. . . . . 6 ⊢ (𝑘 = 𝐿 → (𝐴‘𝑘) = (𝐴‘𝐿))
3	2	breq2d 5109	. . . . 5 ⊢ (𝑘 = 𝐿 → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘𝐿)))
4	2	breq1d 5107	. . . . 5 ⊢ (𝑘 = 𝐿 → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘𝐿) ≤ 1))
5	3, 4	anbi12d 641	. . . 4 ⊢ (𝑘 = 𝐿 → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘𝐿) ∧ (𝐴‘𝐿) ≤ 1)))
6	5	imbi2d 342	. . 3 ⊢ (𝑘 = 𝐿 → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘𝐿) ∧ (𝐴‘𝐿) ≤ 1))))
7		fveq2 6862	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
8	7	breq2d 5109	. . . . 5 ⊢ (𝑘 = 𝑗 → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘𝑗)))
9	7	breq1d 5107	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘𝑗) ≤ 1))
10	8, 9	anbi12d 641	. . . 4 ⊢ (𝑘 = 𝑗 → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)))
11	10	imbi2d 342	. . 3 ⊢ (𝑘 = 𝑗 → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))))
12		fveq2 6862	. . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑗 + 1)))
13	12	breq2d 5109	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘(𝑗 + 1))))
14	12	breq1d 5107	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘(𝑗 + 1)) ≤ 1))
15	13, 14	anbi12d 641	. . . 4 ⊢ (𝑘 = (𝑗 + 1) → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1)))
16	15	imbi2d 342	. . 3 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))))
17		fveq2 6862	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐴‘𝑘) = (𝐴‘𝐾))
18	17	breq2d 5109	. . . . 5 ⊢ (𝑘 = 𝐾 → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘𝐾)))
19	17	breq1d 5107	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘𝐾) ≤ 1))
20	18, 19	anbi12d 641	. . . 4 ⊢ (𝑘 = 𝐾 → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1)))
21	20	imbi2d 342	. . 3 ⊢ (𝑘 = 𝐾 → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1))))
22		fmul01.4	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℤ)
23		fmul01.5	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐿))
24		eluzelz 12843	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝑀 ∈ ℤ)
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
26	22	zred 12671	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ℝ)
27	26	leidd 11747	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ≤ 𝐿)
28		eluz 12847	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (ℤ_≥‘𝐿) ↔ 𝐿 ≤ 𝑀))
29	22, 25, 28	syl2anc 593	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ (ℤ_≥‘𝐿) ↔ 𝐿 ≤ 𝑀))
30	23, 29	mpbid 234	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ≤ 𝑀)
31	22, 25, 22, 27, 30	elfzd 13514	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (𝐿...𝑀))
32	31	ancli 556	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)))
33		fmul01.2	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝜑
34		nfv 1933	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐿 ∈ (𝐿...𝑀)
35	33, 34	nfan 1918	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))
36		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑖0
37		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑖 ≤
38		fmul01.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝐵
39		nfcv 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝐿
40	38, 39	nffv 6872	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝐵‘𝐿)
41	36, 37, 40	nfbr 5144	. . . . . . . . 9 ⊢ Ⅎ𝑖0 ≤ (𝐵‘𝐿)
42	35, 41	nfim 1915	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿))
43		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑖 = 𝐿 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝐿 ∈ (𝐿...𝑀)))
44	43	anbi2d 639	. . . . . . . . 9 ⊢ (𝑖 = 𝐿 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))))
45		fveq2 6862	. . . . . . . . . 10 ⊢ (𝑖 = 𝐿 → (𝐵‘𝑖) = (𝐵‘𝐿))
46	45	breq2d 5109	. . . . . . . . 9 ⊢ (𝑖 = 𝐿 → (0 ≤ (𝐵‘𝑖) ↔ 0 ≤ (𝐵‘𝐿)))
47	44, 46	imbi12d 346	. . . . . . . 8 ⊢ (𝑖 = 𝐿 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) ↔ ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿))))
48		fmul01.8	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
49	42, 47, 48	vtoclg1f 3534	. . . . . . 7 ⊢ (𝐿 ∈ (𝐿...𝑀) → ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿)))
50	31, 32, 49	sylc 65	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐵‘𝐿))
51		fmul01.3	. . . . . . . 8 ⊢ 𝐴 = seq𝐿( · , 𝐵)
52	51	fveq1i 6863	. . . . . . 7 ⊢ (𝐴‘𝐿) = (seq𝐿( · , 𝐵)‘𝐿)
53		seq1 14021	. . . . . . . 8 ⊢ (𝐿 ∈ ℤ → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
54	22, 53	syl 17	. . . . . . 7 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
55	52, 54	eqtrid 2808	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝐿) = (𝐵‘𝐿))
56	50, 55	breqtrrd 5125	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐴‘𝐿))
57		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑖1
58	40, 37, 57	nfbr 5144	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐵‘𝐿) ≤ 1
59	35, 58	nfim 1915	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ≤ 1)
60	45	breq1d 5107	. . . . . . . . 9 ⊢ (𝑖 = 𝐿 → ((𝐵‘𝑖) ≤ 1 ↔ (𝐵‘𝐿) ≤ 1))
61	44, 60	imbi12d 346	. . . . . . . 8 ⊢ (𝑖 = 𝐿 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) ↔ ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ≤ 1)))
62		fmul01.9	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)
63	59, 61, 62	vtoclg1f 3534	. . . . . . 7 ⊢ (𝐿 ∈ (𝐿...𝑀) → ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ≤ 1))
64	31, 32, 63	sylc 65	. . . . . 6 ⊢ (𝜑 → (𝐵‘𝐿) ≤ 1)
65	55, 64	eqbrtrd 5119	. . . . 5 ⊢ (𝜑 → (𝐴‘𝐿) ≤ 1)
66	56, 65	jca 519	. . . 4 ⊢ (𝜑 → (0 ≤ (𝐴‘𝐿) ∧ (𝐴‘𝐿) ≤ 1))
67	66	a1i 11	. . 3 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → (𝜑 → (0 ≤ (𝐴‘𝐿) ∧ (𝐴‘𝐿) ≤ 1)))
68		elfzouz 13663	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝐿..^𝑀) → 𝑗 ∈ (ℤ_≥‘𝐿))
69	68	3ad2ant1 1145	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (ℤ_≥‘𝐿))
70		simpl3 1206	. . . . . . . . . 10 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝜑)
71		elfzouz2 13674	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝐿..^𝑀) → 𝑀 ∈ (ℤ_≥‘𝑗))
72		fzss2 13563	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (ℤ_≥‘𝑗) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
73	71, 72	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝐿..^𝑀) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
74	73	3ad2ant1 1145	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
75	74	sselda 3934	. . . . . . . . . 10 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝑘 ∈ (𝐿...𝑀))
76		nfv 1933	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑘 ∈ (𝐿...𝑀)
77	33, 76	nfan 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑘 ∈ (𝐿...𝑀))
78		nfcv 2923	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝑘
79	38, 78	nffv 6872	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘𝑘)
80	79	nfel1 2939	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐵‘𝑘) ∈ ℝ
81	77, 80	nfim 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑘 ∈ (𝐿...𝑀)) → (𝐵‘𝑘) ∈ ℝ)
82		eleq1 2849	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑘 ∈ (𝐿...𝑀)))
83	82	anbi2d 639	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝑘 ∈ (𝐿...𝑀))))
84		fveq2 6862	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝐵‘𝑖) = (𝐵‘𝑘))
85	84	eleq1d 2846	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝑘) ∈ ℝ))
86	83, 85	imbi12d 346	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝑘 ∈ (𝐿...𝑀)) → (𝐵‘𝑘) ∈ ℝ)))
87		fmul01.7	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
88	81, 86, 87	chvarfv 2274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐿...𝑀)) → (𝐵‘𝑘) ∈ ℝ)
89	70, 75, 88	syl2anc 593	. . . . . . . . 9 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → (𝐵‘𝑘) ∈ ℝ)
90		remulcl 11152	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑘 · 𝑙) ∈ ℝ)
91	90	adantl 485	. . . . . . . . 9 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ (𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ)) → (𝑘 · 𝑙) ∈ ℝ)
92	69, 89, 91	seqcl 14029	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℝ)
93		simp3 1150	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 𝜑)
94		fzofzp1 13764	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝐿..^𝑀) → (𝑗 + 1) ∈ (𝐿...𝑀))
95	94	3ad2ant1 1145	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝑗 + 1) ∈ (𝐿...𝑀))
96		nfv 1933	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑗 + 1) ∈ (𝐿...𝑀)
97	33, 96	nfan 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))
98		nfcv 2923	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(𝑗 + 1)
99	38, 98	nffv 6872	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘(𝑗 + 1))
100	99	nfel1 2939	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐵‘(𝑗 + 1)) ∈ ℝ
101	97, 100	nfim 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
102		eleq1 2849	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 + 1) → (𝑖 ∈ (𝐿...𝑀) ↔ (𝑗 + 1) ∈ (𝐿...𝑀)))
103	102	anbi2d 639	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))))
104		fveq2 6862	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 + 1) → (𝐵‘𝑖) = (𝐵‘(𝑗 + 1)))
105	104	eleq1d 2846	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘(𝑗 + 1)) ∈ ℝ))
106	103, 105	imbi12d 346	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 + 1) → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)))
107	101, 106, 87	vtoclg1f 3534	. . . . . . . . . 10 ⊢ ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ))
108	107	anabsi7 681	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
109	93, 95, 108	syl2anc 593	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
110		pm3.35 812	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))) → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))
111	110	ancoms 462	. . . . . . . . . . 11 ⊢ (((𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))
112		simpl 486	. . . . . . . . . . 11 ⊢ ((0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1) → 0 ≤ (𝐴‘𝑗))
113	111, 112	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘𝑗))
114	113	3adant1 1142	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘𝑗))
115	51	fveq1i 6863	. . . . . . . . 9 ⊢ (𝐴‘𝑗) = (seq𝐿( · , 𝐵)‘𝑗)
116	114, 115	breqtrdi 5138	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘𝑗))
117		simp1 1148	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (𝐿..^𝑀))
118	94	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → (𝑗 + 1) ∈ (𝐿...𝑀))
119		simpl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → 𝜑)
120	119, 118	jca 519	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
121	36, 37, 99	nfbr 5144	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖0 ≤ (𝐵‘(𝑗 + 1))
122	97, 121	nfim 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))
123	104	breq2d 5109	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → (0 ≤ (𝐵‘𝑖) ↔ 0 ≤ (𝐵‘(𝑗 + 1))))
124	103, 123	imbi12d 346	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 + 1) → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))))
125	122, 124, 48	vtoclg1f 3534	. . . . . . . . . 10 ⊢ ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1))))
126	118, 120, 125	sylc 65	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))
127	93, 117, 126	syl2anc 593	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐵‘(𝑗 + 1)))
128	92, 109, 116, 127	mulge0d 11758	. . . . . . 7 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
129		seqp1 14023	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ_≥‘𝐿) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) = ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
130	69, 129	syl 17	. . . . . . 7 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) = ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
131	128, 130	breqtrrd 5125	. . . . . 6 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘(𝑗 + 1)))
132	51	fveq1i 6863	. . . . . 6 ⊢ (𝐴‘(𝑗 + 1)) = (seq𝐿( · , 𝐵)‘(𝑗 + 1))
133	131, 132	breqtrrdi 5139	. . . . 5 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘(𝑗 + 1)))
134	92, 109	remulcld 11206	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ∈ ℝ)
135		1red 11176	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 1 ∈ ℝ)
136	93, 95	jca 519	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
137	99, 37, 57	nfbr 5144	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘(𝑗 + 1)) ≤ 1
138	97, 137	nfim 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1)
139	104	breq1d 5107	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 + 1) → ((𝐵‘𝑖) ≤ 1 ↔ (𝐵‘(𝑗 + 1)) ≤ 1))
140	103, 139	imbi12d 346	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1)))
141	138, 140, 62	vtoclg1f 3534	. . . . . . . . . . 11 ⊢ ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1))
142	95, 136, 141	sylc 65	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐵‘(𝑗 + 1)) ≤ 1)
143	109, 135, 92, 116, 142	lemul2ad 12126	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ ((seq𝐿( · , 𝐵)‘𝑗) · 1))
144	92	recnd 11204	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℂ)
145	144	mulridd 11193	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · 1) = (seq𝐿( · , 𝐵)‘𝑗))
146	143, 145	breqtrd 5123	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ (seq𝐿( · , 𝐵)‘𝑗))
147		simp2 1149	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)))
148	110	simprd 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))) → (𝐴‘𝑗) ≤ 1)
149	93, 147, 148	syl2anc 593	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐴‘𝑗) ≤ 1)
150	115, 149	eqbrtrrid 5133	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ≤ 1)
151	134, 92, 135, 146, 150	letrd 11334	. . . . . . 7 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ 1)
152	130, 151	eqbrtrd 5119	. . . . . 6 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) ≤ 1)
153	132, 152	eqbrtrid 5132	. . . . 5 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐴‘(𝑗 + 1)) ≤ 1)
154	133, 153	jca 519	. . . 4 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))
155	154	3exp 1131	. . 3 ⊢ (𝑗 ∈ (𝐿..^𝑀) → ((𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) → (𝜑 → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))))
156	6, 11, 16, 21, 67, 155	fzind2 13788	. 2 ⊢ (𝐾 ∈ (𝐿...𝑀) → (𝜑 → (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1)))
157	1, 156	mpcom 38	1 ⊢ (𝜑 → (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 Ⅎwnf 1802 ∈ wcel 2141 Ⅎwnfc 2908 ⊆ wss 3902 class class class wbr 5097 ‘cfv 6516 (class class class)co 7391 ℝcr 11066 0cc0 11067 1c1 11068 + caddc 11070 · cmul 11072 ≤ cle 11211 ℤcz 12562 ℤ_≥cuz 12833 ...cfz 13506 ..^cfzo 13653 seqcseq 14008

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-n0 12476 df-z 12563 df-uz 12834 df-fz 13507 df-fzo 13654 df-seq 14009

This theorem is referenced by: fmul01lt1lem1 46121 fmul01lt1lem2 46122

W3C validator