fmul01 - Mathbox for Glauco Siliprandi

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Theorem fmul01 45551

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 6840	. . . . . 6 ⊢ (𝑘 = 𝐿 → (𝐴‘𝑘) = (𝐴‘𝐿))
3	2	breq2d 5114	. . . . 5 ⊢ (𝑘 = 𝐿 → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘𝐿)))
4	2	breq1d 5112	. . . . 5 ⊢ (𝑘 = 𝐿 → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘𝐿) ≤ 1))
5	3, 4	anbi12d 632	. . . 4 ⊢ (𝑘 = 𝐿 → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘𝐿) ∧ (𝐴‘𝐿) ≤ 1)))
6	5	imbi2d 340	. . 3 ⊢ (𝑘 = 𝐿 → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘𝐿) ∧ (𝐴‘𝐿) ≤ 1))))
7		fveq2 6840	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
8	7	breq2d 5114	. . . . 5 ⊢ (𝑘 = 𝑗 → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘𝑗)))
9	7	breq1d 5112	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘𝑗) ≤ 1))
10	8, 9	anbi12d 632	. . . 4 ⊢ (𝑘 = 𝑗 → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)))
11	10	imbi2d 340	. . 3 ⊢ (𝑘 = 𝑗 → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))))
12		fveq2 6840	. . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑗 + 1)))
13	12	breq2d 5114	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘(𝑗 + 1))))
14	12	breq1d 5112	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘(𝑗 + 1)) ≤ 1))
15	13, 14	anbi12d 632	. . . 4 ⊢ (𝑘 = (𝑗 + 1) → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1)))
16	15	imbi2d 340	. . 3 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))))
17		fveq2 6840	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐴‘𝑘) = (𝐴‘𝐾))
18	17	breq2d 5114	. . . . 5 ⊢ (𝑘 = 𝐾 → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘𝐾)))
19	17	breq1d 5112	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘𝐾) ≤ 1))
20	18, 19	anbi12d 632	. . . 4 ⊢ (𝑘 = 𝐾 → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1)))
21	20	imbi2d 340	. . 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 12779	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝑀 ∈ ℤ)
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
26	22	zred 12614	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ℝ)
27	26	leidd 11720	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ≤ 𝐿)
28		eluz 12783	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (ℤ_≥‘𝐿) ↔ 𝐿 ≤ 𝑀))
29	22, 25, 28	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ (ℤ_≥‘𝐿) ↔ 𝐿 ≤ 𝑀))
30	23, 29	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ≤ 𝑀)
31	22, 25, 22, 27, 30	elfzd 13452	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (𝐿...𝑀))
32	31	ancli 548	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)))
33		fmul01.2	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝜑
34		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐿 ∈ (𝐿...𝑀)
35	33, 34	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))
36		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑖0
37		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑖 ≤
38		fmul01.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝐵
39		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝐿
40	38, 39	nffv 6850	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝐵‘𝐿)
41	36, 37, 40	nfbr 5149	. . . . . . . . 9 ⊢ Ⅎ𝑖0 ≤ (𝐵‘𝐿)
42	35, 41	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿))
43		eleq1 2816	. . . . . . . . . 10 ⊢ (𝑖 = 𝐿 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝐿 ∈ (𝐿...𝑀)))
44	43	anbi2d 630	. . . . . . . . 9 ⊢ (𝑖 = 𝐿 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))))
45		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑖 = 𝐿 → (𝐵‘𝑖) = (𝐵‘𝐿))
46	45	breq2d 5114	. . . . . . . . 9 ⊢ (𝑖 = 𝐿 → (0 ≤ (𝐵‘𝑖) ↔ 0 ≤ (𝐵‘𝐿)))
47	44, 46	imbi12d 344	. . . . . . . 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 3533	. . . . . . 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 6841	. . . . . . 7 ⊢ (𝐴‘𝐿) = (seq𝐿( · , 𝐵)‘𝐿)
53		seq1 13955	. . . . . . . 8 ⊢ (𝐿 ∈ ℤ → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
54	22, 53	syl 17	. . . . . . 7 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
55	52, 54	eqtrid 2776	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝐿) = (𝐵‘𝐿))
56	50, 55	breqtrrd 5130	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐴‘𝐿))
57		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑖1
58	40, 37, 57	nfbr 5149	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐵‘𝐿) ≤ 1
59	35, 58	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ≤ 1)
60	45	breq1d 5112	. . . . . . . . 9 ⊢ (𝑖 = 𝐿 → ((𝐵‘𝑖) ≤ 1 ↔ (𝐵‘𝐿) ≤ 1))
61	44, 60	imbi12d 344	. . . . . . . 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 3533	. . . . . . 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 5124	. . . . 5 ⊢ (𝜑 → (𝐴‘𝐿) ≤ 1)
66	56, 65	jca 511	. . . 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 13600	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝐿..^𝑀) → 𝑗 ∈ (ℤ_≥‘𝐿))
69	68	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (ℤ_≥‘𝐿))
70		simpl3 1194	. . . . . . . . . 10 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝜑)
71		elfzouz2 13611	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝐿..^𝑀) → 𝑀 ∈ (ℤ_≥‘𝑗))
72		fzss2 13501	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (ℤ_≥‘𝑗) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
73	71, 72	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝐿..^𝑀) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
74	73	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
75	74	sselda 3943	. . . . . . . . . 10 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝑘 ∈ (𝐿...𝑀))
76		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑘 ∈ (𝐿...𝑀)
77	33, 76	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑘 ∈ (𝐿...𝑀))
78		nfcv 2891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝑘
79	38, 78	nffv 6850	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘𝑘)
80	79	nfel1 2908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐵‘𝑘) ∈ ℝ
81	77, 80	nfim 1896	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑘 ∈ (𝐿...𝑀)) → (𝐵‘𝑘) ∈ ℝ)
82		eleq1 2816	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑘 ∈ (𝐿...𝑀)))
83	82	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝑘 ∈ (𝐿...𝑀))))
84		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝐵‘𝑖) = (𝐵‘𝑘))
85	84	eleq1d 2813	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝑘) ∈ ℝ))
86	83, 85	imbi12d 344	. . . . . . . . . . 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 2241	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐿...𝑀)) → (𝐵‘𝑘) ∈ ℝ)
89	70, 75, 88	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → (𝐵‘𝑘) ∈ ℝ)
90		remulcl 11129	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑘 · 𝑙) ∈ ℝ)
91	90	adantl 481	. . . . . . . . 9 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ (𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ)) → (𝑘 · 𝑙) ∈ ℝ)
92	69, 89, 91	seqcl 13963	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℝ)
93		simp3 1138	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 𝜑)
94		fzofzp1 13701	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝐿..^𝑀) → (𝑗 + 1) ∈ (𝐿...𝑀))
95	94	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝑗 + 1) ∈ (𝐿...𝑀))
96		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑗 + 1) ∈ (𝐿...𝑀)
97	33, 96	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))
98		nfcv 2891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(𝑗 + 1)
99	38, 98	nffv 6850	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘(𝑗 + 1))
100	99	nfel1 2908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐵‘(𝑗 + 1)) ∈ ℝ
101	97, 100	nfim 1896	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
102		eleq1 2816	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 + 1) → (𝑖 ∈ (𝐿...𝑀) ↔ (𝑗 + 1) ∈ (𝐿...𝑀)))
103	102	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))))
104		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 + 1) → (𝐵‘𝑖) = (𝐵‘(𝑗 + 1)))
105	104	eleq1d 2813	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘(𝑗 + 1)) ∈ ℝ))
106	103, 105	imbi12d 344	. . . . . . . . . . 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 3533	. . . . . . . . . 10 ⊢ ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ))
108	107	anabsi7 671	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
109	93, 95, 108	syl2anc 584	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
110		pm3.35 802	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))) → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))
111	110	ancoms 458	. . . . . . . . . . 11 ⊢ (((𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))
112		simpl 482	. . . . . . . . . . 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 1130	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘𝑗))
115	51	fveq1i 6841	. . . . . . . . 9 ⊢ (𝐴‘𝑗) = (seq𝐿( · , 𝐵)‘𝑗)
116	114, 115	breqtrdi 5143	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘𝑗))
117		simp1 1136	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (𝐿..^𝑀))
118	94	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → (𝑗 + 1) ∈ (𝐿...𝑀))
119		simpl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → 𝜑)
120	119, 118	jca 511	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
121	36, 37, 99	nfbr 5149	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖0 ≤ (𝐵‘(𝑗 + 1))
122	97, 121	nfim 1896	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))
123	104	breq2d 5114	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → (0 ≤ (𝐵‘𝑖) ↔ 0 ≤ (𝐵‘(𝑗 + 1))))
124	103, 123	imbi12d 344	. . . . . . . . . . 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 3533	. . . . . . . . . 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 584	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐵‘(𝑗 + 1)))
128	92, 109, 116, 127	mulge0d 11731	. . . . . . 7 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
129		seqp1 13957	. . . . . . . 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 5130	. . . . . 6 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘(𝑗 + 1)))
132	51	fveq1i 6841	. . . . . 6 ⊢ (𝐴‘(𝑗 + 1)) = (seq𝐿( · , 𝐵)‘(𝑗 + 1))
133	131, 132	breqtrrdi 5144	. . . . 5 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘(𝑗 + 1)))
134	92, 109	remulcld 11180	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ∈ ℝ)
135		1red 11151	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 1 ∈ ℝ)
136	93, 95	jca 511	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
137	99, 37, 57	nfbr 5149	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘(𝑗 + 1)) ≤ 1
138	97, 137	nfim 1896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1)
139	104	breq1d 5112	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 + 1) → ((𝐵‘𝑖) ≤ 1 ↔ (𝐵‘(𝑗 + 1)) ≤ 1))
140	103, 139	imbi12d 344	. . . . . . . . . . . 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 3533	. . . . . . . . . . 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 12099	. . . . . . . . 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 11178	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℂ)
145	144	mulridd 11167	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · 1) = (seq𝐿( · , 𝐵)‘𝑗))
146	143, 145	breqtrd 5128	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ (seq𝐿( · , 𝐵)‘𝑗))
147		simp2 1137	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)))
148	110	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))) → (𝐴‘𝑗) ≤ 1)
149	93, 147, 148	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐴‘𝑗) ≤ 1)
150	115, 149	eqbrtrrid 5138	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ≤ 1)
151	134, 92, 135, 146, 150	letrd 11307	. . . . . . 7 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ 1)
152	130, 151	eqbrtrd 5124	. . . . . 6 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) ≤ 1)
153	132, 152	eqbrtrid 5137	. . . . 5 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐴‘(𝑗 + 1)) ≤ 1)
154	133, 153	jca 511	. . . 4 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))
155	154	3exp 1119	. . 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 13722	. 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2109 Ⅎwnfc 2876 ⊆ wss 3911 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 ≤ cle 11185 ℤcz 12505 ℤ_≥cuz 12769 ...cfz 13444 ..^cfzo 13591 seqcseq 13942

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-fzo 13592 df-seq 13943

This theorem is referenced by: fmul01lt1lem1 45555 fmul01lt1lem2 45556

W3C validator