fmul01 - Mathbox for Glauco Siliprandi

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Theorem fmul01 43121

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 6774	. . . . . 6 ⊢ (𝑘 = 𝐿 → (𝐴‘𝑘) = (𝐴‘𝐿))
3	2	breq2d 5086	. . . . 5 ⊢ (𝑘 = 𝐿 → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘𝐿)))
4	2	breq1d 5084	. . . . 5 ⊢ (𝑘 = 𝐿 → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘𝐿) ≤ 1))
5	3, 4	anbi12d 631	. . . 4 ⊢ (𝑘 = 𝐿 → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘𝐿) ∧ (𝐴‘𝐿) ≤ 1)))
6	5	imbi2d 341	. . 3 ⊢ (𝑘 = 𝐿 → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘𝐿) ∧ (𝐴‘𝐿) ≤ 1))))
7		fveq2 6774	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
8	7	breq2d 5086	. . . . 5 ⊢ (𝑘 = 𝑗 → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘𝑗)))
9	7	breq1d 5084	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘𝑗) ≤ 1))
10	8, 9	anbi12d 631	. . . 4 ⊢ (𝑘 = 𝑗 → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)))
11	10	imbi2d 341	. . 3 ⊢ (𝑘 = 𝑗 → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))))
12		fveq2 6774	. . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑗 + 1)))
13	12	breq2d 5086	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘(𝑗 + 1))))
14	12	breq1d 5084	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘(𝑗 + 1)) ≤ 1))
15	13, 14	anbi12d 631	. . . 4 ⊢ (𝑘 = (𝑗 + 1) → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1)))
16	15	imbi2d 341	. . 3 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → (0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))))
17		fveq2 6774	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐴‘𝑘) = (𝐴‘𝐾))
18	17	breq2d 5086	. . . . 5 ⊢ (𝑘 = 𝐾 → (0 ≤ (𝐴‘𝑘) ↔ 0 ≤ (𝐴‘𝐾)))
19	17	breq1d 5084	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝐴‘𝑘) ≤ 1 ↔ (𝐴‘𝐾) ≤ 1))
20	18, 19	anbi12d 631	. . . 4 ⊢ (𝑘 = 𝐾 → ((0 ≤ (𝐴‘𝑘) ∧ (𝐴‘𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1)))
21	20	imbi2d 341	. . 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 12592	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝑀 ∈ ℤ)
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
26	22	zred 12426	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ℝ)
27	26	leidd 11541	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ≤ 𝐿)
28		eluz 12596	. . . . . . . . . 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 231	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ≤ 𝑀)
31	22, 25, 22, 27, 30	elfzd 13247	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (𝐿...𝑀))
32	31	ancli 549	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)))
33		fmul01.2	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝜑
34		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐿 ∈ (𝐿...𝑀)
35	33, 34	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))
36		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑖0
37		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑖 ≤
38		fmul01.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝐵
39		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝐿
40	38, 39	nffv 6784	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝐵‘𝐿)
41	36, 37, 40	nfbr 5121	. . . . . . . . 9 ⊢ Ⅎ𝑖0 ≤ (𝐵‘𝐿)
42	35, 41	nfim 1899	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿))
43		eleq1 2826	. . . . . . . . . 10 ⊢ (𝑖 = 𝐿 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝐿 ∈ (𝐿...𝑀)))
44	43	anbi2d 629	. . . . . . . . 9 ⊢ (𝑖 = 𝐿 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))))
45		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑖 = 𝐿 → (𝐵‘𝑖) = (𝐵‘𝐿))
46	45	breq2d 5086	. . . . . . . . 9 ⊢ (𝑖 = 𝐿 → (0 ≤ (𝐵‘𝑖) ↔ 0 ≤ (𝐵‘𝐿)))
47	44, 46	imbi12d 345	. . . . . . . 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 3504	. . . . . . 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 6775	. . . . . . 7 ⊢ (𝐴‘𝐿) = (seq𝐿( · , 𝐵)‘𝐿)
53		seq1 13734	. . . . . . . 8 ⊢ (𝐿 ∈ ℤ → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
54	22, 53	syl 17	. . . . . . 7 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
55	52, 54	eqtrid 2790	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝐿) = (𝐵‘𝐿))
56	50, 55	breqtrrd 5102	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐴‘𝐿))
57		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑖1
58	40, 37, 57	nfbr 5121	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐵‘𝐿) ≤ 1
59	35, 58	nfim 1899	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ≤ 1)
60	45	breq1d 5084	. . . . . . . . 9 ⊢ (𝑖 = 𝐿 → ((𝐵‘𝑖) ≤ 1 ↔ (𝐵‘𝐿) ≤ 1))
61	44, 60	imbi12d 345	. . . . . . . 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 3504	. . . . . . 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 5096	. . . . 5 ⊢ (𝜑 → (𝐴‘𝐿) ≤ 1)
66	56, 65	jca 512	. . . 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 13391	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝐿..^𝑀) → 𝑗 ∈ (ℤ_≥‘𝐿))
69	68	3ad2ant1 1132	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (ℤ_≥‘𝐿))
70		simpl3 1192	. . . . . . . . . 10 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝜑)
71		elfzouz2 13402	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝐿..^𝑀) → 𝑀 ∈ (ℤ_≥‘𝑗))
72		fzss2 13296	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (ℤ_≥‘𝑗) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
73	71, 72	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝐿..^𝑀) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
74	73	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
75	74	sselda 3921	. . . . . . . . . 10 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝑘 ∈ (𝐿...𝑀))
76		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑘 ∈ (𝐿...𝑀)
77	33, 76	nfan 1902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑘 ∈ (𝐿...𝑀))
78		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝑘
79	38, 78	nffv 6784	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘𝑘)
80	79	nfel1 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐵‘𝑘) ∈ ℝ
81	77, 80	nfim 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑘 ∈ (𝐿...𝑀)) → (𝐵‘𝑘) ∈ ℝ)
82		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑘 ∈ (𝐿...𝑀)))
83	82	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝑘 ∈ (𝐿...𝑀))))
84		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝐵‘𝑖) = (𝐵‘𝑘))
85	84	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝑘) ∈ ℝ))
86	83, 85	imbi12d 345	. . . . . . . . . . 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 2233	. . . . . . . . . 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 10956	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑘 · 𝑙) ∈ ℝ)
91	90	adantl 482	. . . . . . . . 9 ⊢ (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) ∧ (𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ)) → (𝑘 · 𝑙) ∈ ℝ)
92	69, 89, 91	seqcl 13743	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℝ)
93		simp3 1137	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 𝜑)
94		fzofzp1 13484	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝐿..^𝑀) → (𝑗 + 1) ∈ (𝐿...𝑀))
95	94	3ad2ant1 1132	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝑗 + 1) ∈ (𝐿...𝑀))
96		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑗 + 1) ∈ (𝐿...𝑀)
97	33, 96	nfan 1902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))
98		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(𝑗 + 1)
99	38, 98	nffv 6784	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘(𝑗 + 1))
100	99	nfel1 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐵‘(𝑗 + 1)) ∈ ℝ
101	97, 100	nfim 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
102		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 + 1) → (𝑖 ∈ (𝐿...𝑀) ↔ (𝑗 + 1) ∈ (𝐿...𝑀)))
103	102	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))))
104		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 + 1) → (𝐵‘𝑖) = (𝐵‘(𝑗 + 1)))
105	104	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘(𝑗 + 1)) ∈ ℝ))
106	103, 105	imbi12d 345	. . . . . . . . . . 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 3504	. . . . . . . . . 10 ⊢ ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ))
108	107	anabsi7 668	. . . . . . . . 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		pm3.35 800	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))) → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))
111	110	ancoms 459	. . . . . . . . . . 11 ⊢ (((𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1))
112		simpl 483	. . . . . . . . . . 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 1129	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘𝑗))
115	51	fveq1i 6775	. . . . . . . . 9 ⊢ (𝐴‘𝑗) = (seq𝐿( · , 𝐵)‘𝑗)
116	114, 115	breqtrdi 5115	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘𝑗))
117		simp1 1135	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (𝐿..^𝑀))
118	94	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → (𝑗 + 1) ∈ (𝐿...𝑀))
119		simpl 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → 𝜑)
120	119, 118	jca 512	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
121	36, 37, 99	nfbr 5121	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖0 ≤ (𝐵‘(𝑗 + 1))
122	97, 121	nfim 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))
123	104	breq2d 5086	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 + 1) → (0 ≤ (𝐵‘𝑖) ↔ 0 ≤ (𝐵‘(𝑗 + 1))))
124	103, 123	imbi12d 345	. . . . . . . . . . 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 3504	. . . . . . . . . 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 11552	. . . . . . 7 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
129		seqp1 13736	. . . . . . . 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 5102	. . . . . 6 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘(𝑗 + 1)))
132	51	fveq1i 6775	. . . . . 6 ⊢ (𝐴‘(𝑗 + 1)) = (seq𝐿( · , 𝐵)‘(𝑗 + 1))
133	131, 132	breqtrrdi 5116	. . . . 5 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘(𝑗 + 1)))
134	92, 109	remulcld 11005	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ∈ ℝ)
135		1red 10976	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → 1 ∈ ℝ)
136	93, 95	jca 512	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
137	99, 37, 57	nfbr 5121	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘(𝑗 + 1)) ≤ 1
138	97, 137	nfim 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1)
139	104	breq1d 5084	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 + 1) → ((𝐵‘𝑖) ≤ 1 ↔ (𝐵‘(𝑗 + 1)) ≤ 1))
140	103, 139	imbi12d 345	. . . . . . . . . . . 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 3504	. . . . . . . . . . 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 11915	. . . . . . . . 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 11003	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℂ)
145	144	mulid1d 10992	. . . . . . . . 9 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · 1) = (seq𝐿( · , 𝐵)‘𝑗))
146	143, 145	breqtrd 5100	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ (seq𝐿( · , 𝐵)‘𝑗))
147		simp2 1136	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)))
148	110	simprd 496	. . . . . . . . . 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 5110	. . . . . . . 8 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ≤ 1)
151	134, 92, 135, 146, 150	letrd 11132	. . . . . . 7 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ 1)
152	130, 151	eqbrtrd 5096	. . . . . 6 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) ≤ 1)
153	132, 152	eqbrtrid 5109	. . . . 5 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (𝐴‘(𝑗 + 1)) ≤ 1)
154	133, 153	jca 512	. . . 4 ⊢ ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴‘𝑗) ∧ (𝐴‘𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))
155	154	3exp 1118	. . 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 13505	. 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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2106 Ⅎwnfc 2887 ⊆ wss 3887 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 ≤ cle 11010 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 ..^cfzo 13382 seqcseq 13721

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-fz 13240 df-fzo 13383 df-seq 13722

This theorem is referenced by: fmul01lt1lem1 43125 fmul01lt1lem2 43126

W3C validator