fmul01lt1lem1 - Mathbox for Glauco Siliprandi

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Theorem fmul01lt1lem1 42226

Description: Given a finite multiplication of values betweeen 0 and 1, a value larger than its first element is larger the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
fmul01lt1lem1.1	⊢ Ⅎ𝑖𝐵
fmul01lt1lem1.2	⊢ Ⅎ𝑖𝜑
fmul01lt1lem1.3	⊢ 𝐴 = seq𝐿( · , 𝐵)
fmul01lt1lem1.4	⊢ (𝜑 → 𝐿 ∈ ℤ)
fmul01lt1lem1.5	⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐿))
fmul01lt1lem1.6	⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
fmul01lt1lem1.7	⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
fmul01lt1lem1.8	⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)
fmul01lt1lem1.9	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
fmul01lt1lem1.10	⊢ (𝜑 → (𝐵‘𝐿) < 𝐸)

**Assertion**
Ref	Expression
fmul01lt1lem1	⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)

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

Proof of Theorem fmul01lt1lem1 Dummy variables 𝑗 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → 𝑀 = 𝐿)
2	1	fveq2d 6649	. . . 4 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝑀) = (𝐴‘𝐿))
3		fmul01lt1lem1.3	. . . . . 6 ⊢ 𝐴 = seq𝐿( · , 𝐵)
4	3	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → 𝐴 = seq𝐿( · , 𝐵))
5	4	fveq1d 6647	. . . 4 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝐿) = (seq𝐿( · , 𝐵)‘𝐿))
6		fmul01lt1lem1.4	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℤ)
7		seq1 13377	. . . . . 6 ⊢ (𝐿 ∈ ℤ → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
9	8	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
10	2, 5, 9	3eqtrd 2837	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝑀) = (𝐵‘𝐿))
11		fmul01lt1lem1.10	. . . 4 ⊢ (𝜑 → (𝐵‘𝐿) < 𝐸)
12	11	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐵‘𝐿) < 𝐸)
13	10, 12	eqbrtrd 5052	. 2 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝑀) < 𝐸)
14		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → ¬ 𝑀 = 𝐿)
15	14	neqned 2994	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → 𝑀 ≠ 𝐿)
16	6	zred 12075	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ)
17		fmul01lt1lem1.5	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐿))
18		eluzelz 12241	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝑀 ∈ ℤ)
19	17, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
20	19	zred 12075	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ)
21		eluzle 12244	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝐿 ≤ 𝑀)
22	17, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿 ≤ 𝑀)
23	16, 20, 22	3jca 1125	. . . . . 6 ⊢ (𝜑 → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝐿 ≤ 𝑀))
24	23	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝐿 ≤ 𝑀))
25		leltne 10719	. . . . 5 ⊢ ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝐿 ≤ 𝑀) → (𝐿 < 𝑀 ↔ 𝑀 ≠ 𝐿))
26	24, 25	syl 17	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → (𝐿 < 𝑀 ↔ 𝑀 ≠ 𝐿))
27	15, 26	mpbird 260	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → 𝐿 < 𝑀)
28	3	fveq1i 6646	. . . 4 ⊢ (𝐴‘𝑀) = (seq𝐿( · , 𝐵)‘𝑀)
29		remulcl 10611	. . . . . . 7 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑗 · 𝑘) ∈ ℝ)
30	29	adantl 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ (𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (𝑗 · 𝑘) ∈ ℝ)
31		recn 10616	. . . . . . . . 9 ⊢ (𝑗 ∈ ℝ → 𝑗 ∈ ℂ)
32	31	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑗 ∈ ℂ)
33		recn 10616	. . . . . . . . 9 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℂ)
34	33	3ad2ant2 1131	. . . . . . . 8 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑘 ∈ ℂ)
35		recn 10616	. . . . . . . . 9 ⊢ (𝑙 ∈ ℝ → 𝑙 ∈ ℂ)
36	35	3ad2ant3 1132	. . . . . . . 8 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑙 ∈ ℂ)
37	32, 34, 36	mulassd 10653	. . . . . . 7 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → ((𝑗 · 𝑘) · 𝑙) = (𝑗 · (𝑘 · 𝑙)))
38	37	adantl 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ (𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ)) → ((𝑗 · 𝑘) · 𝑙) = (𝑗 · (𝑘 · 𝑙)))
39		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 < 𝑀)
40	39	olcd 871	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 < 𝐿 ∨ 𝐿 < 𝑀))
41	20, 16	jca 515	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
42	41	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
43		lttri2 10712	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 ≠ 𝐿 ↔ (𝑀 < 𝐿 ∨ 𝐿 < 𝑀)))
44	42, 43	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 ≠ 𝐿 ↔ (𝑀 < 𝐿 ∨ 𝐿 < 𝑀)))
45	40, 44	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ≠ 𝐿)
46	45	neneqd 2992	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ¬ 𝑀 = 𝐿)
47		uzp1 12267	. . . . . . . . . 10 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
48	17, 47	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
49	48	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
50	49	ord 861	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (¬ 𝑀 = 𝐿 → 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
51	46, 50	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ (ℤ_≥‘(𝐿 + 1)))
52	6	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 ∈ ℤ)
53		uzid 12246	. . . . . . 7 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ (ℤ_≥‘𝐿))
54	52, 53	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 ∈ (ℤ_≥‘𝐿))
55		fmul01lt1lem1.2	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝜑
56		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝑗 ∈ (𝐿...𝑀)
57	55, 56	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ (𝐿...𝑀))
58		fmul01lt1lem1.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝐵
59		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝑗
60	58, 59	nffv 6655	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝐵‘𝑗)
61	60	nfel1 2971	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐵‘𝑗) ∈ ℝ
62	57, 61	nfim 1897	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
63		eleq1 2877	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑗 ∈ (𝐿...𝑀)))
64	63	anbi2d 631	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (𝐿...𝑀))))
65		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝐵‘𝑖) = (𝐵‘𝑗))
66	65	eleq1d 2874	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝑗) ∈ ℝ))
67	64, 66	imbi12d 348	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ)))
68		fmul01lt1lem1.6	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
69	62, 67, 68	chvarfv 2240	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
70	69	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
71	30, 38, 51, 54, 70	seqsplit 13399	. . . . 5 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝑀) = ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)))
72		eluzfz1 12909	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝐿 ∈ (𝐿...𝑀))
73	17, 72	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (𝐿...𝑀))
74	73	ancli 552	. . . . . . . . . 10 ⊢ (𝜑 → (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)))
75		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝐿 ∈ (𝐿...𝑀)
76	55, 75	nfan 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))
77		nfcv 2955	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝐿
78	58, 77	nffv 6655	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘𝐿)
79	78	nfel1 2971	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐵‘𝐿) ∈ ℝ
80	76, 79	nfim 1897	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ∈ ℝ)
81		eleq1 2877	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐿 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝐿 ∈ (𝐿...𝑀)))
82	81	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐿 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))))
83		fveq2 6645	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐿 → (𝐵‘𝑖) = (𝐵‘𝐿))
84	83	eleq1d 2874	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐿 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝐿) ∈ ℝ))
85	82, 84	imbi12d 348	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐿 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ∈ ℝ)))
86	80, 85, 68	vtoclg1f 3514	. . . . . . . . . 10 ⊢ (𝐿 ∈ (𝐿...𝑀) → ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ∈ ℝ))
87	73, 74, 86	sylc 65	. . . . . . . . 9 ⊢ (𝜑 → (𝐵‘𝐿) ∈ ℝ)
88	8, 87	eqeltrd 2890	. . . . . . . 8 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) ∈ ℝ)
89	88	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝐿) ∈ ℝ)
90	6	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℤ)
91	19	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑀 ∈ ℤ)
92		elfzelz 12902	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → 𝑗 ∈ ℤ)
93	92	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ∈ ℤ)
94	90, 91, 93	3jca 1125	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ))
95	16	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℝ)
96		peano2re 10802	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ ℝ → (𝐿 + 1) ∈ ℝ)
97	16, 96	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐿 + 1) ∈ ℝ)
98	97	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ∈ ℝ)
99	92	zred 12075	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → 𝑗 ∈ ℝ)
100	99	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ∈ ℝ)
101	16	lep1d 11560	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿 ≤ (𝐿 + 1))
102	101	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ (𝐿 + 1))
103		elfzle1 12905	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → (𝐿 + 1) ≤ 𝑗)
104	103	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ≤ 𝑗)
105	95, 98, 100, 102, 104	letrd 10786	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ 𝑗)
106		elfzle2 12906	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → 𝑗 ≤ 𝑀)
107	106	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ≤ 𝑀)
108	105, 107	jca 515	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀))
109		elfz2 12892	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝐿...𝑀) ↔ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (𝐿 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))
110	94, 108, 109	sylanbrc 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ∈ (𝐿...𝑀))
111	110, 69	syldan 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
112	111	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
113	51, 112, 30	seqcl 13386	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ∈ ℝ)
114	89, 113	remulcld 10660	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) ∈ ℝ)
115		fmul01lt1lem1.9	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
116	115	rpred 12419	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
117	116	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐸 ∈ ℝ)
118		1red 10631	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 1 ∈ ℝ)
119		nfcv 2955	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖0
120		nfcv 2955	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 ≤
121	119, 120, 78	nfbr 5077	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖0 ≤ (𝐵‘𝐿)
122	76, 121	nfim 1897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿))
123	83	breq2d 5042	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐿 → (0 ≤ (𝐵‘𝑖) ↔ 0 ≤ (𝐵‘𝐿)))
124	82, 123	imbi12d 348	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐿 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) ↔ ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿))))
125		fmul01lt1lem1.7	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
126	122, 124, 125	vtoclg1f 3514	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (𝐿...𝑀) → ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿)))
127	73, 74, 126	sylc 65	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝐵‘𝐿))
128	127, 8	breqtrrd 5058	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (seq𝐿( · , 𝐵)‘𝐿))
129	128	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 0 ≤ (seq𝐿( · , 𝐵)‘𝐿))
130		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝐿 < 𝑀
131	55, 130	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ 𝐿 < 𝑀)
132		eqid 2798	. . . . . . . . . 10 ⊢ seq(𝐿 + 1)( · , 𝐵) = seq(𝐿 + 1)( · , 𝐵)
133	6	peano2zd 12078	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 + 1) ∈ ℤ)
134	133	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐿 + 1) ∈ ℤ)
135	16	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 ∈ ℝ)
136	135, 39	gtned 10764	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ≠ 𝐿)
137	136	neneqd 2992	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ¬ 𝑀 = 𝐿)
138	17	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ (ℤ_≥‘𝐿))
139	138, 47	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
140		orel1 886	. . . . . . . . . . 11 ⊢ (¬ 𝑀 = 𝐿 → ((𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))) → 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
141	137, 139, 140	sylc 65	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ (ℤ_≥‘(𝐿 + 1)))
142	19	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ ℤ)
143	134, 142, 142	3jca 1125	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((𝐿 + 1) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
144		zltp1le 12020	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 < 𝑀 ↔ (𝐿 + 1) ≤ 𝑀))
145	52, 142, 144	syl2anc 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐿 < 𝑀 ↔ (𝐿 + 1) ≤ 𝑀))
146	39, 145	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐿 + 1) ≤ 𝑀)
147	20	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ ℝ)
148	147	leidd 11195	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ≤ 𝑀)
149	146, 148	jca 515	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((𝐿 + 1) ≤ 𝑀 ∧ 𝑀 ≤ 𝑀))
150		elfz2 12892	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ((𝐿 + 1)...𝑀) ↔ (((𝐿 + 1) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ((𝐿 + 1) ≤ 𝑀 ∧ 𝑀 ≤ 𝑀)))
151	143, 149, 150	sylanbrc 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ ((𝐿 + 1)...𝑀))
152	6	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℤ)
153	19	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑀 ∈ ℤ)
154		elfzelz 12902	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → 𝑖 ∈ ℤ)
155	154	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℤ)
156	152, 153, 155	3jca 1125	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ))
157	16	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℝ)
158	157, 96	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ∈ ℝ)
159	154	zred 12075	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → 𝑖 ∈ ℝ)
160	159	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℝ)
161	101	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ (𝐿 + 1))
162		elfzle1 12905	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → (𝐿 + 1) ≤ 𝑖)
163	162	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ≤ 𝑖)
164	157, 158, 160, 161, 163	letrd 10786	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ 𝑖)
165		elfzle2 12906	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → 𝑖 ≤ 𝑀)
166	165	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ≤ 𝑀)
167	164, 166	jca 515	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀))
168		elfz2 12892	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (𝐿...𝑀) ↔ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝐿 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)))
169	156, 167, 168	sylanbrc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ (𝐿...𝑀))
170	169, 68	syldan 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
171	170	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
172		simpll 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝜑)
173	6	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℤ)
174	19	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑀 ∈ ℤ)
175	154	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℤ)
176	173, 174, 175	3jca 1125	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ))
177	16	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℝ)
178	97	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ∈ ℝ)
179	159	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℝ)
180	101	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ (𝐿 + 1))
181	162	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ≤ 𝑖)
182	177, 178, 179, 180, 181	letrd 10786	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ 𝑖)
183	165	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ≤ 𝑀)
184	182, 183	jca 515	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀))
185	176, 184, 168	sylanbrc 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ (𝐿...𝑀))
186	172, 185, 125	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 0 ≤ (𝐵‘𝑖))
187		fmul01lt1lem1.8	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)
188	172, 185, 187	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑖) ≤ 1)
189	58, 131, 132, 134, 141, 151, 171, 186, 188	fmul01 42222	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (0 ≤ (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ∧ (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ≤ 1))
190	189	simprd 499	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ≤ 1)
191	113, 118, 89, 129, 190	lemul2ad 11569	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) ≤ ((seq𝐿( · , 𝐵)‘𝐿) · 1))
192	88	recnd 10658	. . . . . . . . 9 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) ∈ ℂ)
193	192	mulid1d 10647	. . . . . . . 8 ⊢ (𝜑 → ((seq𝐿( · , 𝐵)‘𝐿) · 1) = (seq𝐿( · , 𝐵)‘𝐿))
194	193	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · 1) = (seq𝐿( · , 𝐵)‘𝐿))
195	191, 194	breqtrd 5056	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) ≤ (seq𝐿( · , 𝐵)‘𝐿))
196	8, 11	eqbrtrd 5052	. . . . . . 7 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) < 𝐸)
197	196	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝐿) < 𝐸)
198	114, 89, 117, 195, 197	lelttrd 10787	. . . . 5 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) < 𝐸)
199	71, 198	eqbrtrd 5052	. . . 4 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝑀) < 𝐸)
200	28, 199	eqbrtrid 5065	. . 3 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐴‘𝑀) < 𝐸)
201	27, 200	syldan 594	. 2 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → (𝐴‘𝑀) < 𝐸)
202	13, 201	pm2.61dan 812	1 ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 ℤcz 11969 ℤ_≥cuz 12231 ℝ⁺crp 12377 ...cfz 12885 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-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365

This theorem is referenced by: fmul01lt1lem2 42227

W3C validator