fmul01lt1lem1 - Mathbox for Glauco Siliprandi

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Theorem fmul01lt1lem1 46320

Description: Given a finite multiplication of values between 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 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → 𝑀 = 𝐿)
2	1	fveq2d 6885	. . . 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 6883	. . . 4 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝐿) = (seq𝐿( · , 𝐵)‘𝐿))
6		fmul01lt1lem1.4	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℤ)
7		seq1 14046	. . . . . 6 ⊢ (𝐿 ∈ ℤ → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
8	6, 7	syl 18	. . . . 5 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
9	8	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
10	2, 5, 9	3eqtrd 2802	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝑀) = (𝐵‘𝐿))
11		fmul01lt1lem1.10	. . . 4 ⊢ (𝜑 → (𝐵‘𝐿) < 𝐸)
12	11	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐵‘𝐿) < 𝐸)
13	10, 12	eqbrtrd 5133	. 2 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝑀) < 𝐸)
14		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → ¬ 𝑀 = 𝐿)
15	14	neqned 2965	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → 𝑀 ≠ 𝐿)
16	6	zred 12695	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ)
17		fmul01lt1lem1.5	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐿))
18		eluzelz 12867	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝑀 ∈ ℤ)
19	17, 18	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
20	19	zred 12695	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ)
21		eluzle 12870	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝐿 ≤ 𝑀)
22	17, 21	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐿 ≤ 𝑀)
23	16, 20, 22	3jca 1146	. . . . . 6 ⊢ (𝜑 → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝐿 ≤ 𝑀))
24	23	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝐿 ≤ 𝑀))
25		leltne 11294	. . . . 5 ⊢ ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝐿 ≤ 𝑀) → (𝐿 < 𝑀 ↔ 𝑀 ≠ 𝐿))
26	24, 25	syl 18	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → (𝐿 < 𝑀 ↔ 𝑀 ≠ 𝐿))
27	15, 26	mpbird 260	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → 𝐿 < 𝑀)
28	3	fveq1i 6882	. . . 4 ⊢ (𝐴‘𝑀) = (seq𝐿( · , 𝐵)‘𝑀)
29		remulcl 11180	. . . . . . 7 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑗 · 𝑘) ∈ ℝ)
30	29	adantl 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ (𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (𝑗 · 𝑘) ∈ ℝ)
31		recn 11185	. . . . . . . . 9 ⊢ (𝑗 ∈ ℝ → 𝑗 ∈ ℂ)
32	31	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑗 ∈ ℂ)
33		recn 11185	. . . . . . . . 9 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℂ)
34	33	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑘 ∈ ℂ)
35		recn 11185	. . . . . . . . 9 ⊢ (𝑙 ∈ ℝ → 𝑙 ∈ ℂ)
36	35	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑙 ∈ ℂ)
37	32, 34, 36	mulassd 11227	. . . . . . 7 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → ((𝑗 · 𝑘) · 𝑙) = (𝑗 · (𝑘 · 𝑙)))
38	37	adantl 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ (𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ)) → ((𝑗 · 𝑘) · 𝑙) = (𝑗 · (𝑘 · 𝑙)))
39		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 < 𝑀)
40	39	olcd 887	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 < 𝐿 ∨ 𝐿 < 𝑀))
41	20, 16	jca 520	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
42	41	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
43		lttri2 11287	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 ≠ 𝐿 ↔ (𝑀 < 𝐿 ∨ 𝐿 < 𝑀)))
44	42, 43	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 ≠ 𝐿 ↔ (𝑀 < 𝐿 ∨ 𝐿 < 𝑀)))
45	40, 44	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ≠ 𝐿)
46	45	neneqd 2963	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ¬ 𝑀 = 𝐿)
47		uzp1 12894	. . . . . . . . . 10 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
48	17, 47	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
49	48	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
50	49	ord 877	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (¬ 𝑀 = 𝐿 → 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
51	46, 50	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ (ℤ_≥‘(𝐿 + 1)))
52	6	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 ∈ ℤ)
53		uzid 12872	. . . . . . 7 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ (ℤ_≥‘𝐿))
54	52, 53	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 ∈ (ℤ_≥‘𝐿))
55		fmul01lt1lem1.2	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝜑
56		nfv 1944	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝑗 ∈ (𝐿...𝑀)
57	55, 56	nfan 1929	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ (𝐿...𝑀))
58		fmul01lt1lem1.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝐵
59		nfcv 2925	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝑗
60	58, 59	nffv 6891	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝐵‘𝑗)
61	60	nfel1 2941	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐵‘𝑗) ∈ ℝ
62	57, 61	nfim 1926	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
63		eleq1 2851	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑗 ∈ (𝐿...𝑀)))
64	63	anbi2d 641	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (𝐿...𝑀))))
65		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝐵‘𝑖) = (𝐵‘𝑗))
66	65	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝑗) ∈ ℝ))
67	64, 66	imbi12d 347	. . . . . . . 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 2276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
70	69	adantlr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
71	30, 38, 51, 54, 70	seqsplit 14067	. . . . 5 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝑀) = ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)))
72		eluzfz1 13554	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝐿 ∈ (𝐿...𝑀))
73	17, 72	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (𝐿...𝑀))
74	73	ancli 557	. . . . . . . . . 10 ⊢ (𝜑 → (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)))
75		nfv 1944	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝐿 ∈ (𝐿...𝑀)
76	55, 75	nfan 1929	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))
77		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝐿
78	58, 77	nffv 6891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘𝐿)
79	78	nfel1 2941	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐵‘𝐿) ∈ ℝ
80	76, 79	nfim 1926	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ∈ ℝ)
81		eleq1 2851	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐿 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝐿 ∈ (𝐿...𝑀)))
82	81	anbi2d 641	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐿 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))))
83		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐿 → (𝐵‘𝑖) = (𝐵‘𝐿))
84	83	eleq1d 2848	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐿 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝐿) ∈ ℝ))
85	82, 84	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐿 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ∈ ℝ)))
86	80, 85, 68	vtoclg1f 3535	. . . . . . . . . 10 ⊢ (𝐿 ∈ (𝐿...𝑀) → ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ∈ ℝ))
87	73, 74, 86	sylc 66	. . . . . . . . 9 ⊢ (𝜑 → (𝐵‘𝐿) ∈ ℝ)
88	8, 87	eqeltrd 2863	. . . . . . . 8 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) ∈ ℝ)
89	88	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝐿) ∈ ℝ)
90	6	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℤ)
91	19	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑀 ∈ ℤ)
92		elfzelz 13547	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → 𝑗 ∈ ℤ)
93	92	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ∈ ℤ)
94	16	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℝ)
95		peano2re 11378	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ ℝ → (𝐿 + 1) ∈ ℝ)
96	16, 95	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐿 + 1) ∈ ℝ)
97	96	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ∈ ℝ)
98	92	zred 12695	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → 𝑗 ∈ ℝ)
99	98	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ∈ ℝ)
100	16	lep1d 12141	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ≤ (𝐿 + 1))
101	100	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ (𝐿 + 1))
102		elfzle1 13550	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → (𝐿 + 1) ≤ 𝑗)
103	102	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ≤ 𝑗)
104	94, 97, 99, 101, 103	letrd 11362	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ 𝑗)
105		elfzle2 13551	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → 𝑗 ≤ 𝑀)
106	105	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ≤ 𝑀)
107	90, 91, 93, 104, 106	elfzd 13538	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ∈ (𝐿...𝑀))
108	107, 69	syldan 602	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
109	108	adantlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
110	51, 109, 30	seqcl 14054	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ∈ ℝ)
111	89, 110	remulcld 11234	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) ∈ ℝ)
112		fmul01lt1lem1.9	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
113	112	rpred 13055	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
114	113	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐸 ∈ ℝ)
115		1red 11204	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 1 ∈ ℝ)
116		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖0
117		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 ≤
118	116, 117, 78	nfbr 5158	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖0 ≤ (𝐵‘𝐿)
119	76, 118	nfim 1926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿))
120	83	breq2d 5121	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐿 → (0 ≤ (𝐵‘𝑖) ↔ 0 ≤ (𝐵‘𝐿)))
121	82, 120	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐿 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) ↔ ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿))))
122		fmul01lt1lem1.7	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
123	119, 121, 122	vtoclg1f 3535	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (𝐿...𝑀) → ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿)))
124	73, 74, 123	sylc 66	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝐵‘𝐿))
125	124, 8	breqtrrd 5139	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (seq𝐿( · , 𝐵)‘𝐿))
126	125	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 0 ≤ (seq𝐿( · , 𝐵)‘𝐿))
127		nfv 1944	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝐿 < 𝑀
128	55, 127	nfan 1929	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ 𝐿 < 𝑀)
129		eqid 2763	. . . . . . . . . 10 ⊢ seq(𝐿 + 1)( · , 𝐵) = seq(𝐿 + 1)( · , 𝐵)
130	6	peano2zd 12698	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 + 1) ∈ ℤ)
131	130	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐿 + 1) ∈ ℤ)
132	16	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 ∈ ℝ)
133	132, 39	gtned 11340	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ≠ 𝐿)
134	133	neneqd 2963	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ¬ 𝑀 = 𝐿)
135	17	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ (ℤ_≥‘𝐿))
136	135, 47	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
137		orel1 901	. . . . . . . . . . 11 ⊢ (¬ 𝑀 = 𝐿 → ((𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))) → 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
138	134, 136, 137	sylc 66	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ (ℤ_≥‘(𝐿 + 1)))
139	19	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ ℤ)
140		zltp1le 12639	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 < 𝑀 ↔ (𝐿 + 1) ≤ 𝑀))
141	52, 139, 140	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐿 < 𝑀 ↔ (𝐿 + 1) ≤ 𝑀))
142	39, 141	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐿 + 1) ≤ 𝑀)
143	20	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ ℝ)
144	143	leidd 11775	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ≤ 𝑀)
145	131, 139, 139, 142, 144	elfzd 13538	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ ((𝐿 + 1)...𝑀))
146	6	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℤ)
147	19	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑀 ∈ ℤ)
148		elfzelz 13547	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → 𝑖 ∈ ℤ)
149	148	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℤ)
150	16	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℝ)
151	150, 95	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ∈ ℝ)
152	148	zred 12695	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → 𝑖 ∈ ℝ)
153	152	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℝ)
154	100	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ (𝐿 + 1))
155		elfzle1 13550	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → (𝐿 + 1) ≤ 𝑖)
156	155	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ≤ 𝑖)
157	150, 151, 153, 154, 156	letrd 11362	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ 𝑖)
158		elfzle2 13551	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → 𝑖 ≤ 𝑀)
159	158	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ≤ 𝑀)
160	146, 147, 149, 157, 159	elfzd 13538	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ (𝐿...𝑀))
161	160, 68	syldan 602	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
162	161	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
163		simpll 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝜑)
164	6	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℤ)
165	19	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑀 ∈ ℤ)
166	148	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℤ)
167	16	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℝ)
168	96	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ∈ ℝ)
169	152	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℝ)
170	100	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ (𝐿 + 1))
171	155	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ≤ 𝑖)
172	167, 168, 169, 170, 171	letrd 11362	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ 𝑖)
173	158	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ≤ 𝑀)
174	164, 165, 166, 172, 173	elfzd 13538	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ (𝐿...𝑀))
175	163, 174, 122	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 0 ≤ (𝐵‘𝑖))
176		fmul01lt1lem1.8	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)
177	163, 174, 176	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑖) ≤ 1)
178	58, 128, 129, 131, 138, 145, 162, 175, 177	fmul01 46316	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (0 ≤ (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ∧ (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ≤ 1))
179	178	simprd 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ≤ 1)
180	110, 115, 89, 126, 179	lemul2ad 12150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) ≤ ((seq𝐿( · , 𝐵)‘𝐿) · 1))
181	88	recnd 11232	. . . . . . . . 9 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) ∈ ℂ)
182	181	mulridd 11221	. . . . . . . 8 ⊢ (𝜑 → ((seq𝐿( · , 𝐵)‘𝐿) · 1) = (seq𝐿( · , 𝐵)‘𝐿))
183	182	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · 1) = (seq𝐿( · , 𝐵)‘𝐿))
184	180, 183	breqtrd 5137	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) ≤ (seq𝐿( · , 𝐵)‘𝐿))
185	8, 11	eqbrtrd 5133	. . . . . . 7 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) < 𝐸)
186	185	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝐿) < 𝐸)
187	111, 89, 114, 184, 186	lelttrd 11363	. . . . 5 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) < 𝐸)
188	71, 187	eqbrtrd 5133	. . . 4 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝑀) < 𝐸)
189	28, 188	eqbrtrid 5146	. . 3 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐴‘𝑀) < 𝐸)
190	27, 189	syldan 602	. 2 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → (𝐴‘𝑀) < 𝐸)
191	13, 190	pm2.61dan 824	1 ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 Ⅎwnf 1813 ∈ wcel 2143 Ⅎwnfc 2910 ≠ wne 2958 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 0cc0 11095 1c1 11096 + caddc 11098 · cmul 11100 < clt 11238 ≤ cle 11239 ℤcz 12586 ℤ_≥cuz 12857 ℝ⁺crp 13011 ...cfz 13530 seqcseq 14033

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-n0 12500 df-z 12587 df-uz 12858 df-rp 13012 df-fz 13531 df-fzo 13679 df-seq 14034

This theorem is referenced by: fmul01lt1lem2 46321

W3C validator