fmul01lt1lem1 - Mathbox for Glauco Siliprandi

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Theorem fmul01lt1lem1 45599

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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → 𝑀 = 𝐿)
2	1	fveq2d 6910	. . . 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 6908	. . . 4 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝐿) = (seq𝐿( · , 𝐵)‘𝐿))
6		fmul01lt1lem1.4	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℤ)
7		seq1 14055	. . . . . 6 ⊢ (𝐿 ∈ ℤ → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿))
10	2, 5, 9	3eqtrd 2781	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝑀) = (𝐵‘𝐿))
11		fmul01lt1lem1.10	. . . 4 ⊢ (𝜑 → (𝐵‘𝐿) < 𝐸)
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐵‘𝐿) < 𝐸)
13	10, 12	eqbrtrd 5165	. 2 ⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝑀) < 𝐸)
14		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → ¬ 𝑀 = 𝐿)
15	14	neqned 2947	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → 𝑀 ≠ 𝐿)
16	6	zred 12722	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ)
17		fmul01lt1lem1.5	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐿))
18		eluzelz 12888	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝑀 ∈ ℤ)
19	17, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
20	19	zred 12722	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ)
21		eluzle 12891	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝐿 ≤ 𝑀)
22	17, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿 ≤ 𝑀)
23	16, 20, 22	3jca 1129	. . . . . 6 ⊢ (𝜑 → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝐿 ≤ 𝑀))
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝐿 ≤ 𝑀))
25		leltne 11350	. . . . 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 257	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → 𝐿 < 𝑀)
28	3	fveq1i 6907	. . . 4 ⊢ (𝐴‘𝑀) = (seq𝐿( · , 𝐵)‘𝑀)
29		remulcl 11240	. . . . . . 7 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑗 · 𝑘) ∈ ℝ)
30	29	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ (𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (𝑗 · 𝑘) ∈ ℝ)
31		recn 11245	. . . . . . . . 9 ⊢ (𝑗 ∈ ℝ → 𝑗 ∈ ℂ)
32	31	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑗 ∈ ℂ)
33		recn 11245	. . . . . . . . 9 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℂ)
34	33	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑘 ∈ ℂ)
35		recn 11245	. . . . . . . . 9 ⊢ (𝑙 ∈ ℝ → 𝑙 ∈ ℂ)
36	35	3ad2ant3 1136	. . . . . . . 8 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑙 ∈ ℂ)
37	32, 34, 36	mulassd 11284	. . . . . . 7 ⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → ((𝑗 · 𝑘) · 𝑙) = (𝑗 · (𝑘 · 𝑙)))
38	37	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ (𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ)) → ((𝑗 · 𝑘) · 𝑙) = (𝑗 · (𝑘 · 𝑙)))
39		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 < 𝑀)
40	39	olcd 875	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 < 𝐿 ∨ 𝐿 < 𝑀))
41	20, 16	jca 511	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
42	41	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
43		lttri2 11343	. . . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ≠ 𝐿)
46	45	neneqd 2945	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ¬ 𝑀 = 𝐿)
47		uzp1 12919	. . . . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
50	49	ord 865	. . . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 ∈ ℤ)
53		uzid 12893	. . . . . . 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 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝑗 ∈ (𝐿...𝑀)
57	55, 56	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ (𝐿...𝑀))
58		fmul01lt1lem1.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝐵
59		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝑗
60	58, 59	nffv 6916	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝐵‘𝑗)
61	60	nfel1 2922	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐵‘𝑗) ∈ ℝ
62	57, 61	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
63		eleq1 2829	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑗 ∈ (𝐿...𝑀)))
64	63	anbi2d 630	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (𝐿...𝑀))))
65		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝐵‘𝑖) = (𝐵‘𝑗))
66	65	eleq1d 2826	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝑗) ∈ ℝ))
67	64, 66	imbi12d 344	. . . . . . . 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 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
71	30, 38, 51, 54, 70	seqsplit 14076	. . . . 5 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝑀) = ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)))
72		eluzfz1 13571	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝐿 ∈ (𝐿...𝑀))
73	17, 72	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (𝐿...𝑀))
74	73	ancli 548	. . . . . . . . . 10 ⊢ (𝜑 → (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)))
75		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝐿 ∈ (𝐿...𝑀)
76	55, 75	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))
77		nfcv 2905	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝐿
78	58, 77	nffv 6916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝐵‘𝐿)
79	78	nfel1 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐵‘𝐿) ∈ ℝ
80	76, 79	nfim 1896	. . . . . . . . . . 11 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ∈ ℝ)
81		eleq1 2829	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐿 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝐿 ∈ (𝐿...𝑀)))
82	81	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐿 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))))
83		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐿 → (𝐵‘𝑖) = (𝐵‘𝐿))
84	83	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐿 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝐿) ∈ ℝ))
85	82, 84	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐿 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ∈ ℝ)))
86	80, 85, 68	vtoclg1f 3570	. . . . . . . . . 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 2841	. . . . . . . 8 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) ∈ ℝ)
89	88	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝐿) ∈ ℝ)
90	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℤ)
91	19	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑀 ∈ ℤ)
92		elfzelz 13564	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → 𝑗 ∈ ℤ)
93	92	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ∈ ℤ)
94	16	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℝ)
95		peano2re 11434	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ ℝ → (𝐿 + 1) ∈ ℝ)
96	16, 95	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐿 + 1) ∈ ℝ)
97	96	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ∈ ℝ)
98	92	zred 12722	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → 𝑗 ∈ ℝ)
99	98	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ∈ ℝ)
100	16	lep1d 12199	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ≤ (𝐿 + 1))
101	100	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ (𝐿 + 1))
102		elfzle1 13567	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → (𝐿 + 1) ≤ 𝑗)
103	102	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ≤ 𝑗)
104	94, 97, 99, 101, 103	letrd 11418	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ 𝑗)
105		elfzle2 13568	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → 𝑗 ≤ 𝑀)
106	105	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ≤ 𝑀)
107	90, 91, 93, 104, 106	elfzd 13555	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ∈ (𝐿...𝑀))
108	107, 69	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
109	108	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑗) ∈ ℝ)
110	51, 109, 30	seqcl 14063	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ∈ ℝ)
111	89, 110	remulcld 11291	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) ∈ ℝ)
112		fmul01lt1lem1.9	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
113	112	rpred 13077	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
114	113	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐸 ∈ ℝ)
115		1red 11262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 1 ∈ ℝ)
116		nfcv 2905	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖0
117		nfcv 2905	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 ≤
118	116, 117, 78	nfbr 5190	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖0 ≤ (𝐵‘𝐿)
119	76, 118	nfim 1896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿))
120	83	breq2d 5155	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐿 → (0 ≤ (𝐵‘𝑖) ↔ 0 ≤ (𝐵‘𝐿)))
121	82, 120	imbi12d 344	. . . . . . . . . . . 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 3570	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (𝐿...𝑀) → ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿)))
124	73, 74, 123	sylc 65	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝐵‘𝐿))
125	124, 8	breqtrrd 5171	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (seq𝐿( · , 𝐵)‘𝐿))
126	125	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 0 ≤ (seq𝐿( · , 𝐵)‘𝐿))
127		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝐿 < 𝑀
128	55, 127	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ 𝐿 < 𝑀)
129		eqid 2737	. . . . . . . . . 10 ⊢ seq(𝐿 + 1)( · , 𝐵) = seq(𝐿 + 1)( · , 𝐵)
130	6	peano2zd 12725	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 + 1) ∈ ℤ)
131	130	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐿 + 1) ∈ ℤ)
132	16	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 ∈ ℝ)
133	132, 39	gtned 11396	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ≠ 𝐿)
134	133	neneqd 2945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ¬ 𝑀 = 𝐿)
135	17	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ (ℤ_≥‘𝐿))
136	135, 47	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
137		orel1 889	. . . . . . . . . . 11 ⊢ (¬ 𝑀 = 𝐿 → ((𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))) → 𝑀 ∈ (ℤ_≥‘(𝐿 + 1))))
138	134, 136, 137	sylc 65	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ (ℤ_≥‘(𝐿 + 1)))
139	19	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ ℤ)
140		zltp1le 12667	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 < 𝑀 ↔ (𝐿 + 1) ≤ 𝑀))
141	52, 139, 140	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐿 < 𝑀 ↔ (𝐿 + 1) ≤ 𝑀))
142	39, 141	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐿 + 1) ≤ 𝑀)
143	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ ℝ)
144	143	leidd 11829	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ≤ 𝑀)
145	131, 139, 139, 142, 144	elfzd 13555	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ ((𝐿 + 1)...𝑀))
146	6	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℤ)
147	19	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑀 ∈ ℤ)
148		elfzelz 13564	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → 𝑖 ∈ ℤ)
149	148	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℤ)
150	16	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℝ)
151	150, 95	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ∈ ℝ)
152	148	zred 12722	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → 𝑖 ∈ ℝ)
153	152	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℝ)
154	100	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ (𝐿 + 1))
155		elfzle1 13567	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → (𝐿 + 1) ≤ 𝑖)
156	155	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ≤ 𝑖)
157	150, 151, 153, 154, 156	letrd 11418	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ 𝑖)
158		elfzle2 13568	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → 𝑖 ≤ 𝑀)
159	158	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ≤ 𝑀)
160	146, 147, 149, 157, 159	elfzd 13555	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ (𝐿...𝑀))
161	160, 68	syldan 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
162	161	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
163		simpll 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝜑)
164	6	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℤ)
165	19	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑀 ∈ ℤ)
166	148	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℤ)
167	16	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℝ)
168	96	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ∈ ℝ)
169	152	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℝ)
170	100	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ (𝐿 + 1))
171	155	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ≤ 𝑖)
172	167, 168, 169, 170, 171	letrd 11418	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ 𝑖)
173	158	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ≤ 𝑀)
174	164, 165, 166, 172, 173	elfzd 13555	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ (𝐿...𝑀))
175	163, 174, 122	syl2anc 584	. . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑖) ≤ 1)
178	58, 128, 129, 131, 138, 145, 162, 175, 177	fmul01 45595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (0 ≤ (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ∧ (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ≤ 1))
179	178	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ≤ 1)
180	110, 115, 89, 126, 179	lemul2ad 12208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) ≤ ((seq𝐿( · , 𝐵)‘𝐿) · 1))
181	88	recnd 11289	. . . . . . . . 9 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) ∈ ℂ)
182	181	mulridd 11278	. . . . . . . 8 ⊢ (𝜑 → ((seq𝐿( · , 𝐵)‘𝐿) · 1) = (seq𝐿( · , 𝐵)‘𝐿))
183	182	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · 1) = (seq𝐿( · , 𝐵)‘𝐿))
184	180, 183	breqtrd 5169	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) ≤ (seq𝐿( · , 𝐵)‘𝐿))
185	8, 11	eqbrtrd 5165	. . . . . . 7 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) < 𝐸)
186	185	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝐿) < 𝐸)
187	111, 89, 114, 184, 186	lelttrd 11419	. . . . 5 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) < 𝐸)
188	71, 187	eqbrtrd 5165	. . . 4 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝑀) < 𝐸)
189	28, 188	eqbrtrid 5178	. . 3 ⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐴‘𝑀) < 𝐸)
190	27, 189	syldan 591	. 2 ⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → (𝐴‘𝑀) < 𝐸)
191	13, 190	pm2.61dan 813	1 ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 ℤcz 12613 ℤ_≥cuz 12878 ℝ⁺crp 13034 ...cfz 13547 seqcseq 14042

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043

This theorem is referenced by: fmul01lt1lem2 45600

W3C validator