fmul01lt1lem2 - Mathbox for Glauco Siliprandi

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Theorem fmul01lt1lem2 43816

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

**Hypotheses**
Ref	Expression
fmul01lt1lem2.1	⊢ Ⅎ𝑖𝐵
fmul01lt1lem2.2	⊢ Ⅎ𝑖𝜑
fmul01lt1lem2.3	⊢ 𝐴 = seq𝐿( · , 𝐵)
fmul01lt1lem2.4	⊢ (𝜑 → 𝐿 ∈ ℤ)
fmul01lt1lem2.5	⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐿))
fmul01lt1lem2.6	⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
fmul01lt1lem2.7	⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
fmul01lt1lem2.8	⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)
fmul01lt1lem2.9	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
fmul01lt1lem2.10	⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑀))
fmul01lt1lem2.11	⊢ (𝜑 → (𝐵‘𝐽) < 𝐸)

**Assertion**
Ref	Expression
fmul01lt1lem2	⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)

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

Proof of Theorem fmul01lt1lem2 Dummy variables 𝑎 𝑏 𝑐 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmul01lt1lem2.1	. . 3 ⊢ Ⅎ𝑖𝐵
2		fmul01lt1lem2.2	. . . 4 ⊢ Ⅎ𝑖𝜑
3		nfv 1917	. . . 4 ⊢ Ⅎ𝑖 𝐽 = 𝐿
4	2, 3	nfan 1902	. . 3 ⊢ Ⅎ𝑖(𝜑 ∧ 𝐽 = 𝐿)
5		fmul01lt1lem2.3	. . 3 ⊢ 𝐴 = seq𝐿( · , 𝐵)
6		fmul01lt1lem2.4	. . . 4 ⊢ (𝜑 → 𝐿 ∈ ℤ)
7	6	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → 𝐿 ∈ ℤ)
8		fmul01lt1lem2.5	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐿))
9	8	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → 𝑀 ∈ (ℤ_≥‘𝐿))
10		fmul01lt1lem2.6	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
11	10	adantlr 713	. . 3 ⊢ (((𝜑 ∧ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
12		fmul01lt1lem2.7	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
13	12	adantlr 713	. . 3 ⊢ (((𝜑 ∧ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
14		fmul01lt1lem2.8	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)
15	14	adantlr 713	. . 3 ⊢ (((𝜑 ∧ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)
16		fmul01lt1lem2.9	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
17	16	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → 𝐸 ∈ ℝ⁺)
18		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → 𝐽 = 𝐿)
19	18	fveq2d 6846	. . . 4 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → (𝐵‘𝐽) = (𝐵‘𝐿))
20		fmul01lt1lem2.11	. . . . 5 ⊢ (𝜑 → (𝐵‘𝐽) < 𝐸)
21	20	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → (𝐵‘𝐽) < 𝐸)
22	19, 21	eqbrtrrd 5129	. . 3 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → (𝐵‘𝐿) < 𝐸)
23	1, 4, 5, 7, 9, 11, 13, 15, 17, 22	fmul01lt1lem1 43815	. 2 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → (𝐴‘𝑀) < 𝐸)
24	5	fveq1i 6843	. . 3 ⊢ (𝐴‘𝑀) = (seq𝐿( · , 𝐵)‘𝑀)
25		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑖 𝑎 ∈ (𝐿...𝑀)
26	2, 25	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑎 ∈ (𝐿...𝑀))
27		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝑎
28	1, 27	nffv 6852	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐵‘𝑎)
29	28	nfel1 2923	. . . . . . . 8 ⊢ Ⅎ𝑖(𝐵‘𝑎) ∈ ℝ
30	26, 29	nfim 1899	. . . . . . 7 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ)
31		eleq1w 2820	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑎 ∈ (𝐿...𝑀)))
32	31	anbi2d 629	. . . . . . . 8 ⊢ (𝑖 = 𝑎 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝑎 ∈ (𝐿...𝑀))))
33		fveq2 6842	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (𝐵‘𝑖) = (𝐵‘𝑎))
34	33	eleq1d 2822	. . . . . . . 8 ⊢ (𝑖 = 𝑎 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝑎) ∈ ℝ))
35	32, 34	imbi12d 344	. . . . . . 7 ⊢ (𝑖 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ)))
36	30, 35, 10	chvarfv 2233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ)
37		remulcl 11136	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑎 · 𝑗) ∈ ℝ)
38	37	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ)
39	8, 36, 38	seqcl 13928	. . . . 5 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝑀) ∈ ℝ)
40	39	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) ∈ ℝ)
41		fmul01lt1lem2.10	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑀))
42		elfzuz3 13438	. . . . . . 7 ⊢ (𝐽 ∈ (𝐿...𝑀) → 𝑀 ∈ (ℤ_≥‘𝐽))
43	41, 42	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐽))
44		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑖 𝑎 ∈ (𝐽...𝑀)
45	2, 44	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑎 ∈ (𝐽...𝑀))
46	45, 29	nfim 1899	. . . . . . 7 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑎 ∈ (𝐽...𝑀)) → (𝐵‘𝑎) ∈ ℝ)
47		eleq1w 2820	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (𝐽...𝑀) ↔ 𝑎 ∈ (𝐽...𝑀)))
48	47	anbi2d 629	. . . . . . . 8 ⊢ (𝑖 = 𝑎 → ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) ↔ (𝜑 ∧ 𝑎 ∈ (𝐽...𝑀))))
49	48, 34	imbi12d 344	. . . . . . 7 ⊢ (𝑖 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝑎 ∈ (𝐽...𝑀)) → (𝐵‘𝑎) ∈ ℝ)))
50	6	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐿 ∈ ℤ)
51		eluzelz 12773	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝑀 ∈ ℤ)
52	8, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
53	52	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑀 ∈ ℤ)
54		elfzelz 13441	. . . . . . . . . 10 ⊢ (𝑖 ∈ (𝐽...𝑀) → 𝑖 ∈ ℤ)
55	54	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑖 ∈ ℤ)
56	6	zred 12607	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ ℝ)
57	56	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐿 ∈ ℝ)
58		elfzelz 13441	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (𝐿...𝑀) → 𝐽 ∈ ℤ)
59	41, 58	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℤ)
60	59	zred 12607	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℝ)
61	60	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐽 ∈ ℝ)
62	54	zred 12607	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐽...𝑀) → 𝑖 ∈ ℝ)
63	62	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑖 ∈ ℝ)
64		elfzle1 13444	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (𝐿...𝑀) → 𝐿 ≤ 𝐽)
65	41, 64	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ≤ 𝐽)
66	65	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐿 ≤ 𝐽)
67		elfzle1 13444	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐽...𝑀) → 𝐽 ≤ 𝑖)
68	67	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐽 ≤ 𝑖)
69	57, 61, 63, 66, 68	letrd 11312	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐿 ≤ 𝑖)
70		elfzle2 13445	. . . . . . . . . 10 ⊢ (𝑖 ∈ (𝐽...𝑀) → 𝑖 ≤ 𝑀)
71	70	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑖 ≤ 𝑀)
72	50, 53, 55, 69, 71	elfzd 13432	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑖 ∈ (𝐿...𝑀))
73	72, 10	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
74	46, 49, 73	chvarfv 2233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐽...𝑀)) → (𝐵‘𝑎) ∈ ℝ)
75	43, 74, 38	seqcl 13928	. . . . 5 ⊢ (𝜑 → (seq𝐽( · , 𝐵)‘𝑀) ∈ ℝ)
76	75	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐽( · , 𝐵)‘𝑀) ∈ ℝ)
77	16	rpred 12957	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ)
78	77	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐸 ∈ ℝ)
79		remulcl 11136	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 · 𝑏) ∈ ℝ)
80	79	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ)) → (𝑎 · 𝑏) ∈ ℝ)
81		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑎 ∈ ℝ)
82	81	recnd 11183	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑎 ∈ ℂ)
83		simp2 1137	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑏 ∈ ℝ)
84	83	recnd 11183	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑏 ∈ ℂ)
85		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
86	85	recnd 11183	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℂ)
87	82, 84, 86	mulassd 11178	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → ((𝑎 · 𝑏) · 𝑐) = (𝑎 · (𝑏 · 𝑐)))
88	87	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ)) → ((𝑎 · 𝑏) · 𝑐) = (𝑎 · (𝑏 · 𝑐)))
89	59	zcnd 12608	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℂ)
90		1cnd 11150	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
91	89, 90	npcand 11516	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐽 − 1) + 1) = 𝐽)
92	91	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ_≥‘((𝐽 − 1) + 1)) = (ℤ_≥‘𝐽))
93	43, 92	eleqtrrd 2841	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘((𝐽 − 1) + 1)))
94	93	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝑀 ∈ (ℤ_≥‘((𝐽 − 1) + 1)))
95	6	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐿 ∈ ℤ)
96	59	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐽 ∈ ℤ)
97		1zzd 12534	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 1 ∈ ℤ)
98	96, 97	zsubcld 12612	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ∈ ℤ)
99		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ¬ 𝐽 = 𝐿)
100		eqcom 2743	. . . . . . . . . . . 12 ⊢ (𝐽 = 𝐿 ↔ 𝐿 = 𝐽)
101	99, 100	sylnib 327	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ¬ 𝐿 = 𝐽)
102	56, 60	leloed 11298	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐿 ≤ 𝐽 ↔ (𝐿 < 𝐽 ∨ 𝐿 = 𝐽)))
103	65, 102	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿 < 𝐽 ∨ 𝐿 = 𝐽))
104	103	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐿 < 𝐽 ∨ 𝐿 = 𝐽))
105		orel2 889	. . . . . . . . . . 11 ⊢ (¬ 𝐿 = 𝐽 → ((𝐿 < 𝐽 ∨ 𝐿 = 𝐽) → 𝐿 < 𝐽))
106	101, 104, 105	sylc 65	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐿 < 𝐽)
107		zltlem1 12556	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐿 < 𝐽 ↔ 𝐿 ≤ (𝐽 − 1)))
108	6, 59, 107	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 < 𝐽 ↔ 𝐿 ≤ (𝐽 − 1)))
109	108	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐿 < 𝐽 ↔ 𝐿 ≤ (𝐽 − 1)))
110	106, 109	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐿 ≤ (𝐽 − 1))
111		eluz2 12769	. . . . . . . . 9 ⊢ ((𝐽 − 1) ∈ (ℤ_≥‘𝐿) ↔ (𝐿 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ ∧ 𝐿 ≤ (𝐽 − 1)))
112	95, 98, 110, 111	syl3anbrc 1343	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ∈ (ℤ_≥‘𝐿))
113		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖 ¬ 𝐽 = 𝐿
114	2, 113	nfan 1902	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝜑 ∧ ¬ 𝐽 = 𝐿)
115	114, 25	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑖((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀))
116	115, 29	nfim 1899	. . . . . . . . 9 ⊢ Ⅎ𝑖(((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ)
117	31	anbi2d 629	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ ((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀))))
118	117, 34	imbi12d 344	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → ((((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ)))
119	10	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
120	116, 118, 119	chvarfv 2233	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ)
121	80, 88, 94, 112, 120	seqsplit 13941	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) = ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq((𝐽 − 1) + 1)( · , 𝐵)‘𝑀)))
122	91	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((𝐽 − 1) + 1) = 𝐽)
123	122	seqeq1d 13912	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → seq((𝐽 − 1) + 1)( · , 𝐵) = seq𝐽( · , 𝐵))
124	123	fveq1d 6844	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq((𝐽 − 1) + 1)( · , 𝐵)‘𝑀) = (seq𝐽( · , 𝐵)‘𝑀))
125	124	oveq2d 7373	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq((𝐽 − 1) + 1)( · , 𝐵)‘𝑀)) = ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq𝐽( · , 𝐵)‘𝑀)))
126	121, 125	eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) = ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq𝐽( · , 𝐵)‘𝑀)))
127		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝑎 ∈ (𝐿...(𝐽 − 1))
128	114, 127	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑖((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1)))
129	128, 29	nfim 1899	. . . . . . . . 9 ⊢ Ⅎ𝑖(((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑎) ∈ ℝ)
130		eleq1w 2820	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (𝐿...(𝐽 − 1)) ↔ 𝑎 ∈ (𝐿...(𝐽 − 1))))
131	130	anbi2d 629	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) ↔ ((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1)))))
132	131, 34	imbi12d 344	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → ((((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑖) ∈ ℝ) ↔ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑎) ∈ ℝ)))
133	6	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝐿 ∈ ℤ)
134	52	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑀 ∈ ℤ)
135		elfzelz 13441	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (𝐿...(𝐽 − 1)) → 𝑖 ∈ ℤ)
136	135	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ∈ ℤ)
137		elfzle1 13444	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (𝐿...(𝐽 − 1)) → 𝐿 ≤ 𝑖)
138	137	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝐿 ≤ 𝑖)
139	135	zred 12607	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (𝐿...(𝐽 − 1)) → 𝑖 ∈ ℝ)
140	139	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ∈ ℝ)
141	60	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝐽 ∈ ℝ)
142	52	zred 12607	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
143	142	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑀 ∈ ℝ)
144		1red 11156	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
145	60, 144	resubcld 11583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐽 − 1) ∈ ℝ)
146	145	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐽 − 1) ∈ ℝ)
147		elfzle2 13445	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (𝐿...(𝐽 − 1)) → 𝑖 ≤ (𝐽 − 1))
148	147	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ≤ (𝐽 − 1))
149	60	lem1d 12088	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐽 − 1) ≤ 𝐽)
150	149	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐽 − 1) ≤ 𝐽)
151	140, 146, 141, 148, 150	letrd 11312	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ≤ 𝐽)
152		elfzle2 13445	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ (𝐿...𝑀) → 𝐽 ≤ 𝑀)
153	41, 152	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ≤ 𝑀)
154	153	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝐽 ≤ 𝑀)
155	140, 141, 143, 151, 154	letrd 11312	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ≤ 𝑀)
156	133, 134, 136, 138, 155	elfzd 13432	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ∈ (𝐿...𝑀))
157	156, 10	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑖) ∈ ℝ)
158	157	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑖) ∈ ℝ)
159	129, 132, 158	chvarfv 2233	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑎) ∈ ℝ)
160	37	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ)
161	112, 159, 160	seqcl 13928	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘(𝐽 − 1)) ∈ ℝ)
162		1red 11156	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 1 ∈ ℝ)
163		eqid 2736	. . . . . . . . 9 ⊢ seq𝐽( · , 𝐵) = seq𝐽( · , 𝐵)
164	43	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝑀 ∈ (ℤ_≥‘𝐽))
165		eluzfz2 13449	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘𝐽) → 𝑀 ∈ (𝐽...𝑀))
166	43, 165	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (𝐽...𝑀))
167	166	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝑀 ∈ (𝐽...𝑀))
168	73	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
169	72, 12	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 0 ≤ (𝐵‘𝑖))
170	169	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐽...𝑀)) → 0 ≤ (𝐵‘𝑖))
171	72, 14	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ≤ 1)
172	171	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ≤ 1)
173	1, 114, 163, 96, 164, 167, 168, 170, 172	fmul01 43811	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (0 ≤ (seq𝐽( · , 𝐵)‘𝑀) ∧ (seq𝐽( · , 𝐵)‘𝑀) ≤ 1))
174	173	simpld 495	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 0 ≤ (seq𝐽( · , 𝐵)‘𝑀))
175		eqid 2736	. . . . . . . . 9 ⊢ seq𝐿( · , 𝐵) = seq𝐿( · , 𝐵)
176	8	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝑀 ∈ (ℤ_≥‘𝐿))
177		1zzd 12534	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℤ)
178	59, 177	zsubcld 12612	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ)
179	6, 52, 178	3jca 1128	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
180	179	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
181	145, 60, 142	3jca 1128	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐽 − 1) ∈ ℝ ∧ 𝐽 ∈ ℝ ∧ 𝑀 ∈ ℝ))
182	181	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((𝐽 − 1) ∈ ℝ ∧ 𝐽 ∈ ℝ ∧ 𝑀 ∈ ℝ))
183	60	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐽 ∈ ℝ)
184	183	lem1d 12088	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ≤ 𝐽)
185	153	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐽 ≤ 𝑀)
186	184, 185	jca 512	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((𝐽 − 1) ≤ 𝐽 ∧ 𝐽 ≤ 𝑀))
187		letr 11249	. . . . . . . . . . . 12 ⊢ (((𝐽 − 1) ∈ ℝ ∧ 𝐽 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((𝐽 − 1) ≤ 𝐽 ∧ 𝐽 ≤ 𝑀) → (𝐽 − 1) ≤ 𝑀))
188	182, 186, 187	sylc 65	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ≤ 𝑀)
189	110, 188	jca 512	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐿 ≤ (𝐽 − 1) ∧ (𝐽 − 1) ≤ 𝑀))
190		elfz2 13431	. . . . . . . . . 10 ⊢ ((𝐽 − 1) ∈ (𝐿...𝑀) ↔ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ) ∧ (𝐿 ≤ (𝐽 − 1) ∧ (𝐽 − 1) ≤ 𝑀)))
191	180, 189, 190	sylanbrc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ∈ (𝐿...𝑀))
192	12	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
193	14	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)
194	1, 114, 175, 95, 176, 191, 119, 192, 193	fmul01 43811	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (0 ≤ (seq𝐿( · , 𝐵)‘(𝐽 − 1)) ∧ (seq𝐿( · , 𝐵)‘(𝐽 − 1)) ≤ 1))
195	194	simprd 496	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘(𝐽 − 1)) ≤ 1)
196	161, 162, 76, 174, 195	lemul1ad 12094	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq𝐽( · , 𝐵)‘𝑀)) ≤ (1 · (seq𝐽( · , 𝐵)‘𝑀)))
197	126, 196	eqbrtrd 5127	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) ≤ (1 · (seq𝐽( · , 𝐵)‘𝑀)))
198	76	recnd 11183	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐽( · , 𝐵)‘𝑀) ∈ ℂ)
199	198	mulid2d 11173	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (1 · (seq𝐽( · , 𝐵)‘𝑀)) = (seq𝐽( · , 𝐵)‘𝑀))
200	197, 199	breqtrd 5131	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) ≤ (seq𝐽( · , 𝐵)‘𝑀))
201	1, 2, 163, 59, 43, 73, 169, 171, 16, 20	fmul01lt1lem1 43815	. . . . 5 ⊢ (𝜑 → (seq𝐽( · , 𝐵)‘𝑀) < 𝐸)
202	201	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐽( · , 𝐵)‘𝑀) < 𝐸)
203	40, 76, 78, 200, 202	lelttrd 11313	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) < 𝐸)
204	24, 203	eqbrtrid 5140	. 2 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐴‘𝑀) < 𝐸)
205	23, 204	pm2.61dan 811	1 ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2887 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 < clt 11189 ≤ cle 11190 − cmin 11385 ℤcz 12499 ℤ_≥cuz 12763 ℝ⁺crp 12915 ...cfz 13424 seqcseq 13906

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-seq 13907

This theorem is referenced by: fmul01lt1 43817

W3C validator