fmul01lt1lem2 - Mathbox for Glauco Siliprandi

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Theorem fmul01lt1lem2 38652

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 1830	. . . 4 ⊢ Ⅎ𝑖 𝐽 = 𝐿
4	2, 3	nfan 1816	. . 3 ⊢ Ⅎ𝑖(𝜑 ∧ 𝐽 = 𝐿)
5		fmul01lt1lem2.3	. . 3 ⊢ 𝐴 = seq𝐿( · , 𝐵)
6		fmul01lt1lem2.4	. . . 4 ⊢ (𝜑 → 𝐿 ∈ ℤ)
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → 𝐿 ∈ ℤ)
8		fmul01lt1lem2.5	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐿))
9	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → 𝑀 ∈ (ℤ_≥‘𝐿))
10		fmul01lt1lem2.6	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
11	10	adantlr 747	. . 3 ⊢ (((𝜑 ∧ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
12		fmul01lt1lem2.7	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
13	12	adantlr 747	. . 3 ⊢ (((𝜑 ∧ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
14		fmul01lt1lem2.8	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)
15	14	adantlr 747	. . 3 ⊢ (((𝜑 ∧ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)
16		fmul01lt1lem2.9	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → 𝐸 ∈ ℝ⁺)
18		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → 𝐽 = 𝐿)
19	18	fveq2d 6107	. . . 4 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → (𝐵‘𝐽) = (𝐵‘𝐿))
20		fmul01lt1lem2.11	. . . . 5 ⊢ (𝜑 → (𝐵‘𝐽) < 𝐸)
21	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → (𝐵‘𝐽) < 𝐸)
22	19, 21	eqbrtrrd 4607	. . 3 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → (𝐵‘𝐿) < 𝐸)
23	1, 4, 5, 7, 9, 11, 13, 15, 17, 22	fmul01lt1lem1 38651	. 2 ⊢ ((𝜑 ∧ 𝐽 = 𝐿) → (𝐴‘𝑀) < 𝐸)
24	5	fveq1i 6104	. . 3 ⊢ (𝐴‘𝑀) = (seq𝐿( · , 𝐵)‘𝑀)
25		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑖 𝑎 ∈ (𝐿...𝑀)
26	2, 25	nfan 1816	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑎 ∈ (𝐿...𝑀))
27		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝑎
28	1, 27	nffv 6110	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐵‘𝑎)
29	28	nfel1 2765	. . . . . . . 8 ⊢ Ⅎ𝑖(𝐵‘𝑎) ∈ ℝ
30	26, 29	nfim 1813	. . . . . . 7 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ)
31		eleq1 2676	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑎 ∈ (𝐿...𝑀)))
32	31	anbi2d 736	. . . . . . . 8 ⊢ (𝑖 = 𝑎 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝑎 ∈ (𝐿...𝑀))))
33		fveq2 6103	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (𝐵‘𝑖) = (𝐵‘𝑎))
34	33	eleq1d 2672	. . . . . . . 8 ⊢ (𝑖 = 𝑎 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝑎) ∈ ℝ))
35	32, 34	imbi12d 333	. . . . . . 7 ⊢ (𝑖 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ)))
36	30, 35, 10	chvar 2250	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ)
37		remulcl 9900	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑎 · 𝑗) ∈ ℝ)
38	37	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ)
39	8, 36, 38	seqcl 12683	. . . . 5 ⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝑀) ∈ ℝ)
40	39	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) ∈ ℝ)
41		fmul01lt1lem2.10	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑀))
42		elfzuz3 12210	. . . . . . 7 ⊢ (𝐽 ∈ (𝐿...𝑀) → 𝑀 ∈ (ℤ_≥‘𝐽))
43	41, 42	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐽))
44		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑖 𝑎 ∈ (𝐽...𝑀)
45	2, 44	nfan 1816	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑎 ∈ (𝐽...𝑀))
46	45, 29	nfim 1813	. . . . . . 7 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑎 ∈ (𝐽...𝑀)) → (𝐵‘𝑎) ∈ ℝ)
47		eleq1 2676	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (𝐽...𝑀) ↔ 𝑎 ∈ (𝐽...𝑀)))
48	47	anbi2d 736	. . . . . . . 8 ⊢ (𝑖 = 𝑎 → ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) ↔ (𝜑 ∧ 𝑎 ∈ (𝐽...𝑀))))
49	48, 34	imbi12d 333	. . . . . . 7 ⊢ (𝑖 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝑎 ∈ (𝐽...𝑀)) → (𝐵‘𝑎) ∈ ℝ)))
50	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐿 ∈ ℤ)
51		eluzelz 11573	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (ℤ_≥‘𝐿) → 𝑀 ∈ ℤ)
52	8, 51	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
53	52	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑀 ∈ ℤ)
54		elfzelz 12213	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐽...𝑀) → 𝑖 ∈ ℤ)
55	54	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑖 ∈ ℤ)
56	50, 53, 55	3jca 1235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ))
57	6	zred 11358	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ ℝ)
58	57	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐿 ∈ ℝ)
59		elfzelz 12213	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (𝐿...𝑀) → 𝐽 ∈ ℤ)
60	41, 59	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ ℤ)
61	60	zred 11358	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℝ)
62	61	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐽 ∈ ℝ)
63	54	zred 11358	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (𝐽...𝑀) → 𝑖 ∈ ℝ)
64	63	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑖 ∈ ℝ)
65		elfzle1 12215	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (𝐿...𝑀) → 𝐿 ≤ 𝐽)
66	41, 65	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ≤ 𝐽)
67	66	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐿 ≤ 𝐽)
68		elfzle1 12215	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (𝐽...𝑀) → 𝐽 ≤ 𝑖)
69	68	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐽 ≤ 𝑖)
70	58, 62, 64, 67, 69	letrd 10073	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐿 ≤ 𝑖)
71		elfzle2 12216	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐽...𝑀) → 𝑖 ≤ 𝑀)
72	71	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑖 ≤ 𝑀)
73	70, 72	jca 553	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐿 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀))
74		elfz2 12204	. . . . . . . . 9 ⊢ (𝑖 ∈ (𝐿...𝑀) ↔ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝐿 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)))
75	56, 73, 74	sylanbrc 695	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑖 ∈ (𝐿...𝑀))
76	75, 10	syldan 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
77	46, 49, 76	chvar 2250	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐽...𝑀)) → (𝐵‘𝑎) ∈ ℝ)
78	43, 77, 38	seqcl 12683	. . . . 5 ⊢ (𝜑 → (seq𝐽( · , 𝐵)‘𝑀) ∈ ℝ)
79	78	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐽( · , 𝐵)‘𝑀) ∈ ℝ)
80	16	rpred 11748	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ)
81	80	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐸 ∈ ℝ)
82		remulcl 9900	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 · 𝑏) ∈ ℝ)
83	82	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ)) → (𝑎 · 𝑏) ∈ ℝ)
84		simp1 1054	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑎 ∈ ℝ)
85	84	recnd 9947	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑎 ∈ ℂ)
86		simp2 1055	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑏 ∈ ℝ)
87	86	recnd 9947	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑏 ∈ ℂ)
88		simp3 1056	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
89	88	recnd 9947	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℂ)
90	85, 87, 89	mulassd 9942	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → ((𝑎 · 𝑏) · 𝑐) = (𝑎 · (𝑏 · 𝑐)))
91	90	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ)) → ((𝑎 · 𝑏) · 𝑐) = (𝑎 · (𝑏 · 𝑐)))
92	60	zcnd 11359	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℂ)
93		1cnd 9935	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
94	92, 93	npcand 10275	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐽 − 1) + 1) = 𝐽)
95	94	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ_≥‘((𝐽 − 1) + 1)) = (ℤ_≥‘𝐽))
96	43, 95	eleqtrrd 2691	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘((𝐽 − 1) + 1)))
97	96	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝑀 ∈ (ℤ_≥‘((𝐽 − 1) + 1)))
98	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐿 ∈ ℤ)
99	60	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐽 ∈ ℤ)
100		1zzd 11285	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 1 ∈ ℤ)
101	99, 100	zsubcld 11363	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ∈ ℤ)
102		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ¬ 𝐽 = 𝐿)
103		eqcom 2617	. . . . . . . . . . . 12 ⊢ (𝐽 = 𝐿 ↔ 𝐿 = 𝐽)
104	102, 103	sylnib 317	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ¬ 𝐿 = 𝐽)
105	57, 61	leloed 10059	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐿 ≤ 𝐽 ↔ (𝐿 < 𝐽 ∨ 𝐿 = 𝐽)))
106	66, 105	mpbid 221	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿 < 𝐽 ∨ 𝐿 = 𝐽))
107	106	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐿 < 𝐽 ∨ 𝐿 = 𝐽))
108		orel2 397	. . . . . . . . . . 11 ⊢ (¬ 𝐿 = 𝐽 → ((𝐿 < 𝐽 ∨ 𝐿 = 𝐽) → 𝐿 < 𝐽))
109	104, 107, 108	sylc 63	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐿 < 𝐽)
110		zltlem1 11307	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐿 < 𝐽 ↔ 𝐿 ≤ (𝐽 − 1)))
111	6, 60, 110	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 < 𝐽 ↔ 𝐿 ≤ (𝐽 − 1)))
112	111	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐿 < 𝐽 ↔ 𝐿 ≤ (𝐽 − 1)))
113	109, 112	mpbid 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐿 ≤ (𝐽 − 1))
114		eluz2 11569	. . . . . . . . 9 ⊢ ((𝐽 − 1) ∈ (ℤ_≥‘𝐿) ↔ (𝐿 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ ∧ 𝐿 ≤ (𝐽 − 1)))
115	98, 101, 113, 114	syl3anbrc 1239	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ∈ (ℤ_≥‘𝐿))
116		nfv 1830	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖 ¬ 𝐽 = 𝐿
117	2, 116	nfan 1816	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝜑 ∧ ¬ 𝐽 = 𝐿)
118	117, 25	nfan 1816	. . . . . . . . . 10 ⊢ Ⅎ𝑖((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀))
119	118, 29	nfim 1813	. . . . . . . . 9 ⊢ Ⅎ𝑖(((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ)
120	31	anbi2d 736	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ ((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀))))
121	120, 34	imbi12d 333	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → ((((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ)))
122	10	adantlr 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
123	119, 121, 122	chvar 2250	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ)
124	83, 91, 97, 115, 123	seqsplit 12696	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) = ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq((𝐽 − 1) + 1)( · , 𝐵)‘𝑀)))
125	94	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((𝐽 − 1) + 1) = 𝐽)
126	125	seqeq1d 12669	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → seq((𝐽 − 1) + 1)( · , 𝐵) = seq𝐽( · , 𝐵))
127	126	fveq1d 6105	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq((𝐽 − 1) + 1)( · , 𝐵)‘𝑀) = (seq𝐽( · , 𝐵)‘𝑀))
128	127	oveq2d 6565	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq((𝐽 − 1) + 1)( · , 𝐵)‘𝑀)) = ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq𝐽( · , 𝐵)‘𝑀)))
129	124, 128	eqtrd 2644	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) = ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq𝐽( · , 𝐵)‘𝑀)))
130		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝑎 ∈ (𝐿...(𝐽 − 1))
131	117, 130	nfan 1816	. . . . . . . . . 10 ⊢ Ⅎ𝑖((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1)))
132	131, 29	nfim 1813	. . . . . . . . 9 ⊢ Ⅎ𝑖(((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑎) ∈ ℝ)
133		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (𝐿...(𝐽 − 1)) ↔ 𝑎 ∈ (𝐿...(𝐽 − 1))))
134	133	anbi2d 736	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) ↔ ((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1)))))
135	134, 34	imbi12d 333	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → ((((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑖) ∈ ℝ) ↔ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑎) ∈ ℝ)))
136	6	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝐿 ∈ ℤ)
137	52	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑀 ∈ ℤ)
138		elfzelz 12213	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (𝐿...(𝐽 − 1)) → 𝑖 ∈ ℤ)
139	138	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ∈ ℤ)
140	136, 137, 139	3jca 1235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ))
141		elfzle1 12215	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (𝐿...(𝐽 − 1)) → 𝐿 ≤ 𝑖)
142	141	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝐿 ≤ 𝑖)
143	138	zred 11358	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (𝐿...(𝐽 − 1)) → 𝑖 ∈ ℝ)
144	143	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ∈ ℝ)
145	61	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝐽 ∈ ℝ)
146	52	zred 11358	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
147	146	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑀 ∈ ℝ)
148		1red 9934	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℝ)
149	61, 148	resubcld 10337	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽 − 1) ∈ ℝ)
150	149	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐽 − 1) ∈ ℝ)
151		elfzle2 12216	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝐿...(𝐽 − 1)) → 𝑖 ≤ (𝐽 − 1))
152	151	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ≤ (𝐽 − 1))
153	61	lem1d 10836	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽 − 1) ≤ 𝐽)
154	153	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐽 − 1) ≤ 𝐽)
155	144, 150, 145, 152, 154	letrd 10073	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ≤ 𝐽)
156		elfzle2 12216	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (𝐿...𝑀) → 𝐽 ≤ 𝑀)
157	41, 156	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ≤ 𝑀)
158	157	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝐽 ≤ 𝑀)
159	144, 145, 147, 155, 158	letrd 10073	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ≤ 𝑀)
160	142, 159	jca 553	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐿 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀))
161	140, 160, 74	sylanbrc 695	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ∈ (𝐿...𝑀))
162	161, 10	syldan 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑖) ∈ ℝ)
163	162	adantlr 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑖) ∈ ℝ)
164	132, 135, 163	chvar 2250	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑎) ∈ ℝ)
165	37	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ)
166	115, 164, 165	seqcl 12683	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘(𝐽 − 1)) ∈ ℝ)
167		1red 9934	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 1 ∈ ℝ)
168		eqid 2610	. . . . . . . . 9 ⊢ seq𝐽( · , 𝐵) = seq𝐽( · , 𝐵)
169	43	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝑀 ∈ (ℤ_≥‘𝐽))
170		eluzfz2 12220	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘𝐽) → 𝑀 ∈ (𝐽...𝑀))
171	43, 170	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (𝐽...𝑀))
172	171	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝑀 ∈ (𝐽...𝑀))
173	76	adantlr 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ∈ ℝ)
174	75, 12	syldan 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 0 ≤ (𝐵‘𝑖))
175	174	adantlr 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐽...𝑀)) → 0 ≤ (𝐵‘𝑖))
176	75, 14	syldan 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ≤ 1)
177	176	adantlr 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ≤ 1)
178	1, 117, 168, 99, 169, 172, 173, 175, 177	fmul01 38647	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (0 ≤ (seq𝐽( · , 𝐵)‘𝑀) ∧ (seq𝐽( · , 𝐵)‘𝑀) ≤ 1))
179	178	simpld 474	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 0 ≤ (seq𝐽( · , 𝐵)‘𝑀))
180		eqid 2610	. . . . . . . . 9 ⊢ seq𝐿( · , 𝐵) = seq𝐿( · , 𝐵)
181	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝑀 ∈ (ℤ_≥‘𝐿))
182		1zzd 11285	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℤ)
183	60, 182	zsubcld 11363	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ)
184	6, 52, 183	3jca 1235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
185	184	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
186	149, 61, 146	3jca 1235	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐽 − 1) ∈ ℝ ∧ 𝐽 ∈ ℝ ∧ 𝑀 ∈ ℝ))
187	186	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((𝐽 − 1) ∈ ℝ ∧ 𝐽 ∈ ℝ ∧ 𝑀 ∈ ℝ))
188	61	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐽 ∈ ℝ)
189	188	lem1d 10836	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ≤ 𝐽)
190	157	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐽 ≤ 𝑀)
191	189, 190	jca 553	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((𝐽 − 1) ≤ 𝐽 ∧ 𝐽 ≤ 𝑀))
192		letr 10010	. . . . . . . . . . . 12 ⊢ (((𝐽 − 1) ∈ ℝ ∧ 𝐽 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((𝐽 − 1) ≤ 𝐽 ∧ 𝐽 ≤ 𝑀) → (𝐽 − 1) ≤ 𝑀))
193	187, 191, 192	sylc 63	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ≤ 𝑀)
194	113, 193	jca 553	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐿 ≤ (𝐽 − 1) ∧ (𝐽 − 1) ≤ 𝑀))
195		elfz2 12204	. . . . . . . . . 10 ⊢ ((𝐽 − 1) ∈ (𝐿...𝑀) ↔ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ) ∧ (𝐿 ≤ (𝐽 − 1) ∧ (𝐽 − 1) ≤ 𝑀)))
196	185, 194, 195	sylanbrc 695	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ∈ (𝐿...𝑀))
197	12	adantlr 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))
198	14	adantlr 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)
199	1, 117, 180, 98, 181, 196, 122, 197, 198	fmul01 38647	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (0 ≤ (seq𝐿( · , 𝐵)‘(𝐽 − 1)) ∧ (seq𝐿( · , 𝐵)‘(𝐽 − 1)) ≤ 1))
200	199	simprd 478	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘(𝐽 − 1)) ≤ 1)
201	166, 167, 79, 179, 200	lemul1ad 10842	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq𝐽( · , 𝐵)‘𝑀)) ≤ (1 · (seq𝐽( · , 𝐵)‘𝑀)))
202	129, 201	eqbrtrd 4605	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) ≤ (1 · (seq𝐽( · , 𝐵)‘𝑀)))
203	79	recnd 9947	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐽( · , 𝐵)‘𝑀) ∈ ℂ)
204	203	mulid2d 9937	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (1 · (seq𝐽( · , 𝐵)‘𝑀)) = (seq𝐽( · , 𝐵)‘𝑀))
205	202, 204	breqtrd 4609	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) ≤ (seq𝐽( · , 𝐵)‘𝑀))
206	1, 2, 168, 60, 43, 76, 174, 176, 16, 20	fmul01lt1lem1 38651	. . . . 5 ⊢ (𝜑 → (seq𝐽( · , 𝐵)‘𝑀) < 𝐸)
207	206	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐽( · , 𝐵)‘𝑀) < 𝐸)
208	40, 79, 81, 205, 207	lelttrd 10074	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) < 𝐸)
209	24, 208	syl5eqbr 4618	. 2 ⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐴‘𝑀) < 𝐸)
210	23, 209	pm2.61dan 828	1 ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 ℤcz 11254 ℤ_≥cuz 11563 ℝ⁺crp 11708 ...cfz 12197 seqcseq 12663

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664

This theorem is referenced by: fmul01lt1 38653

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