trilpolemisumle - Mathbox for Jim Kingdon

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Theorem trilpolemisumle 14442

Description: Lemma for trilpo 14447. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.)

**Hypotheses**
Ref	Expression
trilpolemgt1.f	⊢ (𝜑 → 𝐹:ℕ⟶{0, 1})
trilpolemgt1.a	⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))
trilpolemisumle.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
trilpolemisumle.m	⊢ (𝜑 → 𝑀 ∈ ℕ)

**Assertion**
Ref	Expression
trilpolemisumle	⊢ (𝜑 → Σ𝑖 ∈ 𝑍 ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ Σ𝑖 ∈ 𝑍 (1 / (2↑𝑖)))

Distinct variable groups: 𝑖,𝐹 𝑖,𝑀 𝑖,𝑍 𝜑,𝑖 Allowed substitution hint: 𝐴(𝑖)

Proof of Theorem trilpolemisumle Dummy variables 𝑛 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		trilpolemisumle.z	. 2 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		trilpolemisumle.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3	2	nnzd 9363	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	1	eleq2i 2244	. . . . 5 ⊢ (𝑖 ∈ 𝑍 ↔ 𝑖 ∈ (ℤ_≥‘𝑀))
5	4	biimpi 120	. . . 4 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ (ℤ_≥‘𝑀))
6		eluznn 9589	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑖 ∈ (ℤ_≥‘𝑀)) → 𝑖 ∈ ℕ)
7	2, 5, 6	syl2an 289	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ ℕ)
8		eqid 2177	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))
9		oveq2 5877	. . . . . 6 ⊢ (𝑛 = 𝑖 → (2↑𝑛) = (2↑𝑖))
10	9	oveq2d 5885	. . . . 5 ⊢ (𝑛 = 𝑖 → (1 / (2↑𝑛)) = (1 / (2↑𝑖)))
11		fveq2 5511	. . . . 5 ⊢ (𝑛 = 𝑖 → (𝐹‘𝑛) = (𝐹‘𝑖))
12	10, 11	oveq12d 5887	. . . 4 ⊢ (𝑛 = 𝑖 → ((1 / (2↑𝑛)) · (𝐹‘𝑛)) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
13		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
14		2rp 9645	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
15	14	a1i 9	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 2 ∈ ℝ⁺)
16	13	nnzd 9363	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
17	15, 16	rpexpcld 10663	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
18	17	rpreccld 9694	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ⁺)
19	18	rpred 9683	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
20		trilpolemgt1.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1})
21		0re 7948	. . . . . . . . 9 ⊢ 0 ∈ ℝ
22		1re 7947	. . . . . . . . 9 ⊢ 1 ∈ ℝ
23		prssi 3749	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ)
24	21, 22, 23	mp2an 426	. . . . . . . 8 ⊢ {0, 1} ⊆ ℝ
25	24	a1i 9	. . . . . . 7 ⊢ (𝜑 → {0, 1} ⊆ ℝ)
26	20, 25	fssd 5374	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
27	26	ffvelcdmda 5647	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ)
28	19, 27	remulcld 7978	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
29	8, 12, 13, 28	fvmptd3 5605	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
30	7, 29	syldan 282	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
31	7, 28	syldan 282	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
32		eqid 2177	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛))) = (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))
33	32, 10, 13, 18	fvmptd3 5605	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) = (1 / (2↑𝑖)))
34	7, 33	syldan 282	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) = (1 / (2↑𝑖)))
35	7, 19	syldan 282	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (1 / (2↑𝑖)) ∈ ℝ)
36		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → (𝐹‘𝑖) = 0)
37	36	oveq2d 5885	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 0))
38	18	rpcnd 9685	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℂ)
39	38	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → (1 / (2↑𝑖)) ∈ ℂ)
40	39	mul01d 8340	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · 0) = 0)
41	37, 40	eqtrd 2210	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = 0)
42	18	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → (1 / (2↑𝑖)) ∈ ℝ⁺)
43	42	rpge0d 9687	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → 0 ≤ (1 / (2↑𝑖)))
44	41, 43	eqbrtrd 4022	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
45		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → (𝐹‘𝑖) = 1)
46	45	oveq2d 5885	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 1))
47	38	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ∈ ℂ)
48	47	mulid1d 7965	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · 1) = (1 / (2↑𝑖)))
49	46, 48	eqtrd 2210	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (1 / (2↑𝑖)))
50	19	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ∈ ℝ)
51	50	leidd 8461	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ≤ (1 / (2↑𝑖)))
52	49, 51	eqbrtrd 4022	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
53	20	ffvelcdmda 5647	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ {0, 1})
54		elpri 3614	. . . . 5 ⊢ ((𝐹‘𝑖) ∈ {0, 1} → ((𝐹‘𝑖) = 0 ∨ (𝐹‘𝑖) = 1))
55	53, 54	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐹‘𝑖) = 0 ∨ (𝐹‘𝑖) = 1))
56	44, 52, 55	mpjaodan 798	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
57	7, 56	syldan 282	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
58	20, 8	trilpolemclim 14440	. . 3 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ )
59		nnuz 9552	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
60	29, 28	eqeltrd 2254	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) ∈ ℝ)
61	60	recnd 7976	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) ∈ ℂ)
62	59, 2, 61	iserex 11331	. . 3 ⊢ (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ ↔ seq𝑀( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ ))
63	58, 62	mpbid 147	. 2 ⊢ (𝜑 → seq𝑀( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ )
64		seqex 10433	. . . 4 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ V
65		rpreccl 9667	. . . . . . . 8 ⊢ (2 ∈ ℝ⁺ → (1 / 2) ∈ ℝ⁺)
66	14, 65	ax-mp 5	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ⁺
67	66	a1i 9	. . . . . 6 ⊢ (𝜑 → (1 / 2) ∈ ℝ⁺)
68		1zzd 9269	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
69	67, 68	rpexpcld 10663	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑1) ∈ ℝ⁺)
70		1mhlfehlf 9126	. . . . . . 7 ⊢ (1 − (1 / 2)) = (1 / 2)
71	70, 66	eqeltri 2250	. . . . . 6 ⊢ (1 − (1 / 2)) ∈ ℝ⁺
72	71	a1i 9	. . . . 5 ⊢ (𝜑 → (1 − (1 / 2)) ∈ ℝ⁺)
73	69, 72	rpdivcld 9701	. . . 4 ⊢ (𝜑 → (((1 / 2)↑1) / (1 − (1 / 2))) ∈ ℝ⁺)
74		halfcn 9122	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
75	74	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
76		halfge0 9124	. . . . . . . 8 ⊢ 0 ≤ (1 / 2)
77		halfre 9121	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
78	77	absidi 11119	. . . . . . . 8 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2))
79	76, 78	ax-mp 5	. . . . . . 7 ⊢ (abs‘(1 / 2)) = (1 / 2)
80		halflt1 9125	. . . . . . 7 ⊢ (1 / 2) < 1
81	79, 80	eqbrtri 4021	. . . . . 6 ⊢ (abs‘(1 / 2)) < 1
82	81	a1i 9	. . . . 5 ⊢ (𝜑 → (abs‘(1 / 2)) < 1)
83		1nn0 9181	. . . . . 6 ⊢ 1 ∈ ℕ₀
84	83	a1i 9	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
85		oveq2 5877	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (2↑𝑛) = (2↑𝑗))
86	85	oveq2d 5885	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (1 / (2↑𝑛)) = (1 / (2↑𝑗)))
87		elnnuz 9553	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ_≥‘1))
88	87	biimpri 133	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ_≥‘1) → 𝑗 ∈ ℕ)
89	88	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 𝑗 ∈ ℕ)
90	14	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 2 ∈ ℝ⁺)
91	89	nnzd 9363	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 𝑗 ∈ ℤ)
92	90, 91	rpexpcld 10663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → (2↑𝑗) ∈ ℝ⁺)
93	92	rpreccld 9694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → (1 / (2↑𝑗)) ∈ ℝ⁺)
94	32, 86, 89, 93	fvmptd3 5605	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑗) = (1 / (2↑𝑗)))
95		2cnd 8981	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 2 ∈ ℂ)
96	90	rpap0d 9689	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 2 # 0)
97	95, 96, 91	exprecapd 10647	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((1 / 2)↑𝑗) = (1 / (2↑𝑗)))
98	94, 97	eqtr4d 2213	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑗) = ((1 / 2)↑𝑗))
99	75, 82, 84, 98	geolim2 11504	. . . 4 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ⇝ (((1 / 2)↑1) / (1 − (1 / 2))))
100		breldmg 4829	. . . 4 ⊢ ((seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ V ∧ (((1 / 2)↑1) / (1 − (1 / 2))) ∈ ℝ⁺ ∧ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ⇝ (((1 / 2)↑1) / (1 − (1 / 2)))) → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ )
101	64, 73, 99, 100	mp3an2i 1342	. . 3 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ )
102	33, 38	eqeltrd 2254	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) ∈ ℂ)
103	59, 2, 102	iserex 11331	. . 3 ⊢ (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ ↔ seq𝑀( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ ))
104	101, 103	mpbid 147	. 2 ⊢ (𝜑 → seq𝑀( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ )
105	1, 3, 30, 31, 34, 35, 57, 63, 104	isumle 11487	1 ⊢ (𝜑 → Σ𝑖 ∈ 𝑍 ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ Σ𝑖 ∈ 𝑍 (1 / (2↑𝑖)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 {cpr 3592 class class class wbr 4000 ↦ cmpt 4061 dom cdm 4623 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 ℂcc 7800 ℝcr 7801 0cc0 7802 1c1 7803 + caddc 7805 · cmul 7807 < clt 7982 ≤ cle 7983 − cmin 8118 / cdiv 8618 ℕcn 8908 2c2 8959 ℕ₀cn0 9165 ℤ_≥cuz 9517 ℝ⁺crp 9640 seqcseq 10431 ↑cexp 10505 abscabs 10990 ⇝ cli 11270 Σcsu 11345

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-frec 6386 df-1o 6411 df-oadd 6415 df-er 6529 df-en 6735 df-dom 6736 df-fin 6737 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-ico 9881 df-fz 9996 df-fzo 10129 df-seqfrec 10432 df-exp 10506 df-ihash 10740 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-clim 11271 df-sumdc 11346

This theorem is referenced by: trilpolemgt1 14443 trilpolemeq1 14444

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