trilpolemisumle - Mathbox for Jim Kingdon

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Theorem trilpolemisumle 15528

Description: Lemma for trilpo 15533. 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 9438	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	1	eleq2i 2260	. . . . 5 ⊢ (𝑖 ∈ 𝑍 ↔ 𝑖 ∈ (ℤ_≥‘𝑀))
5	4	biimpi 120	. . . 4 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ (ℤ_≥‘𝑀))
6		eluznn 9665	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑖 ∈ (ℤ_≥‘𝑀)) → 𝑖 ∈ ℕ)
7	2, 5, 6	syl2an 289	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ ℕ)
8		eqid 2193	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))
9		oveq2 5926	. . . . . 6 ⊢ (𝑛 = 𝑖 → (2↑𝑛) = (2↑𝑖))
10	9	oveq2d 5934	. . . . 5 ⊢ (𝑛 = 𝑖 → (1 / (2↑𝑛)) = (1 / (2↑𝑖)))
11		fveq2 5554	. . . . 5 ⊢ (𝑛 = 𝑖 → (𝐹‘𝑛) = (𝐹‘𝑖))
12	10, 11	oveq12d 5936	. . . 4 ⊢ (𝑛 = 𝑖 → ((1 / (2↑𝑛)) · (𝐹‘𝑛)) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
13		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
14		2rp 9724	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
15	14	a1i 9	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 2 ∈ ℝ⁺)
16	13	nnzd 9438	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
17	15, 16	rpexpcld 10768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
18	17	rpreccld 9773	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ⁺)
19	18	rpred 9762	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
20		trilpolemgt1.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1})
21		0re 8019	. . . . . . . . 9 ⊢ 0 ∈ ℝ
22		1re 8018	. . . . . . . . 9 ⊢ 1 ∈ ℝ
23		prssi 3776	. . . . . . . . 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 5416	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
27	26	ffvelcdmda 5693	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ)
28	19, 27	remulcld 8050	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
29	8, 12, 13, 28	fvmptd3 5651	. . 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 2193	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛))) = (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))
33	32, 10, 13, 18	fvmptd3 5651	. . 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 5934	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 0))
38	18	rpcnd 9764	. . . . . . . 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 8412	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · 0) = 0)
41	37, 40	eqtrd 2226	. . . . 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 9766	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → 0 ≤ (1 / (2↑𝑖)))
44	41, 43	eqbrtrd 4051	. . . 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 5934	. . . . . 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	mulridd 8036	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · 1) = (1 / (2↑𝑖)))
49	46, 48	eqtrd 2226	. . . . 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 8533	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ≤ (1 / (2↑𝑖)))
52	49, 51	eqbrtrd 4051	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
53	20	ffvelcdmda 5693	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ {0, 1})
54		elpri 3641	. . . . 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 799	. . 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 15526	. . 3 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ )
59		nnuz 9628	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
60	29, 28	eqeltrd 2270	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) ∈ ℝ)
61	60	recnd 8048	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) ∈ ℂ)
62	59, 2, 61	iserex 11482	. . 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 10520	. . . 4 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ V
65		rpreccl 9746	. . . . . . . 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 9344	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
69	67, 68	rpexpcld 10768	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑1) ∈ ℝ⁺)
70		1mhlfehlf 9200	. . . . . . 7 ⊢ (1 − (1 / 2)) = (1 / 2)
71	70, 66	eqeltri 2266	. . . . . 6 ⊢ (1 − (1 / 2)) ∈ ℝ⁺
72	71	a1i 9	. . . . 5 ⊢ (𝜑 → (1 − (1 / 2)) ∈ ℝ⁺)
73	69, 72	rpdivcld 9780	. . . 4 ⊢ (𝜑 → (((1 / 2)↑1) / (1 − (1 / 2))) ∈ ℝ⁺)
74		halfcn 9196	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
75	74	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
76		halfge0 9198	. . . . . . . 8 ⊢ 0 ≤ (1 / 2)
77		halfre 9195	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
78	77	absidi 11270	. . . . . . . 8 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2))
79	76, 78	ax-mp 5	. . . . . . 7 ⊢ (abs‘(1 / 2)) = (1 / 2)
80		halflt1 9199	. . . . . . 7 ⊢ (1 / 2) < 1
81	79, 80	eqbrtri 4050	. . . . . 6 ⊢ (abs‘(1 / 2)) < 1
82	81	a1i 9	. . . . 5 ⊢ (𝜑 → (abs‘(1 / 2)) < 1)
83		1nn0 9256	. . . . . 6 ⊢ 1 ∈ ℕ₀
84	83	a1i 9	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
85		oveq2 5926	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (2↑𝑛) = (2↑𝑗))
86	85	oveq2d 5934	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (1 / (2↑𝑛)) = (1 / (2↑𝑗)))
87		elnnuz 9629	. . . . . . . . 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 9438	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 𝑗 ∈ ℤ)
92	90, 91	rpexpcld 10768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → (2↑𝑗) ∈ ℝ⁺)
93	92	rpreccld 9773	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → (1 / (2↑𝑗)) ∈ ℝ⁺)
94	32, 86, 89, 93	fvmptd3 5651	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑗) = (1 / (2↑𝑗)))
95		2cnd 9055	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 2 ∈ ℂ)
96	90	rpap0d 9768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 2 # 0)
97	95, 96, 91	exprecapd 10752	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((1 / 2)↑𝑗) = (1 / (2↑𝑗)))
98	94, 97	eqtr4d 2229	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑗) = ((1 / 2)↑𝑗))
99	75, 82, 84, 98	geolim2 11655	. . . 4 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ⇝ (((1 / 2)↑1) / (1 − (1 / 2))))
100		breldmg 4868	. . . 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 1353	. . 3 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ )
102	33, 38	eqeltrd 2270	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) ∈ ℂ)
103	59, 2, 102	iserex 11482	. . 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 11638	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 709 = wceq 1364 ∈ wcel 2164 Vcvv 2760 ⊆ wss 3153 {cpr 3619 class class class wbr 4029 ↦ cmpt 4090 dom cdm 4659 ⟶wf 5250 ‘cfv 5254 (class class class)co 5918 ℂcc 7870 ℝcr 7871 0cc0 7872 1c1 7873 + caddc 7875 · cmul 7877 < clt 8054 ≤ cle 8055 − cmin 8190 / cdiv 8691 ℕcn 8982 2c2 9033 ℕ₀cn0 9240 ℤ_≥cuz 9592 ℝ⁺crp 9719 seqcseq 10518 ↑cexp 10609 abscabs 11141 ⇝ cli 11421 Σcsu 11496

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-ico 9960 df-fz 10075 df-fzo 10209 df-seqfrec 10519 df-exp 10610 df-ihash 10847 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-sumdc 11497

This theorem is referenced by: trilpolemgt1 15529 trilpolemeq1 15530

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