trilpolemisumle - Mathbox for Jim Kingdon

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Theorem trilpolemisumle 16578

Description: Lemma for trilpo 16583. 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 9591	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	1	eleq2i 2296	. . . . 5 ⊢ (𝑖 ∈ 𝑍 ↔ 𝑖 ∈ (ℤ_≥‘𝑀))
5	4	biimpi 120	. . . 4 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ (ℤ_≥‘𝑀))
6		eluznn 9824	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑖 ∈ (ℤ_≥‘𝑀)) → 𝑖 ∈ ℕ)
7	2, 5, 6	syl2an 289	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ ℕ)
8		eqid 2229	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))
9		oveq2 6021	. . . . . 6 ⊢ (𝑛 = 𝑖 → (2↑𝑛) = (2↑𝑖))
10	9	oveq2d 6029	. . . . 5 ⊢ (𝑛 = 𝑖 → (1 / (2↑𝑛)) = (1 / (2↑𝑖)))
11		fveq2 5635	. . . . 5 ⊢ (𝑛 = 𝑖 → (𝐹‘𝑛) = (𝐹‘𝑖))
12	10, 11	oveq12d 6031	. . . 4 ⊢ (𝑛 = 𝑖 → ((1 / (2↑𝑛)) · (𝐹‘𝑛)) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
13		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
14		2rp 9883	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
15	14	a1i 9	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 2 ∈ ℝ⁺)
16	13	nnzd 9591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
17	15, 16	rpexpcld 10949	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
18	17	rpreccld 9932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ⁺)
19	18	rpred 9921	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
20		trilpolemgt1.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1})
21		0re 8169	. . . . . . . . 9 ⊢ 0 ∈ ℝ
22		1re 8168	. . . . . . . . 9 ⊢ 1 ∈ ℝ
23		prssi 3829	. . . . . . . . 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 5492	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
27	26	ffvelcdmda 5778	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ)
28	19, 27	remulcld 8200	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
29	8, 12, 13, 28	fvmptd3 5736	. . 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 2229	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛))) = (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))
33	32, 10, 13, 18	fvmptd3 5736	. . 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 6029	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 0))
38	18	rpcnd 9923	. . . . . . . 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 8562	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · 0) = 0)
41	37, 40	eqtrd 2262	. . . . 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 9925	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → 0 ≤ (1 / (2↑𝑖)))
44	41, 43	eqbrtrd 4108	. . . 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 6029	. . . . . 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 8186	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · 1) = (1 / (2↑𝑖)))
49	46, 48	eqtrd 2262	. . . . 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 8684	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ≤ (1 / (2↑𝑖)))
52	49, 51	eqbrtrd 4108	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
53	20	ffvelcdmda 5778	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ {0, 1})
54		elpri 3690	. . . . 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 803	. . 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 16576	. . 3 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ )
59		nnuz 9782	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
60	29, 28	eqeltrd 2306	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) ∈ ℝ)
61	60	recnd 8198	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) ∈ ℂ)
62	59, 2, 61	iserex 11890	. . 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 10701	. . . 4 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ V
65		rpreccl 9905	. . . . . . . 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 9496	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
69	67, 68	rpexpcld 10949	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑1) ∈ ℝ⁺)
70		1mhlfehlf 9352	. . . . . . 7 ⊢ (1 − (1 / 2)) = (1 / 2)
71	70, 66	eqeltri 2302	. . . . . 6 ⊢ (1 − (1 / 2)) ∈ ℝ⁺
72	71	a1i 9	. . . . 5 ⊢ (𝜑 → (1 − (1 / 2)) ∈ ℝ⁺)
73	69, 72	rpdivcld 9939	. . . 4 ⊢ (𝜑 → (((1 / 2)↑1) / (1 − (1 / 2))) ∈ ℝ⁺)
74		halfcn 9348	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
75	74	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
76		halfge0 9350	. . . . . . . 8 ⊢ 0 ≤ (1 / 2)
77		halfre 9347	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
78	77	absidi 11677	. . . . . . . 8 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2))
79	76, 78	ax-mp 5	. . . . . . 7 ⊢ (abs‘(1 / 2)) = (1 / 2)
80		halflt1 9351	. . . . . . 7 ⊢ (1 / 2) < 1
81	79, 80	eqbrtri 4107	. . . . . 6 ⊢ (abs‘(1 / 2)) < 1
82	81	a1i 9	. . . . 5 ⊢ (𝜑 → (abs‘(1 / 2)) < 1)
83		1nn0 9408	. . . . . 6 ⊢ 1 ∈ ℕ₀
84	83	a1i 9	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
85		oveq2 6021	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (2↑𝑛) = (2↑𝑗))
86	85	oveq2d 6029	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (1 / (2↑𝑛)) = (1 / (2↑𝑗)))
87		elnnuz 9783	. . . . . . . . 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 9591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 𝑗 ∈ ℤ)
92	90, 91	rpexpcld 10949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → (2↑𝑗) ∈ ℝ⁺)
93	92	rpreccld 9932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → (1 / (2↑𝑗)) ∈ ℝ⁺)
94	32, 86, 89, 93	fvmptd3 5736	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑗) = (1 / (2↑𝑗)))
95		2cnd 9206	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 2 ∈ ℂ)
96	90	rpap0d 9927	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 2 # 0)
97	95, 96, 91	exprecapd 10933	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((1 / 2)↑𝑗) = (1 / (2↑𝑗)))
98	94, 97	eqtr4d 2265	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑗) = ((1 / 2)↑𝑗))
99	75, 82, 84, 98	geolim2 12063	. . . 4 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ⇝ (((1 / 2)↑1) / (1 − (1 / 2))))
100		breldmg 4935	. . . 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 1376	. . 3 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ )
102	33, 38	eqeltrd 2306	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) ∈ ℂ)
103	59, 2, 102	iserex 11890	. . 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 12046	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 713 = wceq 1395 ∈ wcel 2200 Vcvv 2800 ⊆ wss 3198 {cpr 3668 class class class wbr 4086 ↦ cmpt 4148 dom cdm 4723 ⟶wf 5320 ‘cfv 5324 (class class class)co 6013 ℂcc 8020 ℝcr 8021 0cc0 8022 1c1 8023 + caddc 8025 · cmul 8027 < clt 8204 ≤ cle 8205 − cmin 8340 / cdiv 8842 ℕcn 9133 2c2 9184 ℕ₀cn0 9392 ℤ_≥cuz 9745 ℝ⁺crp 9878 seqcseq 10699 ↑cexp 10790 abscabs 11548 ⇝ cli 11829 Σcsu 11904

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 ax-pre-mulext 8140 ax-arch 8141 ax-caucvg 8142

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-oadd 6581 df-er 6697 df-en 6905 df-dom 6906 df-fin 6907 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752 df-div 8843 df-inn 9134 df-2 9192 df-3 9193 df-4 9194 df-n0 9393 df-z 9470 df-uz 9746 df-q 9844 df-rp 9879 df-ico 10119 df-fz 10234 df-fzo 10368 df-seqfrec 10700 df-exp 10791 df-ihash 11028 df-cj 11393 df-re 11394 df-im 11395 df-rsqrt 11549 df-abs 11550 df-clim 11830 df-sumdc 11905

This theorem is referenced by: trilpolemgt1 16579 trilpolemeq1 16580

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