trilpolemisumle - Mathbox for Jim Kingdon

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Theorem trilpolemisumle 13406

Description: Lemma for trilpo 13411. 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 9196	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	1	eleq2i 2207	. . . . 5 ⊢ (𝑖 ∈ 𝑍 ↔ 𝑖 ∈ (ℤ_≥‘𝑀))
5	4	biimpi 119	. . . 4 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ (ℤ_≥‘𝑀))
6		eluznn 9421	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑖 ∈ (ℤ_≥‘𝑀)) → 𝑖 ∈ ℕ)
7	2, 5, 6	syl2an 287	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ ℕ)
8		eqid 2140	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))
9		oveq2 5790	. . . . . 6 ⊢ (𝑛 = 𝑖 → (2↑𝑛) = (2↑𝑖))
10	9	oveq2d 5798	. . . . 5 ⊢ (𝑛 = 𝑖 → (1 / (2↑𝑛)) = (1 / (2↑𝑖)))
11		fveq2 5429	. . . . 5 ⊢ (𝑛 = 𝑖 → (𝐹‘𝑛) = (𝐹‘𝑖))
12	10, 11	oveq12d 5800	. . . 4 ⊢ (𝑛 = 𝑖 → ((1 / (2↑𝑛)) · (𝐹‘𝑛)) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
13		simpr 109	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
14		2rp 9475	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
15	14	a1i 9	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 2 ∈ ℝ⁺)
16	13	nnzd 9196	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
17	15, 16	rpexpcld 10479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
18	17	rpreccld 9524	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ⁺)
19	18	rpred 9513	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
20		trilpolemgt1.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1})
21		0re 7790	. . . . . . . . 9 ⊢ 0 ∈ ℝ
22		1re 7789	. . . . . . . . 9 ⊢ 1 ∈ ℝ
23		prssi 3686	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ)
24	21, 22, 23	mp2an 423	. . . . . . . 8 ⊢ {0, 1} ⊆ ℝ
25	24	a1i 9	. . . . . . 7 ⊢ (𝜑 → {0, 1} ⊆ ℝ)
26	20, 25	fssd 5293	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
27	26	ffvelrnda 5563	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ)
28	19, 27	remulcld 7820	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
29	8, 12, 13, 28	fvmptd3 5522	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
30	7, 29	syldan 280	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
31	7, 28	syldan 280	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
32		eqid 2140	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛))) = (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))
33	32, 10, 13, 18	fvmptd3 5522	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) = (1 / (2↑𝑖)))
34	7, 33	syldan 280	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) = (1 / (2↑𝑖)))
35	7, 19	syldan 280	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (1 / (2↑𝑖)) ∈ ℝ)
36		simpr 109	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → (𝐹‘𝑖) = 0)
37	36	oveq2d 5798	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 0))
38	18	rpcnd 9515	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℂ)
39	38	adantr 274	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → (1 / (2↑𝑖)) ∈ ℂ)
40	39	mul01d 8179	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · 0) = 0)
41	37, 40	eqtrd 2173	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = 0)
42	18	adantr 274	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → (1 / (2↑𝑖)) ∈ ℝ⁺)
43	42	rpge0d 9517	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → 0 ≤ (1 / (2↑𝑖)))
44	41, 43	eqbrtrd 3958	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
45		simpr 109	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → (𝐹‘𝑖) = 1)
46	45	oveq2d 5798	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 1))
47	38	adantr 274	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ∈ ℂ)
48	47	mulid1d 7807	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · 1) = (1 / (2↑𝑖)))
49	46, 48	eqtrd 2173	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (1 / (2↑𝑖)))
50	19	adantr 274	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ∈ ℝ)
51	50	leidd 8300	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ≤ (1 / (2↑𝑖)))
52	49, 51	eqbrtrd 3958	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
53	20	ffvelrnda 5563	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ {0, 1})
54		elpri 3555	. . . . 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 788	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
57	7, 56	syldan 280	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
58	20, 8	trilpolemclim 13404	. . 3 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ )
59		nnuz 9385	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
60	29, 28	eqeltrd 2217	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) ∈ ℝ)
61	60	recnd 7818	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) ∈ ℂ)
62	59, 2, 61	iserex 11140	. . 3 ⊢ (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ ↔ seq𝑀( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ ))
63	58, 62	mpbid 146	. 2 ⊢ (𝜑 → seq𝑀( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ )
64		seqex 10251	. . . 4 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ V
65		rpreccl 9497	. . . . . . . 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 9105	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
69	67, 68	rpexpcld 10479	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑1) ∈ ℝ⁺)
70		1mhlfehlf 8962	. . . . . . 7 ⊢ (1 − (1 / 2)) = (1 / 2)
71	70, 66	eqeltri 2213	. . . . . 6 ⊢ (1 − (1 / 2)) ∈ ℝ⁺
72	71	a1i 9	. . . . 5 ⊢ (𝜑 → (1 − (1 / 2)) ∈ ℝ⁺)
73	69, 72	rpdivcld 9531	. . . 4 ⊢ (𝜑 → (((1 / 2)↑1) / (1 − (1 / 2))) ∈ ℝ⁺)
74		halfcn 8958	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
75	74	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
76		halfge0 8960	. . . . . . . 8 ⊢ 0 ≤ (1 / 2)
77		halfre 8957	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
78	77	absidi 10930	. . . . . . . 8 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2))
79	76, 78	ax-mp 5	. . . . . . 7 ⊢ (abs‘(1 / 2)) = (1 / 2)
80		halflt1 8961	. . . . . . 7 ⊢ (1 / 2) < 1
81	79, 80	eqbrtri 3957	. . . . . 6 ⊢ (abs‘(1 / 2)) < 1
82	81	a1i 9	. . . . 5 ⊢ (𝜑 → (abs‘(1 / 2)) < 1)
83		1nn0 9017	. . . . . 6 ⊢ 1 ∈ ℕ₀
84	83	a1i 9	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
85		oveq2 5790	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (2↑𝑛) = (2↑𝑗))
86	85	oveq2d 5798	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (1 / (2↑𝑛)) = (1 / (2↑𝑗)))
87		elnnuz 9386	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ_≥‘1))
88	87	biimpri 132	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ_≥‘1) → 𝑗 ∈ ℕ)
89	88	adantl 275	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 𝑗 ∈ ℕ)
90	14	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 2 ∈ ℝ⁺)
91	89	nnzd 9196	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 𝑗 ∈ ℤ)
92	90, 91	rpexpcld 10479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → (2↑𝑗) ∈ ℝ⁺)
93	92	rpreccld 9524	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → (1 / (2↑𝑗)) ∈ ℝ⁺)
94	32, 86, 89, 93	fvmptd3 5522	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑗) = (1 / (2↑𝑗)))
95		2cnd 8817	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 2 ∈ ℂ)
96	90	rpap0d 9519	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 2 # 0)
97	95, 96, 91	exprecapd 10463	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((1 / 2)↑𝑗) = (1 / (2↑𝑗)))
98	94, 97	eqtr4d 2176	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑗) = ((1 / 2)↑𝑗))
99	75, 82, 84, 98	geolim2 11313	. . . 4 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ⇝ (((1 / 2)↑1) / (1 − (1 / 2))))
100		breldmg 4753	. . . 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 1321	. . 3 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ )
102	33, 38	eqeltrd 2217	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) ∈ ℂ)
103	59, 2, 102	iserex 11140	. . 3 ⊢ (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ ↔ seq𝑀( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ ))
104	101, 103	mpbid 146	. 2 ⊢ (𝜑 → seq𝑀( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ )
105	1, 3, 30, 31, 34, 35, 57, 63, 104	isumle 11296	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 103 ∨ wo 698 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 {cpr 3533 class class class wbr 3937 ↦ cmpt 3997 dom cdm 4547 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 ℝcr 7643 0cc0 7644 1c1 7645 + caddc 7647 · cmul 7649 < clt 7824 ≤ cle 7825 − cmin 7957 / cdiv 8456 ℕcn 8744 2c2 8795 ℕ₀cn0 9001 ℤ_≥cuz 9350 ℝ⁺crp 9470 seqcseq 10249 ↑cexp 10323 abscabs 10801 ⇝ cli 11079 Σcsu 11154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-oadd 6325 df-er 6437 df-en 6643 df-dom 6644 df-fin 6645 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-ico 9707 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-ihash 10554 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 df-sumdc 11155

This theorem is referenced by: trilpolemgt1 13407 trilpolemeq1 13408

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