trilpolemisumle - Mathbox for Jim Kingdon

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Theorem trilpolemisumle 13231

Description: Lemma for trilpo 13236. 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 9172	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	1	eleq2i 2206	. . . . 5 ⊢ (𝑖 ∈ 𝑍 ↔ 𝑖 ∈ (ℤ_≥‘𝑀))
5	4	biimpi 119	. . . 4 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ (ℤ_≥‘𝑀))
6		eluznn 9394	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑖 ∈ (ℤ_≥‘𝑀)) → 𝑖 ∈ ℕ)
7	2, 5, 6	syl2an 287	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ ℕ)
8		eqid 2139	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))
9		oveq2 5782	. . . . . 6 ⊢ (𝑛 = 𝑖 → (2↑𝑛) = (2↑𝑖))
10	9	oveq2d 5790	. . . . 5 ⊢ (𝑛 = 𝑖 → (1 / (2↑𝑛)) = (1 / (2↑𝑖)))
11		fveq2 5421	. . . . 5 ⊢ (𝑛 = 𝑖 → (𝐹‘𝑛) = (𝐹‘𝑖))
12	10, 11	oveq12d 5792	. . . 4 ⊢ (𝑛 = 𝑖 → ((1 / (2↑𝑛)) · (𝐹‘𝑛)) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
13		simpr 109	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
14		2rp 9446	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
15	14	a1i 9	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 2 ∈ ℝ⁺)
16	13	nnzd 9172	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
17	15, 16	rpexpcld 10448	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
18	17	rpreccld 9494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ⁺)
19	18	rpred 9483	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
20		trilpolemgt1.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1})
21		0re 7766	. . . . . . . . 9 ⊢ 0 ∈ ℝ
22		1re 7765	. . . . . . . . 9 ⊢ 1 ∈ ℝ
23		prssi 3678	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ)
24	21, 22, 23	mp2an 422	. . . . . . . 8 ⊢ {0, 1} ⊆ ℝ
25	24	a1i 9	. . . . . . 7 ⊢ (𝜑 → {0, 1} ⊆ ℝ)
26	20, 25	fssd 5285	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
27	26	ffvelrnda 5555	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ)
28	19, 27	remulcld 7796	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
29	8, 12, 13, 28	fvmptd3 5514	. . 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 2139	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛))) = (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))
33	32, 10, 13, 18	fvmptd3 5514	. . 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 5790	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 0))
38	18	rpcnd 9485	. . . . . . . 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 8155	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · 0) = 0)
41	37, 40	eqtrd 2172	. . . . 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 9487	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 0) → 0 ≤ (1 / (2↑𝑖)))
44	41, 43	eqbrtrd 3950	. . . 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 5790	. . . . . 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 7783	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · 1) = (1 / (2↑𝑖)))
49	46, 48	eqtrd 2172	. . . . 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 8276	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ≤ (1 / (2↑𝑖)))
52	49, 51	eqbrtrd 3950	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
53	20	ffvelrnda 5555	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ {0, 1})
54		elpri 3550	. . . . 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 787	. . 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 13229	. . 3 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ )
59		nnuz 9361	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
60	29, 28	eqeltrd 2216	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) ∈ ℝ)
61	60	recnd 7794	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) ∈ ℂ)
62	59, 2, 61	iserex 11108	. . 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 10220	. . . 4 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ V
65		rpreccl 9468	. . . . . . . 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 9081	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
69	67, 68	rpexpcld 10448	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑1) ∈ ℝ⁺)
70		1mhlfehlf 8938	. . . . . . 7 ⊢ (1 − (1 / 2)) = (1 / 2)
71	70, 66	eqeltri 2212	. . . . . 6 ⊢ (1 − (1 / 2)) ∈ ℝ⁺
72	71	a1i 9	. . . . 5 ⊢ (𝜑 → (1 − (1 / 2)) ∈ ℝ⁺)
73	69, 72	rpdivcld 9501	. . . 4 ⊢ (𝜑 → (((1 / 2)↑1) / (1 − (1 / 2))) ∈ ℝ⁺)
74		halfcn 8934	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
75	74	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
76		halfge0 8936	. . . . . . . 8 ⊢ 0 ≤ (1 / 2)
77		halfre 8933	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
78	77	absidi 10898	. . . . . . . 8 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2))
79	76, 78	ax-mp 5	. . . . . . 7 ⊢ (abs‘(1 / 2)) = (1 / 2)
80		halflt1 8937	. . . . . . 7 ⊢ (1 / 2) < 1
81	79, 80	eqbrtri 3949	. . . . . 6 ⊢ (abs‘(1 / 2)) < 1
82	81	a1i 9	. . . . 5 ⊢ (𝜑 → (abs‘(1 / 2)) < 1)
83		1nn0 8993	. . . . . 6 ⊢ 1 ∈ ℕ₀
84	83	a1i 9	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
85		oveq2 5782	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (2↑𝑛) = (2↑𝑗))
86	85	oveq2d 5790	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (1 / (2↑𝑛)) = (1 / (2↑𝑗)))
87		elnnuz 9362	. . . . . . . . 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 9172	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 𝑗 ∈ ℤ)
92	90, 91	rpexpcld 10448	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → (2↑𝑗) ∈ ℝ⁺)
93	92	rpreccld 9494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → (1 / (2↑𝑗)) ∈ ℝ⁺)
94	32, 86, 89, 93	fvmptd3 5514	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑗) = (1 / (2↑𝑗)))
95		2cnd 8793	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 2 ∈ ℂ)
96	90	rpap0d 9489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → 2 # 0)
97	95, 96, 91	exprecapd 10432	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((1 / 2)↑𝑗) = (1 / (2↑𝑗)))
98	94, 97	eqtr4d 2175	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑗) = ((1 / 2)↑𝑗))
99	75, 82, 84, 98	geolim2 11281	. . . 4 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ⇝ (((1 / 2)↑1) / (1 − (1 / 2))))
100		breldmg 4745	. . . 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 1320	. . 3 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ )
102	33, 38	eqeltrd 2216	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) ∈ ℂ)
103	59, 2, 102	iserex 11108	. . 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 11264	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 697 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 {cpr 3528 class class class wbr 3929 ↦ cmpt 3989 dom cdm 4539 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 ℝcr 7619 0cc0 7620 1c1 7621 + caddc 7623 · cmul 7625 < clt 7800 ≤ cle 7801 − cmin 7933 / cdiv 8432 ℕcn 8720 2c2 8771 ℕ₀cn0 8977 ℤ_≥cuz 9326 ℝ⁺crp 9441 seqcseq 10218 ↑cexp 10292 abscabs 10769 ⇝ cli 11047 Σcsu 11122

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-ico 9677 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123

This theorem is referenced by: trilpolemgt1 13232 trilpolemeq1 13233

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