lgsfcl2 - Intuitionistic Logic Explorer

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Theorem lgsfcl2 14074

Description: The function 𝐹 is closed in integers with absolute value less than 1 (namely {-1, 0, 1}, see zabsle1 14067). (Contributed by Mario Carneiro, 4-Feb-2015.)

**Hypotheses**
Ref	Expression
lgsval.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))
lgsfcl2.z	⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}

**Assertion**
Ref	Expression
lgsfcl2	⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶𝑍)

Distinct variable groups: 𝐴,𝑛,𝑥 𝑥,𝐹 𝑛,𝑁,𝑥 𝑛,𝑍 Allowed substitution hints: 𝐹(𝑛) 𝑍(𝑥)

Proof of Theorem lgsfcl2 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0z 9253	. . . . . . . . 9 ⊢ 0 ∈ ℤ
2		0le1 8428	. . . . . . . . 9 ⊢ 0 ≤ 1
3		fveq2 5511	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (abs‘𝑥) = (abs‘0))
4		abs0 11051	. . . . . . . . . . . 12 ⊢ (abs‘0) = 0
5	3, 4	eqtrdi 2226	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (abs‘𝑥) = 0)
6	5	breq1d 4010	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ((abs‘𝑥) ≤ 1 ↔ 0 ≤ 1))
7		lgsfcl2.z	. . . . . . . . . 10 ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}
8	6, 7	elrab2 2896	. . . . . . . . 9 ⊢ (0 ∈ 𝑍 ↔ (0 ∈ ℤ ∧ 0 ≤ 1))
9	1, 2, 8	mpbir2an 942	. . . . . . . 8 ⊢ 0 ∈ 𝑍
10	9	a1i 9	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 0 ∈ 𝑍)
11		1z 9268	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
12		1le1 8519	. . . . . . . . . 10 ⊢ 1 ≤ 1
13		fveq2 5511	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
14		abs1 11065	. . . . . . . . . . . . 13 ⊢ (abs‘1) = 1
15	13, 14	eqtrdi 2226	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
16	15	breq1d 4010	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1))
17	16, 7	elrab2 2896	. . . . . . . . . 10 ⊢ (1 ∈ 𝑍 ↔ (1 ∈ ℤ ∧ 1 ≤ 1))
18	11, 12, 17	mpbir2an 942	. . . . . . . . 9 ⊢ 1 ∈ 𝑍
19	18	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 1 ∈ 𝑍)
20		neg1z 9274	. . . . . . . . . 10 ⊢ -1 ∈ ℤ
21		fveq2 5511	. . . . . . . . . . . . 13 ⊢ (𝑥 = -1 → (abs‘𝑥) = (abs‘-1))
22		ax-1cn 7895	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
23	22	absnegi 11140	. . . . . . . . . . . . . 14 ⊢ (abs‘-1) = (abs‘1)
24	23, 14	eqtri 2198	. . . . . . . . . . . . 13 ⊢ (abs‘-1) = 1
25	21, 24	eqtrdi 2226	. . . . . . . . . . . 12 ⊢ (𝑥 = -1 → (abs‘𝑥) = 1)
26	25	breq1d 4010	. . . . . . . . . . 11 ⊢ (𝑥 = -1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1))
27	26, 7	elrab2 2896	. . . . . . . . . 10 ⊢ (-1 ∈ 𝑍 ↔ (-1 ∈ ℤ ∧ 1 ≤ 1))
28	20, 12, 27	mpbir2an 942	. . . . . . . . 9 ⊢ -1 ∈ 𝑍
29	28	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → -1 ∈ 𝑍)
30		simp1 997	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℤ)
31		8nn 9075	. . . . . . . . . . . . . 14 ⊢ 8 ∈ ℕ
32	31	a1i 9	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 8 ∈ ℕ)
33	30, 32	zmodcld 10331	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 mod 8) ∈ ℕ₀)
34	33	nn0zd 9362	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 mod 8) ∈ ℤ)
35		zdceq 9317	. . . . . . . . . . 11 ⊢ (((𝐴 mod 8) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝐴 mod 8) = 1)
36	34, 11, 35	sylancl 413	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → DECID (𝐴 mod 8) = 1)
37		7nn 9074	. . . . . . . . . . . 12 ⊢ 7 ∈ ℕ
38	37	nnzi 9263	. . . . . . . . . . 11 ⊢ 7 ∈ ℤ
39		zdceq 9317	. . . . . . . . . . 11 ⊢ (((𝐴 mod 8) ∈ ℤ ∧ 7 ∈ ℤ) → DECID (𝐴 mod 8) = 7)
40	34, 38, 39	sylancl 413	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → DECID (𝐴 mod 8) = 7)
41		dcor 935	. . . . . . . . . 10 ⊢ (DECID (𝐴 mod 8) = 1 → (DECID (𝐴 mod 8) = 7 → DECID ((𝐴 mod 8) = 1 ∨ (𝐴 mod 8) = 7)))
42	36, 40, 41	sylc 62	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → DECID ((𝐴 mod 8) = 1 ∨ (𝐴 mod 8) = 7))
43		elprg 3611	. . . . . . . . . . 11 ⊢ ((𝐴 mod 8) ∈ ℕ₀ → ((𝐴 mod 8) ∈ {1, 7} ↔ ((𝐴 mod 8) = 1 ∨ (𝐴 mod 8) = 7)))
44	33, 43	syl 14	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → ((𝐴 mod 8) ∈ {1, 7} ↔ ((𝐴 mod 8) = 1 ∨ (𝐴 mod 8) = 7)))
45	44	dcbid 838	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (DECID (𝐴 mod 8) ∈ {1, 7} ↔ DECID ((𝐴 mod 8) = 1 ∨ (𝐴 mod 8) = 7)))
46	42, 45	mpbird 167	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → DECID (𝐴 mod 8) ∈ {1, 7})
47	19, 29, 46	ifcldcd 3569	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → if((𝐴 mod 8) ∈ {1, 7}, 1, -1) ∈ 𝑍)
48		2nn 9069	. . . . . . . . 9 ⊢ 2 ∈ ℕ
49	48	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 2 ∈ ℕ)
50		dvdsdc 11789	. . . . . . . 8 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℤ) → DECID 2 ∥ 𝐴)
51	49, 30, 50	syl2anc 411	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → DECID 2 ∥ 𝐴)
52	10, 47, 51	ifcldcd 3569	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) ∈ 𝑍)
53	52	ad3antrrr 492	. . . . 5 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 = 2) → if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) ∈ 𝑍)
54		simpl1 1000	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℤ)
55	54	ad2antrr 488	. . . . . 6 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → 𝐴 ∈ ℤ)
56		simplr 528	. . . . . . 7 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → 𝑛 ∈ ℙ)
57		simpr 110	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2)
58	57	neqned 2354	. . . . . . 7 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → 𝑛 ≠ 2)
59		eldifsn 3718	. . . . . . 7 ⊢ (𝑛 ∈ (ℙ ∖ {2}) ↔ (𝑛 ∈ ℙ ∧ 𝑛 ≠ 2))
60	56, 58, 59	sylanbrc 417	. . . . . 6 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → 𝑛 ∈ (ℙ ∖ {2}))
61	7	lgslem4 14071	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (ℙ ∖ {2})) → ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1) ∈ 𝑍)
62	55, 60, 61	syl2anc 411	. . . . 5 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1) ∈ 𝑍)
63		simplr 528	. . . . . . 7 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℕ)
64	63	nnzd 9363	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℤ)
65		2z 9270	. . . . . 6 ⊢ 2 ∈ ℤ
66		zdceq 9317	. . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 2 ∈ ℤ) → DECID 𝑛 = 2)
67	64, 65, 66	sylancl 413	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → DECID 𝑛 = 2)
68	53, 62, 67	ifcldadc 3563	. . . 4 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1)) ∈ 𝑍)
69		simpr 110	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℙ)
70		simpll2 1037	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑁 ∈ ℤ)
71		simpll3 1038	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑁 ≠ 0)
72		pczcl 12281	. . . . 5 ⊢ ((𝑛 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑛 pCnt 𝑁) ∈ ℕ₀)
73	69, 70, 71, 72	syl12anc 1236	. . . 4 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑁) ∈ ℕ₀)
74	7	ssrab3 3241	. . . . . 6 ⊢ 𝑍 ⊆ ℤ
75		zsscn 9250	. . . . . 6 ⊢ ℤ ⊆ ℂ
76	74, 75	sstri 3164	. . . . 5 ⊢ 𝑍 ⊆ ℂ
77	7	lgslem3 14070	. . . . 5 ⊢ ((𝑎 ∈ 𝑍 ∧ 𝑏 ∈ 𝑍) → (𝑎 · 𝑏) ∈ 𝑍)
78	76, 77, 18	expcllem 10517	. . . 4 ⊢ ((if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1)) ∈ 𝑍 ∧ (𝑛 pCnt 𝑁) ∈ ℕ₀) → (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)) ∈ 𝑍)
79	68, 73, 78	syl2anc 411	. . 3 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)) ∈ 𝑍)
80	18	a1i 9	. . 3 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ ℙ) → 1 ∈ 𝑍)
81		simpr 110	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
82		prmdc 12113	. . . 4 ⊢ (𝑛 ∈ ℕ → DECID 𝑛 ∈ ℙ)
83	81, 82	syl 14	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) → DECID 𝑛 ∈ ℙ)
84	79, 80, 83	ifcldadc 3563	. 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) → if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1) ∈ 𝑍)
85		lgsval.1	. 2 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))
86	84, 85	fmptd 5666	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶𝑍)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 DECID wdc 834 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 {crab 2459 ∖ cdif 3126 ifcif 3534 {csn 3591 {cpr 3592 class class class wbr 4000 ↦ cmpt 4061 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 ℂcc 7800 0cc0 7802 1c1 7803 + caddc 7805 ≤ cle 7983 − cmin 8118 -cneg 8119 / cdiv 8618 ℕcn 8908 2c2 8959 7c7 8964 8c8 8965 ℕ₀cn0 9165 ℤcz 9242 mod cmo 10308 ↑cexp 10505 abscabs 10990 ∥ cdvds 11778 ℙcprime 12090 pCnt cpc 12267

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-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-xor 1376 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-2o 6412 df-oadd 6415 df-er 6529 df-en 6735 df-dom 6736 df-fin 6737 df-sup 6977 df-inf 6978 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-5 8970 df-6 8971 df-7 8972 df-8 8973 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-fz 9996 df-fzo 10129 df-fl 10256 df-mod 10309 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-proddc 11543 df-dvds 11779 df-gcd 11927 df-prm 12091 df-phi 12194 df-pc 12268

This theorem is referenced by: lgscllem 14075 lgsfcl 14076 lgsfle1 14077

W3C validator