lgsfcl2 - Intuitionistic Logic Explorer

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Theorem lgsfcl2 13701

Description: The function 𝐹 is closed in integers with absolute value less than 1 (namely {-1, 0, 1}, see zabsle1 13694). (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 9223	. . . . . . . . 9 ⊢ 0 ∈ ℤ
2		0le1 8400	. . . . . . . . 9 ⊢ 0 ≤ 1
3		fveq2 5496	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (abs‘𝑥) = (abs‘0))
4		abs0 11022	. . . . . . . . . . . 12 ⊢ (abs‘0) = 0
5	3, 4	eqtrdi 2219	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (abs‘𝑥) = 0)
6	5	breq1d 3999	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ((abs‘𝑥) ≤ 1 ↔ 0 ≤ 1))
7		lgsfcl2.z	. . . . . . . . . 10 ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}
8	6, 7	elrab2 2889	. . . . . . . . 9 ⊢ (0 ∈ 𝑍 ↔ (0 ∈ ℤ ∧ 0 ≤ 1))
9	1, 2, 8	mpbir2an 937	. . . . . . . 8 ⊢ 0 ∈ 𝑍
10	9	a1i 9	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 0 ∈ 𝑍)
11		1z 9238	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
12		1le1 8491	. . . . . . . . . 10 ⊢ 1 ≤ 1
13		fveq2 5496	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
14		abs1 11036	. . . . . . . . . . . . 13 ⊢ (abs‘1) = 1
15	13, 14	eqtrdi 2219	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
16	15	breq1d 3999	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1))
17	16, 7	elrab2 2889	. . . . . . . . . 10 ⊢ (1 ∈ 𝑍 ↔ (1 ∈ ℤ ∧ 1 ≤ 1))
18	11, 12, 17	mpbir2an 937	. . . . . . . . 9 ⊢ 1 ∈ 𝑍
19	18	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 1 ∈ 𝑍)
20		neg1z 9244	. . . . . . . . . 10 ⊢ -1 ∈ ℤ
21		fveq2 5496	. . . . . . . . . . . . 13 ⊢ (𝑥 = -1 → (abs‘𝑥) = (abs‘-1))
22		ax-1cn 7867	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
23	22	absnegi 11111	. . . . . . . . . . . . . 14 ⊢ (abs‘-1) = (abs‘1)
24	23, 14	eqtri 2191	. . . . . . . . . . . . 13 ⊢ (abs‘-1) = 1
25	21, 24	eqtrdi 2219	. . . . . . . . . . . 12 ⊢ (𝑥 = -1 → (abs‘𝑥) = 1)
26	25	breq1d 3999	. . . . . . . . . . 11 ⊢ (𝑥 = -1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1))
27	26, 7	elrab2 2889	. . . . . . . . . 10 ⊢ (-1 ∈ 𝑍 ↔ (-1 ∈ ℤ ∧ 1 ≤ 1))
28	20, 12, 27	mpbir2an 937	. . . . . . . . 9 ⊢ -1 ∈ 𝑍
29	28	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → -1 ∈ 𝑍)
30		simp1 992	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℤ)
31		8nn 9045	. . . . . . . . . . . . . 14 ⊢ 8 ∈ ℕ
32	31	a1i 9	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 8 ∈ ℕ)
33	30, 32	zmodcld 10301	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 mod 8) ∈ ℕ₀)
34	33	nn0zd 9332	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 mod 8) ∈ ℤ)
35		zdceq 9287	. . . . . . . . . . 11 ⊢ (((𝐴 mod 8) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝐴 mod 8) = 1)
36	34, 11, 35	sylancl 411	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → DECID (𝐴 mod 8) = 1)
37		7nn 9044	. . . . . . . . . . . 12 ⊢ 7 ∈ ℕ
38	37	nnzi 9233	. . . . . . . . . . 11 ⊢ 7 ∈ ℤ
39		zdceq 9287	. . . . . . . . . . 11 ⊢ (((𝐴 mod 8) ∈ ℤ ∧ 7 ∈ ℤ) → DECID (𝐴 mod 8) = 7)
40	34, 38, 39	sylancl 411	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → DECID (𝐴 mod 8) = 7)
41		dcor 930	. . . . . . . . . 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 3603	. . . . . . . . . . 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 833	. . . . . . . . 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 166	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → DECID (𝐴 mod 8) ∈ {1, 7})
47	19, 29, 46	ifcldcd 3561	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → if((𝐴 mod 8) ∈ {1, 7}, 1, -1) ∈ 𝑍)
48		2nn 9039	. . . . . . . . 9 ⊢ 2 ∈ ℕ
49	48	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 2 ∈ ℕ)
50		dvdsdc 11760	. . . . . . . 8 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℤ) → DECID 2 ∥ 𝐴)
51	49, 30, 50	syl2anc 409	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → DECID 2 ∥ 𝐴)
52	10, 47, 51	ifcldcd 3561	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) ∈ 𝑍)
53	52	ad3antrrr 489	. . . . 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 995	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℤ)
55	54	ad2antrr 485	. . . . . 6 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → 𝐴 ∈ ℤ)
56		simplr 525	. . . . . . 7 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → 𝑛 ∈ ℙ)
57		simpr 109	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2)
58	57	neqned 2347	. . . . . . 7 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → 𝑛 ≠ 2)
59		eldifsn 3710	. . . . . . 7 ⊢ (𝑛 ∈ (ℙ ∖ {2}) ↔ (𝑛 ∈ ℙ ∧ 𝑛 ≠ 2))
60	56, 58, 59	sylanbrc 415	. . . . . 6 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → 𝑛 ∈ (ℙ ∖ {2}))
61	7	lgslem4 13698	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (ℙ ∖ {2})) → ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1) ∈ 𝑍)
62	55, 60, 61	syl2anc 409	. . . . 5 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1) ∈ 𝑍)
63		simplr 525	. . . . . . 7 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℕ)
64	63	nnzd 9333	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℤ)
65		2z 9240	. . . . . 6 ⊢ 2 ∈ ℤ
66		zdceq 9287	. . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 2 ∈ ℤ) → DECID 𝑛 = 2)
67	64, 65, 66	sylancl 411	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → DECID 𝑛 = 2)
68	53, 62, 67	ifcldadc 3555	. . . 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 109	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℙ)
70		simpll2 1032	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑁 ∈ ℤ)
71		simpll3 1033	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑁 ≠ 0)
72		pczcl 12252	. . . . 5 ⊢ ((𝑛 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑛 pCnt 𝑁) ∈ ℕ₀)
73	69, 70, 71, 72	syl12anc 1231	. . . 4 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑁) ∈ ℕ₀)
74	7	ssrab3 3233	. . . . . 6 ⊢ 𝑍 ⊆ ℤ
75		zsscn 9220	. . . . . 6 ⊢ ℤ ⊆ ℂ
76	74, 75	sstri 3156	. . . . 5 ⊢ 𝑍 ⊆ ℂ
77	7	lgslem3 13697	. . . . 5 ⊢ ((𝑎 ∈ 𝑍 ∧ 𝑏 ∈ 𝑍) → (𝑎 · 𝑏) ∈ 𝑍)
78	76, 77, 18	expcllem 10487	. . . 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 409	. . 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 109	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
82		prmdc 12084	. . . 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 3555	. 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 5650	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶𝑍)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 703 DECID wdc 829 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 {crab 2452 ∖ cdif 3118 ifcif 3526 {csn 3583 {cpr 3584 class class class wbr 3989 ↦ cmpt 4050 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 0cc0 7774 1c1 7775 + caddc 7777 ≤ cle 7955 − cmin 8090 -cneg 8091 / cdiv 8589 ℕcn 8878 2c2 8929 7c7 8934 8c8 8935 ℕ₀cn0 9135 ℤcz 9212 mod cmo 10278 ↑cexp 10475 abscabs 10961 ∥ cdvds 11749 ℙcprime 12061 pCnt cpc 12238

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-xor 1371 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-2o 6396 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-proddc 11514 df-dvds 11750 df-gcd 11898 df-prm 12062 df-phi 12165 df-pc 12239

This theorem is referenced by: lgscllem 13702 lgsfcl 13703 lgsfle1 13704

W3C validator