lgsfcl2 - Intuitionistic Logic Explorer

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Theorem lgsfcl2 14703

Description: The function 𝐹 is closed in integers with absolute value less than 1 (namely {-1, 0, 1}, see zabsle1 14696). (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 9278	. . . . . . . . 9 ⊢ 0 ∈ ℤ
2		0le1 8452	. . . . . . . . 9 ⊢ 0 ≤ 1
3		fveq2 5527	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (abs‘𝑥) = (abs‘0))
4		abs0 11081	. . . . . . . . . . . 12 ⊢ (abs‘0) = 0
5	3, 4	eqtrdi 2236	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (abs‘𝑥) = 0)
6	5	breq1d 4025	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ((abs‘𝑥) ≤ 1 ↔ 0 ≤ 1))
7		lgsfcl2.z	. . . . . . . . . 10 ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}
8	6, 7	elrab2 2908	. . . . . . . . 9 ⊢ (0 ∈ 𝑍 ↔ (0 ∈ ℤ ∧ 0 ≤ 1))
9	1, 2, 8	mpbir2an 943	. . . . . . . 8 ⊢ 0 ∈ 𝑍
10	9	a1i 9	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 0 ∈ 𝑍)
11		1z 9293	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
12		1le1 8543	. . . . . . . . . 10 ⊢ 1 ≤ 1
13		fveq2 5527	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
14		abs1 11095	. . . . . . . . . . . . 13 ⊢ (abs‘1) = 1
15	13, 14	eqtrdi 2236	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
16	15	breq1d 4025	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1))
17	16, 7	elrab2 2908	. . . . . . . . . 10 ⊢ (1 ∈ 𝑍 ↔ (1 ∈ ℤ ∧ 1 ≤ 1))
18	11, 12, 17	mpbir2an 943	. . . . . . . . 9 ⊢ 1 ∈ 𝑍
19	18	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 1 ∈ 𝑍)
20		neg1z 9299	. . . . . . . . . 10 ⊢ -1 ∈ ℤ
21		fveq2 5527	. . . . . . . . . . . . 13 ⊢ (𝑥 = -1 → (abs‘𝑥) = (abs‘-1))
22		ax-1cn 7918	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
23	22	absnegi 11170	. . . . . . . . . . . . . 14 ⊢ (abs‘-1) = (abs‘1)
24	23, 14	eqtri 2208	. . . . . . . . . . . . 13 ⊢ (abs‘-1) = 1
25	21, 24	eqtrdi 2236	. . . . . . . . . . . 12 ⊢ (𝑥 = -1 → (abs‘𝑥) = 1)
26	25	breq1d 4025	. . . . . . . . . . 11 ⊢ (𝑥 = -1 → ((abs‘𝑥) ≤ 1 ↔ 1 ≤ 1))
27	26, 7	elrab2 2908	. . . . . . . . . 10 ⊢ (-1 ∈ 𝑍 ↔ (-1 ∈ ℤ ∧ 1 ≤ 1))
28	20, 12, 27	mpbir2an 943	. . . . . . . . 9 ⊢ -1 ∈ 𝑍
29	28	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → -1 ∈ 𝑍)
30		simp1 998	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℤ)
31		8nn 9100	. . . . . . . . . . . . . 14 ⊢ 8 ∈ ℕ
32	31	a1i 9	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 8 ∈ ℕ)
33	30, 32	zmodcld 10359	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 mod 8) ∈ ℕ₀)
34	33	nn0zd 9387	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 mod 8) ∈ ℤ)
35		zdceq 9342	. . . . . . . . . . 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 9099	. . . . . . . . . . . 12 ⊢ 7 ∈ ℕ
38	37	nnzi 9288	. . . . . . . . . . 11 ⊢ 7 ∈ ℤ
39		zdceq 9342	. . . . . . . . . . 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 936	. . . . . . . . . 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 3624	. . . . . . . . . . 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 839	. . . . . . . . 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 3582	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → if((𝐴 mod 8) ∈ {1, 7}, 1, -1) ∈ 𝑍)
48		2nn 9094	. . . . . . . . 9 ⊢ 2 ∈ ℕ
49	48	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 2 ∈ ℕ)
50		dvdsdc 11819	. . . . . . . 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 3582	. . . . . 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 1001	. . . . . . 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 2364	. . . . . . 7 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 = 2) → 𝑛 ≠ 2)
59		eldifsn 3731	. . . . . . 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 14700	. . . . . 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 9388	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℤ)
65		2z 9295	. . . . . 6 ⊢ 2 ∈ ℤ
66		zdceq 9342	. . . . . 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 3575	. . . 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 1038	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑁 ∈ ℤ)
71		simpll3 1039	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → 𝑁 ≠ 0)
72		pczcl 12312	. . . . 5 ⊢ ((𝑛 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑛 pCnt 𝑁) ∈ ℕ₀)
73	69, 70, 71, 72	syl12anc 1246	. . . 4 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑁) ∈ ℕ₀)
74	7	ssrab3 3253	. . . . . 6 ⊢ 𝑍 ⊆ ℤ
75		zsscn 9275	. . . . . 6 ⊢ ℤ ⊆ ℂ
76	74, 75	sstri 3176	. . . . 5 ⊢ 𝑍 ⊆ ℂ
77	7	lgslem3 14699	. . . . 5 ⊢ ((𝑎 ∈ 𝑍 ∧ 𝑏 ∈ 𝑍) → (𝑎 · 𝑏) ∈ 𝑍)
78	76, 77, 18	expcllem 10545	. . . 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 12144	. . . 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 3575	. 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 5683	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 709 DECID wdc 835 ∧ w3a 979 = wceq 1363 ∈ wcel 2158 ≠ wne 2357 {crab 2469 ∖ cdif 3138 ifcif 3546 {csn 3604 {cpr 3605 class class class wbr 4015 ↦ cmpt 4076 ⟶wf 5224 ‘cfv 5228 (class class class)co 5888 ℂcc 7823 0cc0 7825 1c1 7826 + caddc 7828 ≤ cle 8007 − cmin 8142 -cneg 8143 / cdiv 8643 ℕcn 8933 2c2 8984 7c7 8989 8c8 8990 ℕ₀cn0 9190 ℤcz 9267 mod cmo 10336 ↑cexp 10533 abscabs 11020 ∥ cdvds 11808 ℙcprime 12121 pCnt cpc 12298

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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-mulrcl 7924 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-precex 7935 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 ax-pre-mulgt0 7942 ax-pre-mulext 7943 ax-arch 7944 ax-caucvg 7945

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-xor 1386 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-recs 6320 df-irdg 6385 df-frec 6406 df-1o 6431 df-2o 6432 df-oadd 6435 df-er 6549 df-en 6755 df-dom 6756 df-fin 6757 df-sup 6997 df-inf 6998 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-reap 8546 df-ap 8553 df-div 8644 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-5 8995 df-6 8996 df-7 8997 df-8 8998 df-n0 9191 df-z 9268 df-uz 9543 df-q 9634 df-rp 9668 df-fz 10023 df-fzo 10157 df-fl 10284 df-mod 10337 df-seqfrec 10460 df-exp 10534 df-ihash 10770 df-cj 10865 df-re 10866 df-im 10867 df-rsqrt 11021 df-abs 11022 df-clim 11301 df-proddc 11573 df-dvds 11809 df-gcd 11958 df-prm 12122 df-phi 12225 df-pc 12299

This theorem is referenced by: lgscllem 14704 lgsfcl 14705 lgsfle1 14706

W3C validator