Theorem mulgfng 13535

Description: Functionality of the group multiple operation. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
mulgfn.b	⊢ 𝐵 = (Base‘𝐺)
mulgfn.t	⊢ · = (.g‘𝐺)

Assertion

Ref	Expression
mulgfng	⊢ (𝐺 ∈ 𝑉 → · Fn (ℤ × 𝐵))

Proof of Theorem mulgfng

Dummy variables 𝑢 𝑣 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2785	. . . . . . 7 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2		fn0g 13282	. . . . . . . 8 ⊢ 0g Fn V
3		funfvex 5606	. . . . . . . . 9 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
4	3	funfni 5385	. . . . . . . 8 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
5	2, 4	mpan 424	. . . . . . 7 ⊢ (𝐺 ∈ V → (0g‘𝐺) ∈ V)
6	1, 5	syl 14	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → (0g‘𝐺) ∈ V)
7	6	ad2antrr 488	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ 𝑛 = 0) → (0g‘𝐺) ∈ V)
8		nnuz 9704	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
9		1zzd 9419	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 1 ∈ ℤ)
10		fvconst2g 5811	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) = 𝑥)
11		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → 𝑥 ∈ 𝐵)
12	10, 11	eqeltrd 2283	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) ∈ 𝐵)
13	12	elexd 2787	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) ∈ V)
14	13	adantll 476	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) ∈ V)
15		simprl 529	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑢 ∈ V)
16		plusgslid 13019	. . . . . . . . . . . . 13 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
17	16	slotex 12934	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝑉 → (+g‘𝐺) ∈ V)
18	17	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (+g‘𝐺) ∈ V)
19		simprr 531	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑣 ∈ V)
20		ovexg 5991	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ V ∧ (+g‘𝐺) ∈ V ∧ 𝑣 ∈ V) → (𝑢(+g‘𝐺)𝑣) ∈ V)
21	15, 18, 19, 20	syl3anc 1250	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢(+g‘𝐺)𝑣) ∈ V)
22	8, 9, 14, 21	seqf 10631	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → seq1((+g‘𝐺), (ℕ × {𝑥})):ℕ⟶V)
23	22	adantrl 478	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → seq1((+g‘𝐺), (ℕ × {𝑥})):ℕ⟶V)
24	23	ad2antrr 488	. . . . . . 7 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ 0 < 𝑛) → seq1((+g‘𝐺), (ℕ × {𝑥})):ℕ⟶V)
25		simprl 529	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → 𝑛 ∈ ℤ)
26	25	ad2antrr 488	. . . . . . . 8 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ 0 < 𝑛) → 𝑛 ∈ ℤ)
27		simpr 110	. . . . . . . 8 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ 0 < 𝑛) → 0 < 𝑛)
28		elnnz 9402	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℤ ∧ 0 < 𝑛))
29	26, 27, 28	sylanbrc 417	. . . . . . 7 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ 0 < 𝑛) → 𝑛 ∈ ℕ)
30	24, 29	ffvelcdmd 5729	. . . . . 6 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ 0 < 𝑛) → (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛) ∈ V)
31		mulgfn.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
32		eqid 2206	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
33	31, 32	grpinvfng 13451	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝑉 → (invg‘𝐺) Fn 𝐵)
34		basfn 12965	. . . . . . . . . . . 12 ⊢ Base Fn V
35		funfvex 5606	. . . . . . . . . . . . 13 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
36	35	funfni 5385	. . . . . . . . . . . 12 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
37	34, 36	mpan 424	. . . . . . . . . . 11 ⊢ (𝐺 ∈ V → (Base‘𝐺) ∈ V)
38	31, 37	eqeltrid 2293	. . . . . . . . . 10 ⊢ (𝐺 ∈ V → 𝐵 ∈ V)
39	1, 38	syl 14	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
40		fnex 5819	. . . . . . . . 9 ⊢ (((invg‘𝐺) Fn 𝐵 ∧ 𝐵 ∈ V) → (invg‘𝐺) ∈ V)
41	33, 39, 40	syl2anc 411	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑉 → (invg‘𝐺) ∈ V)
42	41	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → (invg‘𝐺) ∈ V)
43	23	ad2antrr 488	. . . . . . . 8 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → seq1((+g‘𝐺), (ℕ × {𝑥})):ℕ⟶V)
44	25	znegcld 9517	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → -𝑛 ∈ ℤ)
45	44	ad2antrr 488	. . . . . . . . 9 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → -𝑛 ∈ ℤ)
46		simplr 528	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → ¬ 𝑛 = 0)
47		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → ¬ 0 < 𝑛)
48		ztri3or0 9434	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (𝑛 < 0 ∨ 𝑛 = 0 ∨ 0 < 𝑛))
49	25, 48	syl 14	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → (𝑛 < 0 ∨ 𝑛 = 0 ∨ 0 < 𝑛))
50	49	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → (𝑛 < 0 ∨ 𝑛 = 0 ∨ 0 < 𝑛))
51	46, 47, 50	ecase23d 1363	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → 𝑛 < 0)
52	25	zred 9515	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → 𝑛 ∈ ℝ)
53	52	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → 𝑛 ∈ ℝ)
54	53	lt0neg1d 8608	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → (𝑛 < 0 ↔ 0 < -𝑛))
55	51, 54	mpbid 147	. . . . . . . . 9 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → 0 < -𝑛)
56		elnnz 9402	. . . . . . . . 9 ⊢ (-𝑛 ∈ ℕ ↔ (-𝑛 ∈ ℤ ∧ 0 < -𝑛))
57	45, 55, 56	sylanbrc 417	. . . . . . . 8 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → -𝑛 ∈ ℕ)
58	43, 57	ffvelcdmd 5729	. . . . . . 7 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → (seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛) ∈ V)
59		fvexg 5608	. . . . . . 7 ⊢ (((invg‘𝐺) ∈ V ∧ (seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛) ∈ V) → ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛)) ∈ V)
60	42, 58, 59	syl2anc 411	. . . . . 6 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛)) ∈ V)
61		0zd 9404	. . . . . . 7 ⊢ (((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) → 0 ∈ ℤ)
62		simplrl 535	. . . . . . 7 ⊢ (((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℤ)
63		zdclt 9470	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑛 ∈ ℤ) → DECID 0 < 𝑛)
64	61, 62, 63	syl2anc 411	. . . . . 6 ⊢ (((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) → DECID 0 < 𝑛)
65	30, 60, 64	ifcldadc 3605	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) → if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛))) ∈ V)
66		0zd 9404	. . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → 0 ∈ ℤ)
67		zdceq 9468	. . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑛 = 0)
68	25, 66, 67	syl2anc 411	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → DECID 𝑛 = 0)
69	7, 65, 68	ifcldadc 3605	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → if(𝑛 = 0, (0g‘𝐺), if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛)))) ∈ V)
70	69	ralrimivva 2589	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∀𝑛 ∈ ℤ ∀𝑥 ∈ 𝐵 if(𝑛 = 0, (0g‘𝐺), if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛)))) ∈ V)
71		eqid 2206	. . . 4 ⊢ (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, (0g‘𝐺), if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛))))) = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, (0g‘𝐺), if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛)))))
72	71	fnmpo 6301	. . 3 ⊢ (∀𝑛 ∈ ℤ ∀𝑥 ∈ 𝐵 if(𝑛 = 0, (0g‘𝐺), if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛)))) ∈ V → (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, (0g‘𝐺), if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵))
73	70, 72	syl 14	. 2 ⊢ (𝐺 ∈ 𝑉 → (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, (0g‘𝐺), if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵))
74		eqid 2206	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
75		eqid 2206	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
76		mulgfn.t	. . . 4 ⊢ · = (.g‘𝐺)
77	31, 74, 75, 32, 76	mulgfvalg 13532	. . 3 ⊢ (𝐺 ∈ 𝑉 → · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, (0g‘𝐺), if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛))))))
78	77	fneq1d 5373	. 2 ⊢ (𝐺 ∈ 𝑉 → ( · Fn (ℤ × 𝐵) ↔ (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, (0g‘𝐺), if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵)))
79	73, 78	mpbird 167	1 ⊢ (𝐺 ∈ 𝑉 → · Fn (ℤ × 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 836   ∨ w3o 980   = wceq 1373   ∈ wcel 2177  ∀wral 2485  Vcvv 2773  ifcif 3575  {csn 3638   class class class wbr 4051   × cxp 4681   Fn wfn 5275  ⟶wf 5276  ‘cfv 5280  (class class class)co 5957   ∈ cmpo 5959  ℝcr 7944  0cc0 7945  1c1 7946   < clt 8127  -cneg 8264  ℕcn 9056  ℤcz 9392  seqcseq 10614  Basecbs 12907  +_gcplusg 12984  0_gc0g 13163  inv_gcminusg 13408  ._gcmg 13530

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-inn 9057  df-2 9115  df-n0 9316  df-z 9393  df-uz 9669  df-seqfrec 10615  df-ndx 12910  df-slot 12911  df-base 12913  df-plusg 12997  df-0g 13165  df-minusg 13411  df-mulg 13531

This theorem is referenced by: (None)