Theorem mulgfng 13710

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 2814	. . . . . . 7 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2		fn0g 13457	. . . . . . . 8 ⊢ 0g Fn V
3		funfvex 5656	. . . . . . . . 9 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
4	3	funfni 5432	. . . . . . . 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 9791	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
9		1zzd 9505	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 1 ∈ ℤ)
10		fvconst2g 5867	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) = 𝑥)
11		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → 𝑥 ∈ 𝐵)
12	10, 11	eqeltrd 2308	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) ∈ 𝐵)
13	12	elexd 2816	. . . . . . . . . . 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 531	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑢 ∈ V)
16		plusgslid 13194	. . . . . . . . . . . . 13 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
17	16	slotex 13108	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝑉 → (+g‘𝐺) ∈ V)
18	17	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (+g‘𝐺) ∈ V)
19		simprr 533	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑣 ∈ V)
20		ovexg 6051	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ V ∧ (+g‘𝐺) ∈ V ∧ 𝑣 ∈ V) → (𝑢(+g‘𝐺)𝑣) ∈ V)
21	15, 18, 19, 20	syl3anc 1273	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢(+g‘𝐺)𝑣) ∈ V)
22	8, 9, 14, 21	seqf 10725	. . . . . . . . 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 531	. . . . . . . . 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 9488	. . . . . . . 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 5783	. . . . . 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 2231	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
33	31, 32	grpinvfng 13626	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝑉 → (invg‘𝐺) Fn 𝐵)
34		basfn 13140	. . . . . . . . . . . 12 ⊢ Base Fn V
35		funfvex 5656	. . . . . . . . . . . . 13 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
36	35	funfni 5432	. . . . . . . . . . . 12 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
37	34, 36	mpan 424	. . . . . . . . . . 11 ⊢ (𝐺 ∈ V → (Base‘𝐺) ∈ V)
38	31, 37	eqeltrid 2318	. . . . . . . . . 10 ⊢ (𝐺 ∈ V → 𝐵 ∈ V)
39	1, 38	syl 14	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
40		fnex 5875	. . . . . . . . 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 9603	. . . . . . . . . 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 529	. . . . . . . . . . 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 9520	. . . . . . . . . . . . 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 1386	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → 𝑛 < 0)
52	25	zred 9601	. . . . . . . . . . . 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 8694	. . . . . . . . . 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 9488	. . . . . . . . 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 5783	. . . . . . 7 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → (seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛) ∈ V)
59		fvexg 5658	. . . . . . 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 9490	. . . . . . 7 ⊢ (((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) → 0 ∈ ℤ)
62		simplrl 537	. . . . . . 7 ⊢ (((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℤ)
63		zdclt 9556	. . . . . . 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 3635	. . . . 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 9490	. . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → 0 ∈ ℤ)
67		zdceq 9554	. . . . . 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 3635	. . . 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 2614	. . 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 2231	. . . 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 6366	. . 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 2231	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
75		eqid 2231	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
76		mulgfn.t	. . . 4 ⊢ · = (.g‘𝐺)
77	31, 74, 75, 32, 76	mulgfvalg 13707	. . 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 5420	. 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 841   ∨ w3o 1003   = wceq 1397   ∈ wcel 2202  ∀wral 2510  Vcvv 2802  ifcif 3605  {csn 3669   class class class wbr 4088   × cxp 4723   Fn wfn 5321  ⟶wf 5322  ‘cfv 5326  (class class class)co 6017   ∈ cmpo 6019  ℝcr 8030  0cc0 8031  1c1 8032   < clt 8213  -cneg 8350  ℕcn 9142  ℤcz 9478  seqcseq 10708  Basecbs 13081  +_gcplusg 13159  0_gc0g 13338  inv_gcminusg 13583  ._gcmg 13705

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-seqfrec 10709  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-minusg 13586  df-mulg 13706

This theorem is referenced by: (None)