Theorem mulgfng 13927

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 2833	. . . . . . 7 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2		fn0g 13695	. . . . . . . 8 ⊢ 0g Fn V
3		funfvex 5712	. . . . . . . . 9 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
4	3	funfni 5483	. . . . . . . 8 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
5	2, 4	mpan 428	. . . . . . 7 ⊢ (𝐺 ∈ V → (0g‘𝐺) ∈ V)
6	1, 5	syl 14	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → (0g‘𝐺) ∈ V)
7	6	ad2antrr 492	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ 𝑛 = 0) → (0g‘𝐺) ∈ V)
8		nnuz 9958	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
9		1zzd 9671	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 1 ∈ ℤ)
10		fvconst2g 5929	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) = 𝑥)
11		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → 𝑥 ∈ 𝐵)
12	10, 11	eqeltrd 2315	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) ∈ 𝐵)
13	12	elexd 2835	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) ∈ V)
14	13	adantll 480	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) ∈ V)
15		simprl 535	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑢 ∈ V)
16		plusgslid 13466	. . . . . . . . . . . . 13 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
17	16	slotex 13379	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝑉 → (+g‘𝐺) ∈ V)
18	17	ad2antrr 492	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (+g‘𝐺) ∈ V)
19		simprr 537	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑣 ∈ V)
20		ovexg 6119	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ V ∧ (+g‘𝐺) ∈ V ∧ 𝑣 ∈ V) → (𝑢(+g‘𝐺)𝑣) ∈ V)
21	15, 18, 19, 20	syl3anc 1278	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢(+g‘𝐺)𝑣) ∈ V)
22	8, 9, 14, 21	seqf 10901	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → seq1((+g‘𝐺), (ℕ × {𝑥})):ℕ⟶V)
23	22	adantrl 482	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → seq1((+g‘𝐺), (ℕ × {𝑥})):ℕ⟶V)
24	23	ad2antrr 492	. . . . . . 7 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ 0 < 𝑛) → seq1((+g‘𝐺), (ℕ × {𝑥})):ℕ⟶V)
25		simprl 535	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → 𝑛 ∈ ℤ)
26	25	ad2antrr 492	. . . . . . . 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 9654	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℤ ∧ 0 < 𝑛))
29	26, 27, 28	sylanbrc 421	. . . . . . 7 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ 0 < 𝑛) → 𝑛 ∈ ℕ)
30	24, 29	ffvelcdmd 5844	. . . . . 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 2238	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
33	31, 32	grpinvfng 13849	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝑉 → (invg‘𝐺) Fn 𝐵)
34		basfn 13411	. . . . . . . . . . . 12 ⊢ Base Fn V
35		funfvex 5712	. . . . . . . . . . . . 13 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
36	35	funfni 5483	. . . . . . . . . . . 12 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
37	34, 36	mpan 428	. . . . . . . . . . 11 ⊢ (𝐺 ∈ V → (Base‘𝐺) ∈ V)
38	31, 37	eqeltrid 2325	. . . . . . . . . 10 ⊢ (𝐺 ∈ V → 𝐵 ∈ V)
39	1, 38	syl 14	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
40		fnex 5937	. . . . . . . . 9 ⊢ (((invg‘𝐺) Fn 𝐵 ∧ 𝐵 ∈ V) → (invg‘𝐺) ∈ V)
41	33, 39, 40	syl2anc 415	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑉 → (invg‘𝐺) ∈ V)
42	41	ad3antrrr 496	. . . . . . 7 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → (invg‘𝐺) ∈ V)
43	23	ad2antrr 492	. . . . . . . 8 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → seq1((+g‘𝐺), (ℕ × {𝑥})):ℕ⟶V)
44	25	znegcld 9770	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → -𝑛 ∈ ℤ)
45	44	ad2antrr 492	. . . . . . . . 9 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → -𝑛 ∈ ℤ)
46		simplr 533	. . . . . . . . . . 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 9686	. . . . . . . . . . . . 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 492	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → (𝑛 < 0 ∨ 𝑛 = 0 ∨ 0 < 𝑛))
51	46, 47, 50	ecase23d 1391	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → 𝑛 < 0)
52	25	zred 9768	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → 𝑛 ∈ ℝ)
53	52	ad2antrr 492	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → 𝑛 ∈ ℝ)
54	53	lt0neg1d 8843	. . . . . . . . . 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 9654	. . . . . . . . 9 ⊢ (-𝑛 ∈ ℕ ↔ (-𝑛 ∈ ℤ ∧ 0 < -𝑛))
57	45, 55, 56	sylanbrc 421	. . . . . . . 8 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → -𝑛 ∈ ℕ)
58	43, 57	ffvelcdmd 5844	. . . . . . 7 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → (seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛) ∈ V)
59		fvexg 5714	. . . . . . 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 415	. . . . . 6 ⊢ ((((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛)) ∈ V)
61		0zd 9656	. . . . . . 7 ⊢ (((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) → 0 ∈ ℤ)
62		simplrl 541	. . . . . . 7 ⊢ (((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℤ)
63		zdclt 9722	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑛 ∈ ℤ) → DECID 0 < 𝑛)
64	61, 62, 63	syl2anc 415	. . . . . 6 ⊢ (((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) ∧ ¬ 𝑛 = 0) → DECID 0 < 𝑛)
65	30, 60, 64	ifcldadc 3670	. . . . 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 9656	. . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → 0 ∈ ℤ)
67		zdceq 9720	. . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑛 = 0)
68	25, 66, 67	syl2anc 415	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵)) → DECID 𝑛 = 0)
69	7, 65, 68	ifcldadc 3670	. . . 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 2632	. . 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 2238	. . . 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 6438	. . 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 2238	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
75		eqid 2238	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
76		mulgfn.t	. . . 4 ⊢ · = (.g‘𝐺)
77	31, 74, 75, 32, 76	mulgfvalg 13924	. . 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 5471	. 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 846   ∨ w3o 1008   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821  ifcif 3638  {csn 3709   class class class wbr 4130   × cxp 4772   Fn wfn 5372  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085   ∈ cmpo 6087  ℝcr 8178  0cc0 8179  1c1 8180   < clt 8360  -cneg 8498  ℕcn 9304  ℤcz 9644  seqcseq 10884  Basecbs 13352  +_gcplusg 13431  0_gc0g 13610  inv_gcminusg 13806  ._gcmg 13922

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-seqfrec 10885  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-0g 13612  df-minusg 13809  df-mulg 13923

This theorem is used by: (None)