Theorem mulgval 13667

Description: Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)

Hypotheses

Ref	Expression
mulgval.b	⊢ 𝐵 = (Base‘𝐺)
mulgval.p	⊢ + = (+g‘𝐺)
mulgval.o	⊢ 0 = (0g‘𝐺)
mulgval.i	⊢ 𝐼 = (invg‘𝐺)
mulgval.t	⊢ · = (.g‘𝐺)
mulgval.s	⊢ 𝑆 = seq1( + , (ℕ × {𝑋}))

Assertion

Ref	Expression
mulgval	⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))))

Proof of Theorem mulgval

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

Step	Hyp	Ref	Expression
1		mulgval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2	1	basmex 13100	. . 3 ⊢ (𝑋 ∈ 𝐵 → 𝐺 ∈ V)
3	2	adantl 277	. 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝐺 ∈ V)
4		mulgval.p	. . . . 5 ⊢ + = (+g‘𝐺)
5		mulgval.o	. . . . 5 ⊢ 0 = (0g‘𝐺)
6		mulgval.i	. . . . 5 ⊢ 𝐼 = (invg‘𝐺)
7		mulgval.t	. . . . 5 ⊢ · = (.g‘𝐺)
8	1, 4, 5, 6, 7	mulgfvalg 13666	. . . 4 ⊢ (𝐺 ∈ V → · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
9	8	adantl 277	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
10		simpl 109	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑛 = 𝑁)
11	10	eqeq1d 2238	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑛 = 0 ↔ 𝑁 = 0))
12	10	breq2d 4095	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0 < 𝑛 ↔ 0 < 𝑁))
13		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
14	13	sneqd 3679	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋})
15	14	xpeq2d 4743	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋}))
16	15	seqeq3d 10685	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑋})))
17		mulgval.s	. . . . . . . 8 ⊢ 𝑆 = seq1( + , (ℕ × {𝑋}))
18	16, 17	eqtr4di 2280	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = 𝑆)
19	18, 10	fveq12d 5636	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆‘𝑁))
20	10	negeqd 8349	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → -𝑛 = -𝑁)
21	18, 20	fveq12d 5636	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁))
22	21	fveq2d 5633	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) = (𝐼‘(𝑆‘-𝑁)))
23	12, 19, 22	ifbieq12d 3629	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) = if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))))
24	11, 23	ifbieq2d 3627	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))))
25	24	adantl 277	. . 3 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ (𝑛 = 𝑁 ∧ 𝑥 = 𝑋)) → if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))))
26		simpll 527	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → 𝑁 ∈ ℤ)
27		simplr 528	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → 𝑋 ∈ 𝐵)
28		fn0g 13416	. . . . . . 7 ⊢ 0g Fn V
29		funfvex 5646	. . . . . . . 8 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
30	29	funfni 5423	. . . . . . 7 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
31	28, 30	mpan 424	. . . . . 6 ⊢ (𝐺 ∈ V → (0g‘𝐺) ∈ V)
32	5, 31	eqeltrid 2316	. . . . 5 ⊢ (𝐺 ∈ V → 0 ∈ V)
33	32	ad2antlr 489	. . . 4 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ 𝑁 = 0) → 0 ∈ V)
34		nnuz 9766	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
35		1zzd 9481	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) → 1 ∈ ℤ)
36		fvconst2g 5857	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) = 𝑋)
37		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → 𝑋 ∈ 𝐵)
38	36, 37	eqeltrd 2306	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ 𝐵)
39	38	elexd 2813	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ V)
40	39	adantlr 477	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ V)
41		simprl 529	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑢 ∈ V)
42		plusgslid 13153	. . . . . . . . . . . . 13 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
43	42	slotex 13067	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ V → (+g‘𝐺) ∈ V)
44	4, 43	eqeltrid 2316	. . . . . . . . . . 11 ⊢ (𝐺 ∈ V → + ∈ V)
45	44	ad2antlr 489	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → + ∈ V)
46		simprr 531	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑣 ∈ V)
47		ovexg 6041	. . . . . . . . . 10 ⊢ ((𝑢 ∈ V ∧ + ∈ V ∧ 𝑣 ∈ V) → (𝑢 + 𝑣) ∈ V)
48	41, 45, 46, 47	syl3anc 1271	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢 + 𝑣) ∈ V)
49	34, 35, 40, 48	seqf 10694	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) → seq1( + , (ℕ × {𝑋})):ℕ⟶V)
50	17	feq1i 5466	. . . . . . . 8 ⊢ (𝑆:ℕ⟶V ↔ seq1( + , (ℕ × {𝑋})):ℕ⟶V)
51	49, 50	sylibr 134	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) → 𝑆:ℕ⟶V)
52	51	ad5ant23 522	. . . . . 6 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑆:ℕ⟶V)
53		simp-4l 541	. . . . . . 7 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑁 ∈ ℤ)
54		simpr 110	. . . . . . 7 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 0 < 𝑁)
55		elnnz 9464	. . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
56	53, 54, 55	sylanbrc 417	. . . . . 6 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑁 ∈ ℕ)
57	52, 56	ffvelcdmd 5773	. . . . 5 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → (𝑆‘𝑁) ∈ V)
58	1, 6	grpinvfng 13585	. . . . . . . 8 ⊢ (𝐺 ∈ V → 𝐼 Fn 𝐵)
59		basfn 13099	. . . . . . . . . 10 ⊢ Base Fn V
60		funfvex 5646	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
61	60	funfni 5423	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
62	59, 61	mpan 424	. . . . . . . . 9 ⊢ (𝐺 ∈ V → (Base‘𝐺) ∈ V)
63	1, 62	eqeltrid 2316	. . . . . . . 8 ⊢ (𝐺 ∈ V → 𝐵 ∈ V)
64		fnex 5865	. . . . . . . 8 ⊢ ((𝐼 Fn 𝐵 ∧ 𝐵 ∈ V) → 𝐼 ∈ V)
65	58, 63, 64	syl2anc 411	. . . . . . 7 ⊢ (𝐺 ∈ V → 𝐼 ∈ V)
66	65	ad3antlr 493	. . . . . 6 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝐼 ∈ V)
67	51	ad5ant23 522	. . . . . . 7 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑆:ℕ⟶V)
68		znegcl 9485	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
69	68	ad4antr 494	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℤ)
70		simplr 528	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 𝑁 = 0)
71		simpr 110	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 < 𝑁)
72		ztri3or0 9496	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
73	72	ad4antr 494	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
74	70, 71, 73	ecase23d 1384	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 < 0)
75		zre 9458	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
76	75	ad4antr 494	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℝ)
77	76	lt0neg1d 8670	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ↔ 0 < -𝑁))
78	74, 77	mpbid 147	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 < -𝑁)
79		elnnz 9464	. . . . . . . 8 ⊢ (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
80	69, 78, 79	sylanbrc 417	. . . . . . 7 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℕ)
81	67, 80	ffvelcdmd 5773	. . . . . 6 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑆‘-𝑁) ∈ V)
82		fvexg 5648	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ (𝑆‘-𝑁) ∈ V) → (𝐼‘(𝑆‘-𝑁)) ∈ V)
83	66, 81, 82	syl2anc 411	. . . . 5 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝐼‘(𝑆‘-𝑁)) ∈ V)
84		0zd 9466	. . . . . 6 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → 0 ∈ ℤ)
85		simplll 533	. . . . . 6 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → 𝑁 ∈ ℤ)
86		zdclt 9532	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 0 < 𝑁)
87	84, 85, 86	syl2anc 411	. . . . 5 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → DECID 0 < 𝑁)
88	57, 83, 87	ifcldadc 3632	. . . 4 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))) ∈ V)
89		0zd 9466	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → 0 ∈ ℤ)
90		zdceq 9530	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
91	26, 89, 90	syl2anc 411	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → DECID 𝑁 = 0)
92	33, 88, 91	ifcldadc 3632	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))) ∈ V)
93	9, 25, 26, 27, 92	ovmpod 6138	. 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))))
94	3, 93	mpdan 421	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 839   ∨ w3o 1001   = wceq 1395   ∈ wcel 2200  Vcvv 2799  ifcif 3602  {csn 3666   class class class wbr 4083   × cxp 4717   Fn wfn 5313  ⟶wf 5314  ‘cfv 5318  (class class class)co 6007   ∈ cmpo 6009  ℝcr 8006  0cc0 8007  1c1 8008   < clt 8189  -cneg 8326  ℕcn 9118  ℤcz 9454  seqcseq 10677  Basecbs 13040  +_gcplusg 13118  0_gc0g 13297  inv_gcminusg 13542  ._gcmg 13664

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-inn 9119  df-2 9177  df-n0 9378  df-z 9455  df-uz 9731  df-seqfrec 10678  df-ndx 13043  df-slot 13044  df-base 13046  df-plusg 13131  df-0g 13299  df-minusg 13545  df-mulg 13665

This theorem is referenced by:  mulg0  13670  mulgnn  13671  mulgnegnn  13677  subgmulg  13733