Theorem mulgval 13192

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 12677	. . 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 13191	. . . 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 2202	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑛 = 0 ↔ 𝑁 = 0))
12	10	breq2d 4041	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0 < 𝑛 ↔ 0 < 𝑁))
13		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
14	13	sneqd 3631	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋})
15	14	xpeq2d 4683	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋}))
16	15	seqeq3d 10526	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑋})))
17		mulgval.s	. . . . . . . 8 ⊢ 𝑆 = seq1( + , (ℕ × {𝑋}))
18	16, 17	eqtr4di 2244	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = 𝑆)
19	18, 10	fveq12d 5561	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆‘𝑁))
20	10	negeqd 8214	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → -𝑛 = -𝑁)
21	18, 20	fveq12d 5561	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁))
22	21	fveq2d 5558	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) = (𝐼‘(𝑆‘-𝑁)))
23	12, 19, 22	ifbieq12d 3583	. . . . 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 3581	. . . 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 12958	. . . . . . 7 ⊢ 0g Fn V
29		funfvex 5571	. . . . . . . 8 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
30	29	funfni 5354	. . . . . . 7 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
31	28, 30	mpan 424	. . . . . 6 ⊢ (𝐺 ∈ V → (0g‘𝐺) ∈ V)
32	5, 31	eqeltrid 2280	. . . . 5 ⊢ (𝐺 ∈ V → 0 ∈ V)
33	32	ad2antlr 489	. . . 4 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ 𝑁 = 0) → 0 ∈ V)
34		nnuz 9628	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
35		1zzd 9344	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) → 1 ∈ ℤ)
36		fvconst2g 5772	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) = 𝑋)
37		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → 𝑋 ∈ 𝐵)
38	36, 37	eqeltrd 2270	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ 𝐵)
39	38	elexd 2773	. . . . . . . . . 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 12730	. . . . . . . . . . . . 13 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
43	42	slotex 12645	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ V → (+g‘𝐺) ∈ V)
44	4, 43	eqeltrid 2280	. . . . . . . . . . 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 5952	. . . . . . . . . 10 ⊢ ((𝑢 ∈ V ∧ + ∈ V ∧ 𝑣 ∈ V) → (𝑢 + 𝑣) ∈ V)
48	41, 45, 46, 47	syl3anc 1249	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢 + 𝑣) ∈ V)
49	34, 35, 40, 48	seqf 10535	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) → seq1( + , (ℕ × {𝑋})):ℕ⟶V)
50	17	feq1i 5396	. . . . . . . 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 9327	. . . . . . 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 5694	. . . . 5 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → (𝑆‘𝑁) ∈ V)
58	1, 6	grpinvfng 13116	. . . . . . . 8 ⊢ (𝐺 ∈ V → 𝐼 Fn 𝐵)
59		basfn 12676	. . . . . . . . . 10 ⊢ Base Fn V
60		funfvex 5571	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
61	60	funfni 5354	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
62	59, 61	mpan 424	. . . . . . . . 9 ⊢ (𝐺 ∈ V → (Base‘𝐺) ∈ V)
63	1, 62	eqeltrid 2280	. . . . . . . 8 ⊢ (𝐺 ∈ V → 𝐵 ∈ V)
64		fnex 5780	. . . . . . . 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 9348	. . . . . . . . 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 9359	. . . . . . . . . . 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 1361	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 < 0)
75		zre 9321	. . . . . . . . . . 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 8534	. . . . . . . . 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 9327	. . . . . . . 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 5694	. . . . . 6 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑆‘-𝑁) ∈ V)
82		fvexg 5573	. . . . . 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 9329	. . . . . 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 9394	. . . . . 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 3586	. . . 4 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))) ∈ V)
89		0zd 9329	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → 0 ∈ ℤ)
90		zdceq 9392	. . . . 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 3586	. . 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 6046	. 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 835   ∨ w3o 979   = wceq 1364   ∈ wcel 2164  Vcvv 2760  ifcif 3557  {csn 3618   class class class wbr 4029   × cxp 4657   Fn wfn 5249  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918   ∈ cmpo 5920  ℝcr 7871  0cc0 7872  1c1 7873   < clt 8054  -cneg 8191  ℕcn 8982  ℤcz 9317  seqcseq 10518  Basecbs 12618  +_gcplusg 12695  0_gc0g 12867  inv_gcminusg 13073  ._gcmg 13189

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-minusg 13076  df-mulg 13190

This theorem is referenced by:  mulg0  13195  mulgnn  13196  mulgnegnn  13202  subgmulg  13258