Theorem mulgval 13680

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 13113	. . 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 13679	. . . 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 4744	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋}))
16	15	seqeq3d 10694	. . . . . . . 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 5639	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆‘𝑁))
20	10	negeqd 8357	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → -𝑛 = -𝑁)
21	18, 20	fveq12d 5639	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁))
22	21	fveq2d 5636	. . . . . 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 13429	. . . . . . 7 ⊢ 0g Fn V
29		funfvex 5649	. . . . . . . 8 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
30	29	funfni 5426	. . . . . . 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 9775	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
35		1zzd 9489	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) → 1 ∈ ℤ)
36		fvconst2g 5860	. . . . . . . . . . . 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 13166	. . . . . . . . . . . . 13 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
43	42	slotex 13080	. . . . . . . . . . . 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 6044	. . . . . . . . . 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 10703	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) → seq1( + , (ℕ × {𝑋})):ℕ⟶V)
50	17	feq1i 5469	. . . . . . . 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 9472	. . . . . . 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 5776	. . . . 5 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → (𝑆‘𝑁) ∈ V)
58	1, 6	grpinvfng 13598	. . . . . . . 8 ⊢ (𝐺 ∈ V → 𝐼 Fn 𝐵)
59		basfn 13112	. . . . . . . . . 10 ⊢ Base Fn V
60		funfvex 5649	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
61	60	funfni 5426	. . . . . . . . . 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 5868	. . . . . . . 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 9493	. . . . . . . . 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 9504	. . . . . . . . . . 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 9466	. . . . . . . . . . 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 8678	. . . . . . . . 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 9472	. . . . . . . 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 5776	. . . . . 6 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑆‘-𝑁) ∈ V)
82		fvexg 5651	. . . . . 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 9474	. . . . . 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 9540	. . . . . 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 9474	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → 0 ∈ ℤ)
90		zdceq 9538	. . . . 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 6141	. 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 4718   Fn wfn 5316  ⟶wf 5317  ‘cfv 5321  (class class class)co 6010   ∈ cmpo 6012  ℝcr 8014  0cc0 8015  1c1 8016   < clt 8197  -cneg 8334  ℕcn 9126  ℤcz 9462  seqcseq 10686  Basecbs 13053  +_gcplusg 13131  0_gc0g 13310  inv_gcminusg 13555  ._gcmg 13677

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 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-ltadd 8131

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 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-2 9185  df-n0 9386  df-z 9463  df-uz 9739  df-seqfrec 10687  df-ndx 13056  df-slot 13057  df-base 13059  df-plusg 13144  df-0g 13312  df-minusg 13558  df-mulg 13678

This theorem is referenced by:  mulg0  13683  mulgnn  13684  mulgnegnn  13690  subgmulg  13746