Theorem mulgval 13252

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 12737	. . 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 13251	. . . 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 2205	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑛 = 0 ↔ 𝑁 = 0))
12	10	breq2d 4045	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0 < 𝑛 ↔ 0 < 𝑁))
13		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
14	13	sneqd 3635	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋})
15	14	xpeq2d 4687	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋}))
16	15	seqeq3d 10547	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑋})))
17		mulgval.s	. . . . . . . 8 ⊢ 𝑆 = seq1( + , (ℕ × {𝑋}))
18	16, 17	eqtr4di 2247	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = 𝑆)
19	18, 10	fveq12d 5565	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆‘𝑁))
20	10	negeqd 8221	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → -𝑛 = -𝑁)
21	18, 20	fveq12d 5565	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁))
22	21	fveq2d 5562	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) = (𝐼‘(𝑆‘-𝑁)))
23	12, 19, 22	ifbieq12d 3587	. . . . 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 3585	. . . 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 13018	. . . . . . 7 ⊢ 0g Fn V
29		funfvex 5575	. . . . . . . 8 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
30	29	funfni 5358	. . . . . . 7 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
31	28, 30	mpan 424	. . . . . 6 ⊢ (𝐺 ∈ V → (0g‘𝐺) ∈ V)
32	5, 31	eqeltrid 2283	. . . . 5 ⊢ (𝐺 ∈ V → 0 ∈ V)
33	32	ad2antlr 489	. . . 4 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ 𝑁 = 0) → 0 ∈ V)
34		nnuz 9637	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
35		1zzd 9353	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) → 1 ∈ ℤ)
36		fvconst2g 5776	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) = 𝑋)
37		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → 𝑋 ∈ 𝐵)
38	36, 37	eqeltrd 2273	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ 𝐵)
39	38	elexd 2776	. . . . . . . . . 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 12790	. . . . . . . . . . . . 13 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
43	42	slotex 12705	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ V → (+g‘𝐺) ∈ V)
44	4, 43	eqeltrid 2283	. . . . . . . . . . 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 5956	. . . . . . . . . 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 10556	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) → seq1( + , (ℕ × {𝑋})):ℕ⟶V)
50	17	feq1i 5400	. . . . . . . 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 9336	. . . . . . 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 5698	. . . . 5 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → (𝑆‘𝑁) ∈ V)
58	1, 6	grpinvfng 13176	. . . . . . . 8 ⊢ (𝐺 ∈ V → 𝐼 Fn 𝐵)
59		basfn 12736	. . . . . . . . . 10 ⊢ Base Fn V
60		funfvex 5575	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
61	60	funfni 5358	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
62	59, 61	mpan 424	. . . . . . . . 9 ⊢ (𝐺 ∈ V → (Base‘𝐺) ∈ V)
63	1, 62	eqeltrid 2283	. . . . . . . 8 ⊢ (𝐺 ∈ V → 𝐵 ∈ V)
64		fnex 5784	. . . . . . . 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 9357	. . . . . . . . 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 9368	. . . . . . . . . . 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 9330	. . . . . . . . . . 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 8542	. . . . . . . . 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 9336	. . . . . . . 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 5698	. . . . . 6 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑆‘-𝑁) ∈ V)
82		fvexg 5577	. . . . . 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 9338	. . . . . 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 9403	. . . . . 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 3590	. . . 4 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))) ∈ V)
89		0zd 9338	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → 0 ∈ ℤ)
90		zdceq 9401	. . . . 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 3590	. . 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 6050	. 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 2167  Vcvv 2763  ifcif 3561  {csn 3622   class class class wbr 4033   × cxp 4661   Fn wfn 5253  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922   ∈ cmpo 5924  ℝcr 7878  0cc0 7879  1c1 7880   < clt 8061  -cneg 8198  ℕcn 8990  ℤcz 9326  seqcseq 10539  Basecbs 12678  +_gcplusg 12755  0_gc0g 12927  inv_gcminusg 13133  ._gcmg 13249

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-2 9049  df-n0 9250  df-z 9327  df-uz 9602  df-seqfrec 10540  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-minusg 13136  df-mulg 13250

This theorem is referenced by:  mulg0  13255  mulgnn  13256  mulgnegnn  13262  subgmulg  13318