Theorem mulgval 13927

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 13414	. . 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 13926	. . . 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 2247	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑛 = 0 ↔ 𝑁 = 0))
12	10	breq2d 4142	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0 < 𝑛 ↔ 0 < 𝑁))
13		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
14	13	sneqd 3722	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋})
15	14	xpeq2d 4798	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋}))
16	15	seqeq3d 10894	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑋})))
17		mulgval.s	. . . . . . . 8 ⊢ 𝑆 = seq1( + , (ℕ × {𝑋}))
18	16, 17	eqtr4di 2289	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = 𝑆)
19	18, 10	fveq12d 5702	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆‘𝑁))
20	10	negeqd 8521	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → -𝑛 = -𝑁)
21	18, 20	fveq12d 5702	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁))
22	21	fveq2d 5699	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) = (𝐼‘(𝑆‘-𝑁)))
23	12, 19, 22	ifbieq12d 3667	. . . . 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 3665	. . . 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 531	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → 𝑁 ∈ ℤ)
27		simplr 533	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → 𝑋 ∈ 𝐵)
28		fn0g 13697	. . . . . . 7 ⊢ 0g Fn V
29		funfvex 5712	. . . . . . . 8 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
30	29	funfni 5483	. . . . . . 7 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
31	28, 30	mpan 428	. . . . . 6 ⊢ (𝐺 ∈ V → (0g‘𝐺) ∈ V)
32	5, 31	eqeltrid 2325	. . . . 5 ⊢ (𝐺 ∈ V → 0 ∈ V)
33	32	ad2antlr 493	. . . 4 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ 𝑁 = 0) → 0 ∈ V)
34		nnuz 9960	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
35		1zzd 9673	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) → 1 ∈ ℤ)
36		fvconst2g 5929	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) = 𝑋)
37		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → 𝑋 ∈ 𝐵)
38	36, 37	eqeltrd 2315	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ 𝐵)
39	38	elexd 2835	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ V)
40	39	adantlr 481	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ V)
41		simprl 535	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑢 ∈ V)
42		plusgslid 13468	. . . . . . . . . . . . 13 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
43	42	slotex 13381	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ V → (+g‘𝐺) ∈ V)
44	4, 43	eqeltrid 2325	. . . . . . . . . . 11 ⊢ (𝐺 ∈ V → + ∈ V)
45	44	ad2antlr 493	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → + ∈ V)
46		simprr 537	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑣 ∈ V)
47		ovexg 6119	. . . . . . . . . 10 ⊢ ((𝑢 ∈ V ∧ + ∈ V ∧ 𝑣 ∈ V) → (𝑢 + 𝑣) ∈ V)
48	41, 45, 46, 47	syl3anc 1278	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢 + 𝑣) ∈ V)
49	34, 35, 40, 48	seqf 10903	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐺 ∈ V) → seq1( + , (ℕ × {𝑋})):ℕ⟶V)
50	17	feq1i 5526	. . . . . . . 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 526	. . . . . 6 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑆:ℕ⟶V)
53		simp-4l 547	. . . . . . 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 9656	. . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
56	53, 54, 55	sylanbrc 421	. . . . . 6 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑁 ∈ ℕ)
57	52, 56	ffvelcdmd 5844	. . . . 5 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → (𝑆‘𝑁) ∈ V)
58	1, 6	grpinvfng 13851	. . . . . . . 8 ⊢ (𝐺 ∈ V → 𝐼 Fn 𝐵)
59		basfn 13413	. . . . . . . . . 10 ⊢ Base Fn V
60		funfvex 5712	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
61	60	funfni 5483	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
62	59, 61	mpan 428	. . . . . . . . 9 ⊢ (𝐺 ∈ V → (Base‘𝐺) ∈ V)
63	1, 62	eqeltrid 2325	. . . . . . . 8 ⊢ (𝐺 ∈ V → 𝐵 ∈ V)
64		fnex 5937	. . . . . . . 8 ⊢ ((𝐼 Fn 𝐵 ∧ 𝐵 ∈ V) → 𝐼 ∈ V)
65	58, 63, 64	syl2anc 415	. . . . . . 7 ⊢ (𝐺 ∈ V → 𝐼 ∈ V)
66	65	ad3antlr 497	. . . . . 6 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝐼 ∈ V)
67	51	ad5ant23 526	. . . . . . 7 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑆:ℕ⟶V)
68		znegcl 9677	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
69	68	ad4antr 498	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℤ)
70		simplr 533	. . . . . . . . . 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 9688	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
73	72	ad4antr 498	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
74	70, 71, 73	ecase23d 1391	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 < 0)
75		zre 9650	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
76	75	ad4antr 498	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℝ)
77	76	lt0neg1d 8843	. . . . . . . . 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 9656	. . . . . . . 8 ⊢ (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
80	69, 78, 79	sylanbrc 421	. . . . . . 7 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℕ)
81	67, 80	ffvelcdmd 5844	. . . . . 6 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑆‘-𝑁) ∈ V)
82		fvexg 5714	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ (𝑆‘-𝑁) ∈ V) → (𝐼‘(𝑆‘-𝑁)) ∈ V)
83	66, 81, 82	syl2anc 415	. . . . 5 ⊢ (((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝐼‘(𝑆‘-𝑁)) ∈ V)
84		0zd 9658	. . . . . 6 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → 0 ∈ ℤ)
85		simplll 539	. . . . . 6 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → 𝑁 ∈ ℤ)
86		zdclt 9724	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 0 < 𝑁)
87	84, 85, 86	syl2anc 415	. . . . 5 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → DECID 0 < 𝑁)
88	57, 83, 87	ifcldadc 3670	. . . 4 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))) ∈ V)
89		0zd 9658	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → 0 ∈ ℤ)
90		zdceq 9722	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
91	26, 89, 90	syl2anc 415	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ 𝐺 ∈ V) → DECID 𝑁 = 0)
92	33, 88, 91	ifcldadc 3670	. . 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 6216	. 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 425	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 846   ∨ w3o 1008   = wceq 1402   ∈ wcel 2209  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 9305  ℤcz 9646  seqcseq 10886  Basecbs 13354  +_gcplusg 13433  0_gc0g 13612  inv_gcminusg 13808  ._gcmg 13924

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 9306  df-2 9364  df-n0 9566  df-z 9647  df-uz 9924  df-seqfrec 10887  df-ndx 13357  df-slot 13358  df-base 13360  df-plusg 13446  df-0g 13614  df-minusg 13811  df-mulg 13925

This theorem is used by:  mulg0  13930  mulgnn  13931  mulgnegnn  13937  subgmulg  13993