Theorem subgmulg 19173

Description: A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)

Hypotheses

Ref	Expression
subgmulgcl.t	⊢ · = (.g‘𝐺)
subgmulg.h	⊢ 𝐻 = (𝐺 ↾s 𝑆)
subgmulg.t	⊢ ∙ = (.g‘𝐻)

Assertion

Ref	Expression
subgmulg	⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) = (𝑁 ∙ 𝑋))

Proof of Theorem subgmulg

Step	Hyp	Ref	Expression
1		subgmulg.h	. . . . . 6 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
2		eqid 2761	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
3	1, 2	subg0 19165	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻))
4	3	3ad2ant1 1145	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (0g‘𝐺) = (0g‘𝐻))
5	4	ifeq1d 4497	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)))) = if(𝑁 = 0, (0g‘𝐻), if(0 < 𝑁, (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)))))
6		eqid 2761	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
7	1, 6	ressplusg 17311	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
8	7	3ad2ant1 1145	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (+g‘𝐺) = (+g‘𝐻))
9	8	seqeq2d 14015	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋})))
10	9	adantr 484	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ ¬ 𝑁 = 0) → seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋})))
11	10	fveq1d 6864	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ ¬ 𝑁 = 0) → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁) = (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁))
12	11	ifeq1d 4497	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁))) = if(0 < 𝑁, (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁))))
13		simp2 1149	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℤ)
14	13	zred 12671	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℝ)
15		0re 11177	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
16		axlttri 11248	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 < 0 ↔ ¬ (𝑁 = 0 ∨ 0 < 𝑁)))
17	14, 15, 16	sylancl 595	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 < 0 ↔ ¬ (𝑁 = 0 ∨ 0 < 𝑁)))
18		ioran 996	. . . . . . . . . . 11 ⊢ (¬ (𝑁 = 0 ∨ 0 < 𝑁) ↔ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁))
19	17, 18	bitrdi 289	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 < 0 ↔ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)))
20	19	biimpar 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → 𝑁 < 0)
21		simpl1 1204	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 𝑆 ∈ (SubGrp‘𝐺))
22	13	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 𝑁 ∈ ℤ)
23	22	znegcld 12673	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℤ)
24	14	lt0neg1d 11750	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 < 0 ↔ 0 < -𝑁))
25	24	biimpa 480	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 0 < -𝑁)
26		elnnz 12572	. . . . . . . . . . . . 13 ⊢ (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
27	23, 25, 26	sylanbrc 592	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℕ)
28		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐺) = (Base‘𝐺)
29	28	subgss 19160	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
30	29	3ad2ant1 1145	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐺))
31		simp3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
32	30, 31	sseldd 3935	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐺))
33	32	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 𝑋 ∈ (Base‘𝐺))
34		subgmulgcl.t	. . . . . . . . . . . . 13 ⊢ · = (.g‘𝐺)
35		eqid 2761	. . . . . . . . . . . . 13 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋}))
36	28, 6, 34, 35	mulgnn 19108	. . . . . . . . . . . 12 ⊢ ((-𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐺)) → (-𝑁 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁))
37	27, 33, 36	syl2anc 593	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁))
38	31	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 𝑋 ∈ 𝑆)
39	34	subgmulgcl 19172	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ -𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (-𝑁 · 𝑋) ∈ 𝑆)
40	21, 23, 38, 39	syl3anc 1389	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) ∈ 𝑆)
41	37, 40	eqeltrrd 2862	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁) ∈ 𝑆)
42		eqid 2761	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
43		eqid 2761	. . . . . . . . . . 11 ⊢ (invg‘𝐻) = (invg‘𝐻)
44	1, 42, 43	subginv 19166	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁) ∈ 𝑆) → ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg‘𝐻)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)))
45	21, 41, 44	syl2anc 593	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg‘𝐻)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)))
46	20, 45	syldan 600	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg‘𝐻)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)))
47	9	adantr 484	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋})))
48	47	fveq1d 6864	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁) = (seq1((+g‘𝐻), (ℕ × {𝑋}))‘-𝑁))
49	48	fveq2d 6866	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ((invg‘𝐻)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑋}))‘-𝑁)))
50	46, 49	eqtrd 2796	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑋}))‘-𝑁)))
51	50	anassrs 471	. . . . . 6 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑋}))‘-𝑁)))
52	51	ifeq2da 4510	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁))) = if(0 < 𝑁, (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑋}))‘-𝑁))))
53	12, 52	eqtrd 2796	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁))) = if(0 < 𝑁, (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑋}))‘-𝑁))))
54	53	ifeq2da 4510	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → if(𝑁 = 0, (0g‘𝐻), if(0 < 𝑁, (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)))) = if(𝑁 = 0, (0g‘𝐻), if(0 < 𝑁, (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑋}))‘-𝑁)))))
55	5, 54	eqtrd 2796	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)))) = if(𝑁 = 0, (0g‘𝐻), if(0 < 𝑁, (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑋}))‘-𝑁)))))
56	28, 6, 2, 42, 34, 35	mulgval 19104	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ (Base‘𝐺)) → (𝑁 · 𝑋) = if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)))))
57	13, 32, 56	syl2anc 593	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) = if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁)))))
58	1	subgbas 19163	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
59	58	3ad2ant1 1145	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑆 = (Base‘𝐻))
60	31, 59	eleqtrd 2863	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐻))
61		eqid 2761	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
62		eqid 2761	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
63		eqid 2761	. . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻)
64		subgmulg.t	. . . 4 ⊢ ∙ = (.g‘𝐻)
65		eqid 2761	. . . 4 ⊢ seq1((+g‘𝐻), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋}))
66	61, 62, 63, 43, 64, 65	mulgval 19104	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑁 ∙ 𝑋) = if(𝑁 = 0, (0g‘𝐻), if(0 < 𝑁, (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑋}))‘-𝑁)))))
67	13, 60, 66	syl2anc 593	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 ∙ 𝑋) = if(𝑁 = 0, (0g‘𝐻), if(0 < 𝑁, (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑋}))‘-𝑁)))))
68	55, 57, 67	3eqtr4d 2806	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) = (𝑁 ∙ 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ⊆ wss 3902  ifcif 4477  {csn 4579   class class class wbr 5097   × cxp 5641  ‘cfv 6516  (class class class)co 7391  ℝcr 11066  0cc0 11067  1c1 11068   < clt 11210  -cneg 11409  ℕcn 12204  ℤcz 12562  seqcseq 14008  Basecbs 17236   ↾_s cress 17257  +_gcplusg 17277  0_gc0g 17459  inv_gcminusg 18967  ._gcmg 19100  SubGrpcsubg 19153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-er 8672  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-n0 12476  df-z 12563  df-uz 12834  df-fz 13507  df-seq 14009  df-sets 17191  df-slot 17209  df-ndx 17221  df-base 17237  df-ress 17258  df-plusg 17290  df-0g 17461  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18969  df-minusg 18970  df-mulg 19101  df-subg 19156

This theorem is referenced by:  cycsubgcyg  19932  ablfac2  20122  zringmulg  21496  zringcyg  21509  remulg  21647  subgmulgcld  33184  rezh  34227