Theorem subgmulg 19061

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 2733	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
3	1, 2	subg0 19053	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻))
4	3	3ad2ant1 1133	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (0g‘𝐺) = (0g‘𝐻))
5	4	ifeq1d 4496	. . 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 2733	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
7	1, 6	ressplusg 17202	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
8	7	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (+g‘𝐺) = (+g‘𝐻))
9	8	seqeq2d 13922	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋})))
10	9	adantr 480	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ ¬ 𝑁 = 0) → seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋})))
11	10	fveq1d 6833	. . . . . 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 4496	. . . . 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 1137	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℤ)
14	13	zred 12587	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℝ)
15		0re 11125	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
16		axlttri 11195	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 < 0 ↔ ¬ (𝑁 = 0 ∨ 0 < 𝑁)))
17	14, 15, 16	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 < 0 ↔ ¬ (𝑁 = 0 ∨ 0 < 𝑁)))
18		ioran 985	. . . . . . . . . . 11 ⊢ (¬ (𝑁 = 0 ∨ 0 < 𝑁) ↔ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁))
19	17, 18	bitrdi 287	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 < 0 ↔ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)))
20	19	biimpar 477	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → 𝑁 < 0)
21		simpl1 1192	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 𝑆 ∈ (SubGrp‘𝐺))
22	13	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 𝑁 ∈ ℤ)
23	22	znegcld 12589	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℤ)
24	14	lt0neg1d 11697	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 < 0 ↔ 0 < -𝑁))
25	24	biimpa 476	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 0 < -𝑁)
26		elnnz 12489	. . . . . . . . . . . . 13 ⊢ (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
27	23, 25, 26	sylanbrc 583	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℕ)
28		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐺) = (Base‘𝐺)
29	28	subgss 19048	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
30	29	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐺))
31		simp3 1138	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
32	30, 31	sseldd 3931	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐺))
33	32	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 𝑋 ∈ (Base‘𝐺))
34		subgmulgcl.t	. . . . . . . . . . . . 13 ⊢ · = (.g‘𝐺)
35		eqid 2733	. . . . . . . . . . . . 13 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋}))
36	28, 6, 34, 35	mulgnn 18996	. . . . . . . . . . . 12 ⊢ ((-𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐺)) → (-𝑁 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁))
37	27, 33, 36	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁))
38	31	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 𝑋 ∈ 𝑆)
39	34	subgmulgcl 19060	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ -𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (-𝑁 · 𝑋) ∈ 𝑆)
40	21, 23, 38, 39	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) ∈ 𝑆)
41	37, 40	eqeltrrd 2834	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁) ∈ 𝑆)
42		eqid 2733	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
43		eqid 2733	. . . . . . . . . . 11 ⊢ (invg‘𝐻) = (invg‘𝐻)
44	1, 42, 43	subginv 19054	. . . . . . . . . 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 584	. . . . . . . . 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 591	. . . . . . . 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 480	. . . . . . . . . 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 6833	. . . . . . . . 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 6835	. . . . . . . 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 2768	. . . . . . 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 467	. . . . . 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 4509	. . . . 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 2768	. . . 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 4509	. . 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 2768	. 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 18992	. . 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 584	. 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 19051	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
59	58	3ad2ant1 1133	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑆 = (Base‘𝐻))
60	31, 59	eleqtrd 2835	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐻))
61		eqid 2733	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
62		eqid 2733	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
63		eqid 2733	. . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻)
64		subgmulg.t	. . . 4 ⊢ ∙ = (.g‘𝐻)
65		eqid 2733	. . . 4 ⊢ seq1((+g‘𝐻), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋}))
66	61, 62, 63, 43, 64, 65	mulgval 18992	. . 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 584	. 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 2778	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) = (𝑁 ∙ 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ⊆ wss 3898  ifcif 4476  {csn 4577   class class class wbr 5095   × cxp 5619  ‘cfv 6489  (class class class)co 7355  ℝcr 11016  0cc0 11017  1c1 11018   < clt 11157  -cneg 11356  ℕcn 12136  ℤcz 12479  seqcseq 13915  Basecbs 17127   ↾_s cress 17148  +_gcplusg 17168  0_gc0g 17350  inv_gcminusg 18855  ._gcmg 18988  SubGrpcsubg 19041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-er 8631  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-2 12199  df-n0 12393  df-z 12480  df-uz 12743  df-fz 13415  df-seq 13916  df-sets 17082  df-slot 17100  df-ndx 17112  df-base 17128  df-ress 17149  df-plusg 17181  df-0g 17352  df-mgm 18556  df-sgrp 18635  df-mnd 18651  df-grp 18857  df-minusg 18858  df-mulg 18989  df-subg 19044

This theorem is referenced by:  cycsubgcyg  19821  ablfac2  20011  zringmulg  21402  zringcyg  21415  remulg  21553  subgmulgcld  33054  rezh  34054