Theorem subgmulg 13466

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 2204	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
3	1, 2	subg0 13458	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻))
4	3	3ad2ant1 1020	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (0g‘𝐺) = (0g‘𝐻))
5	4	ifeq1d 3587	. . 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	1	a1i 9	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 = (𝐺 ↾s 𝑆))
7		eqid 2204	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
8	7	a1i 9	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐺))
9		id 19	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
10		subgrcl 13457	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
11	6, 8, 9, 10	ressplusgd 12903	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
12	11	3ad2ant1 1020	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (+g‘𝐺) = (+g‘𝐻))
13	12	seqeq2d 10597	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋})))
14	13	adantr 276	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ ¬ 𝑁 = 0) → seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋})))
15	14	fveq1d 5577	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ ¬ 𝑁 = 0) → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁) = (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁))
16	15	ifeq1d 3587	. . . . 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))))
17		simprl 529	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ¬ 𝑁 = 0)
18		simprr 531	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ¬ 0 < 𝑁)
19		simp2 1000	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℤ)
20		ztri3or0 9413	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
21	19, 20	syl 14	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
22	21	adantr 276	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
23	17, 18, 22	ecase23d 1362	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → 𝑁 < 0)
24		simpl1 1002	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 𝑆 ∈ (SubGrp‘𝐺))
25	19	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 𝑁 ∈ ℤ)
26	25	znegcld 9496	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℤ)
27	19	zred 9494	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℝ)
28	27	lt0neg1d 8587	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 < 0 ↔ 0 < -𝑁))
29	28	biimpa 296	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 0 < -𝑁)
30		elnnz 9381	. . . . . . . . . . . . 13 ⊢ (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
31	26, 29, 30	sylanbrc 417	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℕ)
32		eqid 2204	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐺) = (Base‘𝐺)
33	32	subgss 13452	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
34	33	3ad2ant1 1020	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐺))
35		simp3 1001	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
36	34, 35	sseldd 3193	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐺))
37	36	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 𝑋 ∈ (Base‘𝐺))
38		subgmulgcl.t	. . . . . . . . . . . . 13 ⊢ · = (.g‘𝐺)
39		eqid 2204	. . . . . . . . . . . . 13 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋}))
40	32, 7, 38, 39	mulgnn 13404	. . . . . . . . . . . 12 ⊢ ((-𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐺)) → (-𝑁 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁))
41	31, 37, 40	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁))
42	35	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → 𝑋 ∈ 𝑆)
43	38	subgmulgcl 13465	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ -𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (-𝑁 · 𝑋) ∈ 𝑆)
44	24, 26, 42, 43	syl3anc 1249	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) ∈ 𝑆)
45	41, 44	eqeltrrd 2282	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁) ∈ 𝑆)
46		eqid 2204	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
47		eqid 2204	. . . . . . . . . . 11 ⊢ (invg‘𝐻) = (invg‘𝐻)
48	1, 46, 47	subginv 13459	. . . . . . . . . 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)))
49	24, 45, 48	syl2anc 411	. . . . . . . . 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)))
50	23, 49	syldan 282	. . . . . . . 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)))
51	13	adantr 276	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋})))
52	51	fveq1d 5577	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁) = (seq1((+g‘𝐻), (ℕ × {𝑋}))‘-𝑁))
53	52	fveq2d 5579	. . . . . . . 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)))
54	50, 53	eqtrd 2237	. . . . . . 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)))
55	54	anassrs 400	. . . . . 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)))
56		0z 9382	. . . . . . 7 ⊢ 0 ∈ ℤ
57	19	adantr 276	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ ¬ 𝑁 = 0) → 𝑁 ∈ ℤ)
58		zdclt 9449	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 0 < 𝑁)
59	56, 57, 58	sylancr 414	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ ¬ 𝑁 = 0) → DECID 0 < 𝑁)
60	55, 59	ifeq2dadc 3601	. . . . 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))))
61	16, 60	eqtrd 2237	. . . 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))))
62		0zd 9383	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 0 ∈ ℤ)
63		zdceq 9447	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
64	19, 62, 63	syl2anc 411	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → DECID 𝑁 = 0)
65	61, 64	ifeq2dadc 3601	. . 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)))))
66	5, 65	eqtrd 2237	. 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)))))
67	32, 7, 2, 46, 38, 39	mulgval 13400	. . 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)))))
68	19, 36, 67	syl2anc 411	. 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)))))
69	1	subgbas 13456	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
70	69	3ad2ant1 1020	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑆 = (Base‘𝐻))
71	35, 70	eleqtrd 2283	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐻))
72		eqid 2204	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
73		eqid 2204	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
74		eqid 2204	. . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻)
75		subgmulg.t	. . . 4 ⊢ ∙ = (.g‘𝐻)
76		eqid 2204	. . . 4 ⊢ seq1((+g‘𝐻), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋}))
77	72, 73, 74, 47, 75, 76	mulgval 13400	. . 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)))))
78	19, 71, 77	syl2anc 411	. 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)))))
79	66, 68, 78	3eqtr4d 2247	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) = (𝑁 ∙ 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 835   ∨ w3o 979   ∧ w3a 980   = wceq 1372   ∈ wcel 2175   ⊆ wss 3165  ifcif 3570  {csn 3632   class class class wbr 4043   × cxp 4672  ‘cfv 5270  (class class class)co 5943  0cc0 7924  1c1 7925   < clt 8106  -cneg 8243  ℕcn 9035  ℤcz 9371  seqcseq 10590  Basecbs 12774   ↾_s cress 12775  +_gcplusg 12851  0_gc0g 13030  Grpcgrp 13274  inv_gcminusg 13275  ._gcmg 13397  SubGrpcsubg 13445

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-inn 9036  df-2 9094  df-n0 9295  df-z 9372  df-uz 9648  df-seqfrec 10591  df-ndx 12777  df-slot 12778  df-base 12780  df-sets 12781  df-iress 12782  df-plusg 12864  df-0g 13032  df-mgm 13130  df-sgrp 13176  df-mnd 13191  df-grp 13277  df-minusg 13278  df-mulg 13398  df-subg 13448

This theorem is referenced by:  zringmulg  14302