Theorem subgmulg 13905

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 2232	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
3	1, 2	subg0 13897	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻))
4	3	3ad2ant1 1045	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (0g‘𝐺) = (0g‘𝐻))
5	4	ifeq1d 3640	. . 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 2232	. . . . . . . . . . . 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 13896	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
11	6, 8, 9, 10	ressplusgd 13342	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
12	11	3ad2ant1 1045	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (+g‘𝐺) = (+g‘𝐻))
13	12	seqeq2d 10816	. . . . . . . 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 5672	. . . . . 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 3640	. . . . 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 531	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ¬ 𝑁 = 0)
18		simprr 533	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ¬ 0 < 𝑁)
19		simp2 1025	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℤ)
20		ztri3or0 9619	. . . . . . . . . . . 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 1387	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → 𝑁 < 0)
24		simpl1 1027	. . . . . . . . . 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 9702	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℤ)
27	19	zred 9700	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℝ)
28	27	lt0neg1d 8789	. . . . . . . . . . . . . 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 9587	. . . . . . . . . . . . 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 2232	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐺) = (Base‘𝐺)
33	32	subgss 13891	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
34	33	3ad2ant1 1045	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐺))
35		simp3 1026	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
36	34, 35	sseldd 3239	. . . . . . . . . . . . 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 2232	. . . . . . . . . . . . 13 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋}))
40	32, 7, 38, 39	mulgnn 13843	. . . . . . . . . . . 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 13904	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ -𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (-𝑁 · 𝑋) ∈ 𝑆)
44	24, 26, 42, 43	syl3anc 1274	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) ∈ 𝑆)
45	41, 44	eqeltrrd 2310	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 < 0) → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘-𝑁) ∈ 𝑆)
46		eqid 2232	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
47		eqid 2232	. . . . . . . . . . 11 ⊢ (invg‘𝐻) = (invg‘𝐻)
48	1, 46, 47	subginv 13898	. . . . . . . . . 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 5672	. . . . . . . . 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 5674	. . . . . . . 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 2265	. . . . . . 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 9588	. . . . . . 7 ⊢ 0 ∈ ℤ
57	19	adantr 276	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ ¬ 𝑁 = 0) → 𝑁 ∈ ℤ)
58		zdclt 9655	. . . . . . 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 3654	. . . . 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 2265	. . . 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 9589	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 0 ∈ ℤ)
63		zdceq 9653	. . . . 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 3654	. . 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 2265	. 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 13839	. . 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 13895	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
70	69	3ad2ant1 1045	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑆 = (Base‘𝐻))
71	35, 70	eleqtrd 2311	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐻))
72		eqid 2232	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
73		eqid 2232	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
74		eqid 2232	. . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻)
75		subgmulg.t	. . . 4 ⊢ ∙ = (.g‘𝐻)
76		eqid 2232	. . . 4 ⊢ seq1((+g‘𝐻), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋}))
77	72, 73, 74, 47, 75, 76	mulgval 13839	. . 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 2275	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) = (𝑁 ∙ 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 842   ∨ w3o 1004   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203   ⊆ wss 3211  ifcif 3620  {csn 3689   class class class wbr 4109   × cxp 4747  ‘cfv 5352  (class class class)co 6050  0cc0 8127  1c1 8128   < clt 8308  -cneg 8445  ℕcn 9237  ℤcz 9577  seqcseq 10809  Basecbs 13212   ↾_s cress 13213  +_gcplusg 13290  0_gc0g 13469  Grpcgrp 13713  inv_gcminusg 13714  ._gcmg 13836  SubGrpcsubg 13884

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854  df-seqfrec 10810  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-mulg 13837  df-subg 13887

This theorem is referenced by:  zringmulg  14746