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Theorem gexdvds 19653

Description: The only 𝑁 that annihilate all the elements of the group are the multiples of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)

**Hypotheses**
Ref	Expression
gexcl.1	⊢ 𝑋 = (Base‘𝐺)
gexcl.2	⊢ 𝐸 = (gEx‘𝐺)
gexid.3	⊢ · = (._g‘𝐺)
gexid.4	⊢ 0 = (0_g‘𝐺)

**Assertion**
Ref	Expression
gexdvds	⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸 ∥ 𝑁 ↔ ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))

Distinct variable groups: 𝑥,𝐸 𝑥,𝐺 𝑥,𝑁 𝑥,𝑋 𝑥, 0 𝑥, ·

Proof of Theorem gexdvds Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gexcl.1	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
2		gexcl.2	. . . . . 6 ⊢ 𝐸 = (gEx‘𝐺)
3		gexid.3	. . . . . 6 ⊢ · = (._g‘𝐺)
4		gexid.4	. . . . . 6 ⊢ 0 = (0_g‘𝐺)
5	1, 2, 3, 4	gexdvdsi 19652	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐸 ∥ 𝑁) → (𝑁 · 𝑥) = 0 )
6	5	3expia 1137	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝐸 ∥ 𝑁 → (𝑁 · 𝑥) = 0 ))
7	6	ralrimdva 3163	. . 3 ⊢ (𝐺 ∈ Grp → (𝐸 ∥ 𝑁 → ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))
8	7	adantr 485	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸 ∥ 𝑁 → ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))
9		noel 4290	. . . . . . 7 ⊢ ¬ (abs‘𝑁) ∈ ∅
10		simprr 784	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)
11	10	eleq2d 2847	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → ((abs‘𝑁) ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } ↔ (abs‘𝑁) ∈ ∅))
12		oveq1 7417	. . . . . . . . . . . 12 ⊢ (𝑦 = (abs‘𝑁) → (𝑦 · 𝑥) = ((abs‘𝑁) · 𝑥))
13	12	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑦 = (abs‘𝑁) → ((𝑦 · 𝑥) = 0 ↔ ((abs‘𝑁) · 𝑥) = 0 ))
14	13	ralbidv 3186	. . . . . . . . . 10 ⊢ (𝑦 = (abs‘𝑁) → (∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 ))
15	14	elrab 3649	. . . . . . . . 9 ⊢ ((abs‘𝑁) ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } ↔ ((abs‘𝑁) ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 ))
16	11, 15	bitr3di 289	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → ((abs‘𝑁) ∈ ∅ ↔ ((abs‘𝑁) ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 )))
17	16	rbaibd 549	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) ∧ ∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 ) → ((abs‘𝑁) ∈ ∅ ↔ (abs‘𝑁) ∈ ℕ))
18	9, 17	mtbii 329	. . . . . 6 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) ∧ ∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 ) → ¬ (abs‘𝑁) ∈ ℕ)
19	18	ex 417	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 → ¬ (abs‘𝑁) ∈ ℕ))
20		nn0abscl 15362	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℕ₀)
21	20	ad2antlr 739	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (abs‘𝑁) ∈ ℕ₀)
22		elnn0 12505	. . . . . . 7 ⊢ ((abs‘𝑁) ∈ ℕ₀ ↔ ((abs‘𝑁) ∈ ℕ ∨ (abs‘𝑁) = 0))
23	21, 22	sylib 221	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → ((abs‘𝑁) ∈ ℕ ∨ (abs‘𝑁) = 0))
24	23	ord 877	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (¬ (abs‘𝑁) ∈ ℕ → (abs‘𝑁) = 0))
25	19, 24	syld 48	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 → (abs‘𝑁) = 0))
26		simpr 489	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = 𝑁) → (abs‘𝑁) = 𝑁)
27	26	oveq1d 7425	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = 𝑁) → ((abs‘𝑁) · 𝑥) = (𝑁 · 𝑥))
28	27	eqeq1d 2763	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = 𝑁) → (((abs‘𝑁) · 𝑥) = 0 ↔ (𝑁 · 𝑥) = 0 ))
29		oveq1 7417	. . . . . . . . 9 ⊢ ((abs‘𝑁) = -𝑁 → ((abs‘𝑁) · 𝑥) = (-𝑁 · 𝑥))
30	29	eqeq1d 2763	. . . . . . . 8 ⊢ ((abs‘𝑁) = -𝑁 → (((abs‘𝑁) · 𝑥) = 0 ↔ (-𝑁 · 𝑥) = 0 ))
31		eqid 2761	. . . . . . . . . . . 12 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
32	1, 3, 31	mulgneg 19157	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑥 ∈ 𝑋) → (-𝑁 · 𝑥) = ((inv_g‘𝐺)‘(𝑁 · 𝑥)))
33	32	3expa 1134	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → (-𝑁 · 𝑥) = ((inv_g‘𝐺)‘(𝑁 · 𝑥)))
34	4, 31	grpinvid 19065	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → ((inv_g‘𝐺)‘ 0 ) = 0 )
35	34	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → ((inv_g‘𝐺)‘ 0 ) = 0 )
36	35	eqcomd 2767	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 0 = ((inv_g‘𝐺)‘ 0 ))
37	33, 36	eqeq12d 2777	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → ((-𝑁 · 𝑥) = 0 ↔ ((inv_g‘𝐺)‘(𝑁 · 𝑥)) = ((inv_g‘𝐺)‘ 0 )))
38		simpll 778	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ Grp)
39	1, 3	mulgcl 19156	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑥 ∈ 𝑋) → (𝑁 · 𝑥) ∈ 𝑋)
40	39	3expa 1134	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → (𝑁 · 𝑥) ∈ 𝑋)
41	1, 4	grpidcl 19031	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
42	41	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 0 ∈ 𝑋)
43	1, 31, 38, 40, 42	grpinv11 19073	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → (((inv_g‘𝐺)‘(𝑁 · 𝑥)) = ((inv_g‘𝐺)‘ 0 ) ↔ (𝑁 · 𝑥) = 0 ))
44	37, 43	bitrd 282	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → ((-𝑁 · 𝑥) = 0 ↔ (𝑁 · 𝑥) = 0 ))
45	30, 44	sylan9bbr 519	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = -𝑁) → (((abs‘𝑁) · 𝑥) = 0 ↔ (𝑁 · 𝑥) = 0 ))
46		zre 12594	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
47	46	ad2antlr 739	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 𝑁 ∈ ℝ)
48	47	absord 15466	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑁) = 𝑁 ∨ (abs‘𝑁) = -𝑁))
49	28, 45, 48	mpjaodan 973	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → (((abs‘𝑁) · 𝑥) = 0 ↔ (𝑁 · 𝑥) = 0 ))
50	49	ralbidva 3184	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))
51	50	adantr 485	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))
52		0dvds 16333	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
53	52	ad2antlr 739	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (0 ∥ 𝑁 ↔ 𝑁 = 0))
54		simprl 782	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → 𝐸 = 0)
55	54	breq1d 5118	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (𝐸 ∥ 𝑁 ↔ 0 ∥ 𝑁))
56		zcn 12595	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
57	56	ad2antlr 739	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → 𝑁 ∈ ℂ)
58	57	abs00ad 15340	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → ((abs‘𝑁) = 0 ↔ 𝑁 = 0))
59	53, 55, 58	3bitr4rd 315	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → ((abs‘𝑁) = 0 ↔ 𝐸 ∥ 𝑁))
60	25, 51, 59	3imtr3d 296	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → 𝐸 ∥ 𝑁))
61		elrabi 3645	. . . 4 ⊢ (𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } → 𝐸 ∈ ℕ)
62	46	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
63		nnrp 13027	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℝ⁺)
64		modval 13903	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (𝑁 mod 𝐸) = (𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))))
65	62, 63, 64	syl2an 607	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) = (𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))))
66	65	adantr 485	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → (𝑁 mod 𝐸) = (𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))))
67	66	oveq1d 7425	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ((𝑁 mod 𝐸) · 𝑥) = ((𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))) · 𝑥))
68		simplll 786	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝐺 ∈ Grp)
69		simpllr 787	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝑁 ∈ ℤ)
70		nnz 12611	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℤ)
71	70	ad2antlr 739	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝐸 ∈ ℤ)
72		rerpdivcl 13047	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (𝑁 / 𝐸) ∈ ℝ)
73	62, 63, 72	syl2an 607	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 / 𝐸) ∈ ℝ)
74	73	flcld 13830	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (⌊‘(𝑁 / 𝐸)) ∈ ℤ)
75	74	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → (⌊‘(𝑁 / 𝐸)) ∈ ℤ)
76	71, 75	zmulcld 12705	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → (𝐸 · (⌊‘(𝑁 / 𝐸))) ∈ ℤ)
77		simprl 782	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝑥 ∈ 𝑋)
78		eqid 2761	. . . . . . . . . . 11 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
79	1, 3, 78	mulgsubdir 19179	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑁 ∈ ℤ ∧ (𝐸 · (⌊‘(𝑁 / 𝐸))) ∈ ℤ ∧ 𝑥 ∈ 𝑋)) → ((𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))) · 𝑥) = ((𝑁 · 𝑥)(-_g‘𝐺)((𝐸 · (⌊‘(𝑁 / 𝐸))) · 𝑥)))
80	68, 69, 76, 77, 79	syl13anc 1397	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ((𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))) · 𝑥) = ((𝑁 · 𝑥)(-_g‘𝐺)((𝐸 · (⌊‘(𝑁 / 𝐸))) · 𝑥)))
81		simprr 784	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → (𝑁 · 𝑥) = 0 )
82		dvdsmul1 16334	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℤ ∧ (⌊‘(𝑁 / 𝐸)) ∈ ℤ) → 𝐸 ∥ (𝐸 · (⌊‘(𝑁 / 𝐸))))
83	71, 75, 82	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝐸 ∥ (𝐸 · (⌊‘(𝑁 / 𝐸))))
84	1, 2, 3, 4	gexdvdsi 19652	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐸 ∥ (𝐸 · (⌊‘(𝑁 / 𝐸)))) → ((𝐸 · (⌊‘(𝑁 / 𝐸))) · 𝑥) = 0 )
85	68, 77, 83, 84	syl3anc 1396	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ((𝐸 · (⌊‘(𝑁 / 𝐸))) · 𝑥) = 0 )
86	81, 85	oveq12d 7428	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ((𝑁 · 𝑥)(-_g‘𝐺)((𝐸 · (⌊‘(𝑁 / 𝐸))) · 𝑥)) = ( 0 (-_g‘𝐺) 0 ))
87		simpll 778	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp)
88	1, 4, 78	grpsubid 19089	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ 𝑋) → ( 0 (-_g‘𝐺) 0 ) = 0 )
89	87, 41, 88	syl2anc2 596	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → ( 0 (-_g‘𝐺) 0 ) = 0 )
90	89	adantr 485	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ( 0 (-_g‘𝐺) 0 ) = 0 )
91	86, 90	eqtrd 2796	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ((𝑁 · 𝑥)(-_g‘𝐺)((𝐸 · (⌊‘(𝑁 / 𝐸))) · 𝑥)) = 0 )
92	67, 80, 91	3eqtrd 2800	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ((𝑁 mod 𝐸) · 𝑥) = 0 )
93	92	expr 461	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → ((𝑁 · 𝑥) = 0 → ((𝑁 mod 𝐸) · 𝑥) = 0 ))
94	93	ralimdva 3175	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ))
95		modlt 13912	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (𝑁 mod 𝐸) < 𝐸)
96	62, 63, 95	syl2an 607	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) < 𝐸)
97		zmodcl 13923	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) ∈ ℕ₀)
98	97	adantll 726	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) ∈ ℕ₀)
99	98	nn0red 12565	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) ∈ ℝ)
100		nnre 12239	. . . . . . . . . 10 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℝ)
101	100	adantl 486	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℝ)
102	99, 101	ltnled 11356	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → ((𝑁 mod 𝐸) < 𝐸 ↔ ¬ 𝐸 ≤ (𝑁 mod 𝐸)))
103	96, 102	mpbid 235	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → ¬ 𝐸 ≤ (𝑁 mod 𝐸))
104	1, 2, 3, 4	gexlem2 19651	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝑁 mod 𝐸) ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ) → 𝐸 ∈ (1...(𝑁 mod 𝐸)))
105		elfzle2 13555	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ (1...(𝑁 mod 𝐸)) → 𝐸 ≤ (𝑁 mod 𝐸))
106	104, 105	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝑁 mod 𝐸) ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ) → 𝐸 ≤ (𝑁 mod 𝐸))
107	106	3expia 1137	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑁 mod 𝐸) ∈ ℕ) → (∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 → 𝐸 ≤ (𝑁 mod 𝐸)))
108	107	impancom 456	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ) → ((𝑁 mod 𝐸) ∈ ℕ → 𝐸 ≤ (𝑁 mod 𝐸)))
109	108	con3d 153	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ) → (¬ 𝐸 ≤ (𝑁 mod 𝐸) → ¬ (𝑁 mod 𝐸) ∈ ℕ))
110	109	ex 417	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 → (¬ 𝐸 ≤ (𝑁 mod 𝐸) → ¬ (𝑁 mod 𝐸) ∈ ℕ)))
111	110	ad2antrr 738	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 → (¬ 𝐸 ≤ (𝑁 mod 𝐸) → ¬ (𝑁 mod 𝐸) ∈ ℕ)))
112	103, 111	mpid 45	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 → ¬ (𝑁 mod 𝐸) ∈ ℕ))
113		elnn0 12505	. . . . . . . 8 ⊢ ((𝑁 mod 𝐸) ∈ ℕ₀ ↔ ((𝑁 mod 𝐸) ∈ ℕ ∨ (𝑁 mod 𝐸) = 0))
114	98, 113	sylib 221	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → ((𝑁 mod 𝐸) ∈ ℕ ∨ (𝑁 mod 𝐸) = 0))
115	114	ord 877	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (¬ (𝑁 mod 𝐸) ∈ ℕ → (𝑁 mod 𝐸) = 0))
116	94, 112, 115	3syld 61	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → (𝑁 mod 𝐸) = 0))
117		simpr 489	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℕ)
118		simplr 780	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝑁 ∈ ℤ)
119		dvdsval3 16313	. . . . . 6 ⊢ ((𝐸 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝐸 ∥ 𝑁 ↔ (𝑁 mod 𝐸) = 0))
120	117, 118, 119	syl2anc 595	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝐸 ∥ 𝑁 ↔ (𝑁 mod 𝐸) = 0))
121	116, 120	sylibrd 262	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → 𝐸 ∥ 𝑁))
122	61, 121	sylan2 604	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → 𝐸 ∥ 𝑁))
123		eqid 2761	. . . . 5 ⊢ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }
124	1, 3, 4, 2, 123	gexlem1 19648	. . . 4 ⊢ (𝐺 ∈ Grp → ((𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅) ∨ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }))
125	124	adantr 485	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → ((𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅) ∨ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }))
126	60, 122, 125	mpjaodan 973	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → 𝐸 ∥ 𝑁))
127	8, 126	impbid 215	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸 ∥ 𝑁 ↔ ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ∀wral 3077 {crab 3414 ∅c0 4285 class class class wbr 5108 ‘cfv 6536 (class class class)co 7410 ℂcc 11097 ℝcr 11098 0cc0 11099 1c1 11100 · cmul 11104 < clt 11242 ≤ cle 11243 − cmin 11440 -cneg 11441 / cdiv 11870 ℕcn 12232 ℕ₀cn0 12503 ℤcz 12590 ℝ⁺crp 13015 ...cfz 13534 ⌊cfl 13822 mod cmo 13901 abscabs 15284 ∥ cdvds 16309 Basecbs 17268 0_gc0g 17491 Grpcgrp 18999 inv_gcminusg 19000 -_gcsg 19001 ._gcmg 19132 gExcgex 19594

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-sup 9401 df-inf 9402 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-n0 12504 df-z 12591 df-uz 12862 df-rp 13016 df-fz 13535 df-fl 13824 df-mod 13902 df-seq 14037 df-exp 14097 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-dvds 16310 df-0g 17493 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-grp 19002 df-minusg 19003 df-sbg 19004 df-mulg 19133 df-gex 19598

This theorem is referenced by: gexdvds2 19654

W3C validator