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Theorem gexdvds 19799

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 19798	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐸 ∥ 𝑁) → (𝑁 · 𝑥) = 0 )
6	5	3expia 1139	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝐸 ∥ 𝑁 → (𝑁 · 𝑥) = 0 ))
7	6	ralrimdva 3162	. . 3 ⊢ (𝐺 ∈ Grp → (𝐸 ∥ 𝑁 → ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))
8	7	adantr 486	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸 ∥ 𝑁 → ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))
9		noel 4283	. . . . . . 7 ⊢ ¬ (abs‘𝑁) ∈ ∅
10		simprr 785	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)
11	10	eleq2d 2846	. . . . . . . . 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 7427	. . . . . . . . . . . 12 ⊢ (𝑦 = (abs‘𝑁) → (𝑦 · 𝑥) = ((abs‘𝑁) · 𝑥))
13	12	eqeq1d 2762	. . . . . . . . . . 11 ⊢ (𝑦 = (abs‘𝑁) → ((𝑦 · 𝑥) = 0 ↔ ((abs‘𝑁) · 𝑥) = 0 ))
14	13	ralbidv 3185	. . . . . . . . . 10 ⊢ (𝑦 = (abs‘𝑁) → (∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 ))
15	14	elrab 3644	. . . . . . . . 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 550	. . . . . . 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 418	. . . . 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 15480	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℕ₀)
21	20	ad2antlr 740	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (abs‘𝑁) ∈ ℕ₀)
22		elnn0 12609	. . . . . . 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 878	. . . . 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 490	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = 𝑁) → (abs‘𝑁) = 𝑁)
27	26	oveq1d 7435	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = 𝑁) → ((abs‘𝑁) · 𝑥) = (𝑁 · 𝑥))
28	27	eqeq1d 2762	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = 𝑁) → (((abs‘𝑁) · 𝑥) = 0 ↔ (𝑁 · 𝑥) = 0 ))
29		oveq1 7427	. . . . . . . . 9 ⊢ ((abs‘𝑁) = -𝑁 → ((abs‘𝑁) · 𝑥) = ( -𝑁 · 𝑥))
30	29	eqeq1d 2762	. . . . . . . 8 ⊢ ((abs‘𝑁) = -𝑁 → (((abs‘𝑁) · 𝑥) = 0 ↔ ( -𝑁 · 𝑥) = 0 ))
31		eqid 2760	. . . . . . . . . . . 12 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
32	1, 3, 31	mulgneg 19303	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑥 ∈ 𝑋) → ( -𝑁 · 𝑥) = ((inv_g‘𝐺)‘(𝑁 · 𝑥)))
33	32	3expa 1136	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → ( -𝑁 · 𝑥) = ((inv_g‘𝐺)‘(𝑁 · 𝑥)))
34	4, 31	grpinvid 19211	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → ((inv_g‘𝐺)‘ 0 ) = 0 )
35	34	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → ((inv_g‘𝐺)‘ 0 ) = 0 )
36	35	eqcomd 2766	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 0 = ((inv_g‘𝐺)‘ 0 ))
37	33, 36	eqeq12d 2776	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → (( -𝑁 · 𝑥) = 0 ↔ ((inv_g‘𝐺)‘(𝑁 · 𝑥)) = ((inv_g‘𝐺)‘ 0 )))
38		simpll 779	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ Grp)
39	1, 3	mulgcl 19302	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑥 ∈ 𝑋) → (𝑁 · 𝑥) ∈ 𝑋)
40	39	3expa 1136	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → (𝑁 · 𝑥) ∈ 𝑋)
41	1, 4	grpidcl 19177	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
42	41	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 0 ∈ 𝑋)
43	1, 31, 38, 40, 42	grpinv11 19219	. . . . . . . . 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 520	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = -𝑁) → (((abs‘𝑁) · 𝑥) = 0 ↔ (𝑁 · 𝑥) = 0 ))
46		zre 12698	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
47	46	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 𝑁 ∈ ℝ)
48	47	absord 15584	. . . . . . 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 3183	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))
51	50	adantr 486	. . . 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 16447	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
53	52	ad2antlr 740	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (0 ∥ 𝑁 ↔ 𝑁 = 0))
54		simprl 783	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → 𝐸 = 0)
55	54	breq1d 5112	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (𝐸 ∥ 𝑁 ↔ 0 ∥ 𝑁))
56		zcn 12699	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
57	56	ad2antlr 740	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → 𝑁 ∈ ℂ)
58	57	abs00ad 15458	. . . . 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 3640	. . . 4 ⊢ (𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } → 𝐸 ∈ ℕ)
62	46	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
63		nnrp 13133	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℝ⁺)
64		modval 14012	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (𝑁 mod 𝐸) = (𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))))
65	62, 63, 64	syl2an 608	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) = (𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))))
66	65	adantr 486	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → (𝑁 mod 𝐸) = (𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))))
67	66	oveq1d 7435	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ((𝑁 mod 𝐸) · 𝑥) = ((𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))) · 𝑥))
68		simplll 787	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝐺 ∈ Grp)
69		simpllr 788	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝑁 ∈ ℤ)
70		nnz 12715	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℤ)
71	70	ad2antlr 740	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝐸 ∈ ℤ)
72		rerpdivcl 13153	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (𝑁 / 𝐸) ∈ ℝ)
73	62, 63, 72	syl2an 608	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 / 𝐸) ∈ ℝ)
74	73	flcld 13939	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (⌊‘(𝑁 / 𝐸)) ∈ ℤ)
75	74	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → (⌊‘(𝑁 / 𝐸)) ∈ ℤ)
76	71, 75	zmulcld 12810	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → (𝐸 · (⌊‘(𝑁 / 𝐸))) ∈ ℤ)
77		simprl 783	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝑥 ∈ 𝑋)
78		eqid 2760	. . . . . . . . . . 11 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
79	1, 3, 78	mulgsubdir 19325	. . . . . . . . . 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 1399	. . . . . . . . 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 785	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → (𝑁 · 𝑥) = 0 )
82		dvdsmul1 16448	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℤ ∧ (⌊‘(𝑁 / 𝐸)) ∈ ℤ) → 𝐸 ∥ (𝐸 · (⌊‘(𝑁 / 𝐸))))
83	71, 75, 82	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝐸 ∥ (𝐸 · (⌊‘(𝑁 / 𝐸))))
84	1, 2, 3, 4	gexdvdsi 19798	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐸 ∥ (𝐸 · (⌊‘(𝑁 / 𝐸)))) → ((𝐸 · (⌊‘(𝑁 / 𝐸))) · 𝑥) = 0 )
85	68, 77, 83, 84	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ((𝐸 · (⌊‘(𝑁 / 𝐸))) · 𝑥) = 0 )
86	81, 85	oveq12d 7438	. . . . . . . . . 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 779	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp)
88	1, 4, 78	grpsubid 19235	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ 𝑋) → ( 0 (-_g‘𝐺) 0 ) = 0 )
89	87, 41, 88	syl2anc2 597	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → ( 0 (-_g‘𝐺) 0 ) = 0 )
90	89	adantr 486	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ( 0 (-_g‘𝐺) 0 ) = 0 )
91	86, 90	eqtrd 2795	. . . . . . . . 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 2799	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ((𝑁 mod 𝐸) · 𝑥) = 0 )
93	92	expr 462	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → ((𝑁 · 𝑥) = 0 → ((𝑁 mod 𝐸) · 𝑥) = 0 ))
94	93	ralimdva 3174	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ))
95		modlt 14021	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (𝑁 mod 𝐸) < 𝐸)
96	62, 63, 95	syl2an 608	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) < 𝐸)
97		zmodcl 14032	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) ∈ ℕ₀)
98	97	adantll 727	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) ∈ ℕ₀)
99	98	nn0red 12669	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) ∈ ℝ)
100		nnre 12343	. . . . . . . . . 10 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℝ)
101	100	adantl 487	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℝ)
102	99, 101	ltnled 11458	. . . . . . . 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 19797	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝑁 mod 𝐸) ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ) → 𝐸 ∈ (1...(𝑁 mod 𝐸)))
105		elfzle2 13662	. . . . . . . . . . . . 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 1139	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑁 mod 𝐸) ∈ ℕ) → (∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 → 𝐸 ≤ (𝑁 mod 𝐸)))
108	107	impancom 457	. . . . . . . . . 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 418	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 → (¬ 𝐸 ≤ (𝑁 mod 𝐸) → ¬ (𝑁 mod 𝐸) ∈ ℕ)))
111	110	ad2antrr 739	. . . . . . 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 12609	. . . . . . . 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 878	. . . . . 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 490	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℕ)
118		simplr 781	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝑁 ∈ ℤ)
119		dvdsval3 16427	. . . . . 6 ⊢ ((𝐸 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝐸 ∥ 𝑁 ↔ (𝑁 mod 𝐸) = 0))
120	117, 118, 119	syl2anc 596	. . . . 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 605	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → 𝐸 ∥ 𝑁))
123		eqid 2760	. . . . 5 ⊢ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }
124	1, 3, 4, 2, 123	gexlem1 19794	. . . 4 ⊢ (𝐺 ∈ Grp → ((𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅) ∨ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }))
125	124	adantr 486	. . 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 {crab 3412 ∅c0 4278 class class class wbr 5102 ‘cfv 6538 (class class class)co 7420 ℂcc 11199 ℝcr 11200 0cc0 11201 1c1 11202 · cmul 11206 < clt 11344 ≤ cle 11345 − cmin 11542 -cneg 11543 / cdiv 11974 ℕcn 12336 ℕ₀cn0 12607 ℤcz 12694 ℝ⁺crp 13121 ...cfz 13640 ⌊cfl 13931 mod cmo 14010 abscabs 15402 ∥ cdvds 16423 Basecbs 17388 0_gc0g 17611 Grpcgrp 19145 inv_gcminusg 19146 -_gcsg 19147 ._gcmg 19278 gExcgex 19740

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278 ax-pre-sup 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-er 8718 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9435 df-inf 9436 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-div 11975 df-nn 12337 df-2 12406 df-3 12407 df-n0 12608 df-z 12695 df-uz 12967 df-rp 13122 df-fz 13641 df-fl 13933 df-mod 14011 df-seq 14146 df-exp 14206 df-cj 15267 df-re 15268 df-im 15269 df-sqrt 15403 df-abs 15404 df-dvds 16424 df-0g 17613 df-mgm 18817 df-sgrp 18909 df-mnd 18925 df-grp 19148 df-minusg 19149 df-sbg 19150 df-mulg 19279 df-gex 19744

This theorem is used by: gexdvds2 19800

W3C validator