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Theorem gexdvds 19514

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 19513	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐸 ∥ 𝑁) → (𝑁 · 𝑥) = 0 )
6	5	3expia 1121	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝐸 ∥ 𝑁 → (𝑁 · 𝑥) = 0 ))
7	6	ralrimdva 3133	. . 3 ⊢ (𝐺 ∈ Grp → (𝐸 ∥ 𝑁 → ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))
8	7	adantr 480	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸 ∥ 𝑁 → ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))
9		noel 4301	. . . . . . 7 ⊢ ¬ (abs‘𝑁) ∈ ∅
10		simprr 772	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)
11	10	eleq2d 2814	. . . . . . . . 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 7394	. . . . . . . . . . . 12 ⊢ (𝑦 = (abs‘𝑁) → (𝑦 · 𝑥) = ((abs‘𝑁) · 𝑥))
13	12	eqeq1d 2731	. . . . . . . . . . 11 ⊢ (𝑦 = (abs‘𝑁) → ((𝑦 · 𝑥) = 0 ↔ ((abs‘𝑁) · 𝑥) = 0 ))
14	13	ralbidv 3156	. . . . . . . . . 10 ⊢ (𝑦 = (abs‘𝑁) → (∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 ))
15	14	elrab 3659	. . . . . . . . 9 ⊢ ((abs‘𝑁) ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } ↔ ((abs‘𝑁) ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 ))
16	11, 15	bitr3di 286	. . . . . . . 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 540	. . . . . . 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 326	. . . . . 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 412	. . . . 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 15278	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℕ₀)
21	20	ad2antlr 727	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (abs‘𝑁) ∈ ℕ₀)
22		elnn0 12444	. . . . . . 7 ⊢ ((abs‘𝑁) ∈ ℕ₀ ↔ ((abs‘𝑁) ∈ ℕ ∨ (abs‘𝑁) = 0))
23	21, 22	sylib 218	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → ((abs‘𝑁) ∈ ℕ ∨ (abs‘𝑁) = 0))
24	23	ord 864	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (¬ (abs‘𝑁) ∈ ℕ → (abs‘𝑁) = 0))
25	19, 24	syld 47	. . . 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 484	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = 𝑁) → (abs‘𝑁) = 𝑁)
27	26	oveq1d 7402	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = 𝑁) → ((abs‘𝑁) · 𝑥) = (𝑁 · 𝑥))
28	27	eqeq1d 2731	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = 𝑁) → (((abs‘𝑁) · 𝑥) = 0 ↔ (𝑁 · 𝑥) = 0 ))
29		oveq1 7394	. . . . . . . . 9 ⊢ ((abs‘𝑁) = -𝑁 → ((abs‘𝑁) · 𝑥) = (-𝑁 · 𝑥))
30	29	eqeq1d 2731	. . . . . . . 8 ⊢ ((abs‘𝑁) = -𝑁 → (((abs‘𝑁) · 𝑥) = 0 ↔ (-𝑁 · 𝑥) = 0 ))
31		eqid 2729	. . . . . . . . . . . 12 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
32	1, 3, 31	mulgneg 19024	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑥 ∈ 𝑋) → (-𝑁 · 𝑥) = ((inv_g‘𝐺)‘(𝑁 · 𝑥)))
33	32	3expa 1118	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → (-𝑁 · 𝑥) = ((inv_g‘𝐺)‘(𝑁 · 𝑥)))
34	4, 31	grpinvid 18931	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → ((inv_g‘𝐺)‘ 0 ) = 0 )
35	34	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → ((inv_g‘𝐺)‘ 0 ) = 0 )
36	35	eqcomd 2735	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 0 = ((inv_g‘𝐺)‘ 0 ))
37	33, 36	eqeq12d 2745	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → ((-𝑁 · 𝑥) = 0 ↔ ((inv_g‘𝐺)‘(𝑁 · 𝑥)) = ((inv_g‘𝐺)‘ 0 )))
38		simpll 766	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ Grp)
39	1, 3	mulgcl 19023	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑥 ∈ 𝑋) → (𝑁 · 𝑥) ∈ 𝑋)
40	39	3expa 1118	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → (𝑁 · 𝑥) ∈ 𝑋)
41	1, 4	grpidcl 18897	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
42	41	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 0 ∈ 𝑋)
43	1, 31, 38, 40, 42	grpinv11 18939	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → (((inv_g‘𝐺)‘(𝑁 · 𝑥)) = ((inv_g‘𝐺)‘ 0 ) ↔ (𝑁 · 𝑥) = 0 ))
44	37, 43	bitrd 279	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → ((-𝑁 · 𝑥) = 0 ↔ (𝑁 · 𝑥) = 0 ))
45	30, 44	sylan9bbr 510	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = -𝑁) → (((abs‘𝑁) · 𝑥) = 0 ↔ (𝑁 · 𝑥) = 0 ))
46		zre 12533	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
47	46	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 𝑁 ∈ ℝ)
48	47	absord 15382	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑁) = 𝑁 ∨ (abs‘𝑁) = -𝑁))
49	28, 45, 48	mpjaodan 960	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → (((abs‘𝑁) · 𝑥) = 0 ↔ (𝑁 · 𝑥) = 0 ))
50	49	ralbidva 3154	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))
51	50	adantr 480	. . . 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 16246	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
53	52	ad2antlr 727	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (0 ∥ 𝑁 ↔ 𝑁 = 0))
54		simprl 770	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → 𝐸 = 0)
55	54	breq1d 5117	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (𝐸 ∥ 𝑁 ↔ 0 ∥ 𝑁))
56		zcn 12534	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
57	56	ad2antlr 727	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → 𝑁 ∈ ℂ)
58	57	abs00ad 15256	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → ((abs‘𝑁) = 0 ↔ 𝑁 = 0))
59	53, 55, 58	3bitr4rd 312	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → ((abs‘𝑁) = 0 ↔ 𝐸 ∥ 𝑁))
60	25, 51, 59	3imtr3d 293	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → 𝐸 ∥ 𝑁))
61		elrabi 3654	. . . 4 ⊢ (𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } → 𝐸 ∈ ℕ)
62	46	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
63		nnrp 12963	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℝ⁺)
64		modval 13833	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (𝑁 mod 𝐸) = (𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))))
65	62, 63, 64	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) = (𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))))
66	65	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → (𝑁 mod 𝐸) = (𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))))
67	66	oveq1d 7402	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ((𝑁 mod 𝐸) · 𝑥) = ((𝑁 − (𝐸 · (⌊‘(𝑁 / 𝐸)))) · 𝑥))
68		simplll 774	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝐺 ∈ Grp)
69		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝑁 ∈ ℤ)
70		nnz 12550	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℤ)
71	70	ad2antlr 727	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝐸 ∈ ℤ)
72		rerpdivcl 12983	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (𝑁 / 𝐸) ∈ ℝ)
73	62, 63, 72	syl2an 596	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 / 𝐸) ∈ ℝ)
74	73	flcld 13760	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (⌊‘(𝑁 / 𝐸)) ∈ ℤ)
75	74	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → (⌊‘(𝑁 / 𝐸)) ∈ ℤ)
76	71, 75	zmulcld 12644	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → (𝐸 · (⌊‘(𝑁 / 𝐸))) ∈ ℤ)
77		simprl 770	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝑥 ∈ 𝑋)
78		eqid 2729	. . . . . . . . . . 11 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
79	1, 3, 78	mulgsubdir 19046	. . . . . . . . . 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 1374	. . . . . . . . 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 772	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → (𝑁 · 𝑥) = 0 )
82		dvdsmul1 16247	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℤ ∧ (⌊‘(𝑁 / 𝐸)) ∈ ℤ) → 𝐸 ∥ (𝐸 · (⌊‘(𝑁 / 𝐸))))
83	71, 75, 82	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → 𝐸 ∥ (𝐸 · (⌊‘(𝑁 / 𝐸))))
84	1, 2, 3, 4	gexdvdsi 19513	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐸 ∥ (𝐸 · (⌊‘(𝑁 / 𝐸)))) → ((𝐸 · (⌊‘(𝑁 / 𝐸))) · 𝑥) = 0 )
85	68, 77, 83, 84	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ((𝐸 · (⌊‘(𝑁 / 𝐸))) · 𝑥) = 0 )
86	81, 85	oveq12d 7405	. . . . . . . . . 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 766	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp)
88	1, 4, 78	grpsubid 18956	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ 𝑋) → ( 0 (-_g‘𝐺) 0 ) = 0 )
89	87, 41, 88	syl2anc2 585	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → ( 0 (-_g‘𝐺) 0 ) = 0 )
90	89	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ( 0 (-_g‘𝐺) 0 ) = 0 )
91	86, 90	eqtrd 2764	. . . . . . . . 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 2768	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑁 · 𝑥) = 0 )) → ((𝑁 mod 𝐸) · 𝑥) = 0 )
93	92	expr 456	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → ((𝑁 · 𝑥) = 0 → ((𝑁 mod 𝐸) · 𝑥) = 0 ))
94	93	ralimdva 3145	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ))
95		modlt 13842	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (𝑁 mod 𝐸) < 𝐸)
96	62, 63, 95	syl2an 596	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) < 𝐸)
97		zmodcl 13853	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) ∈ ℕ₀)
98	97	adantll 714	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) ∈ ℕ₀)
99	98	nn0red 12504	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) ∈ ℝ)
100		nnre 12193	. . . . . . . . . 10 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℝ)
101	100	adantl 481	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℝ)
102	99, 101	ltnled 11321	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → ((𝑁 mod 𝐸) < 𝐸 ↔ ¬ 𝐸 ≤ (𝑁 mod 𝐸)))
103	96, 102	mpbid 232	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → ¬ 𝐸 ≤ (𝑁 mod 𝐸))
104	1, 2, 3, 4	gexlem2 19512	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝑁 mod 𝐸) ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ) → 𝐸 ∈ (1...(𝑁 mod 𝐸)))
105		elfzle2 13489	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ (1...(𝑁 mod 𝐸)) → 𝐸 ≤ (𝑁 mod 𝐸))
106	104, 105	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝑁 mod 𝐸) ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ) → 𝐸 ≤ (𝑁 mod 𝐸))
107	106	3expia 1121	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑁 mod 𝐸) ∈ ℕ) → (∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 → 𝐸 ≤ (𝑁 mod 𝐸)))
108	107	impancom 451	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ) → ((𝑁 mod 𝐸) ∈ ℕ → 𝐸 ≤ (𝑁 mod 𝐸)))
109	108	con3d 152	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ) → (¬ 𝐸 ≤ (𝑁 mod 𝐸) → ¬ (𝑁 mod 𝐸) ∈ ℕ))
110	109	ex 412	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 → (¬ 𝐸 ≤ (𝑁 mod 𝐸) → ¬ (𝑁 mod 𝐸) ∈ ℕ)))
111	110	ad2antrr 726	. . . . . . 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 44	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 → ¬ (𝑁 mod 𝐸) ∈ ℕ))
113		elnn0 12444	. . . . . . . 8 ⊢ ((𝑁 mod 𝐸) ∈ ℕ₀ ↔ ((𝑁 mod 𝐸) ∈ ℕ ∨ (𝑁 mod 𝐸) = 0))
114	98, 113	sylib 218	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → ((𝑁 mod 𝐸) ∈ ℕ ∨ (𝑁 mod 𝐸) = 0))
115	114	ord 864	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (¬ (𝑁 mod 𝐸) ∈ ℕ → (𝑁 mod 𝐸) = 0))
116	94, 112, 115	3syld 60	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → (𝑁 mod 𝐸) = 0))
117		simpr 484	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℕ)
118		simplr 768	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝑁 ∈ ℤ)
119		dvdsval3 16226	. . . . . 6 ⊢ ((𝐸 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝐸 ∥ 𝑁 ↔ (𝑁 mod 𝐸) = 0))
120	117, 118, 119	syl2anc 584	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝐸 ∥ 𝑁 ↔ (𝑁 mod 𝐸) = 0))
121	116, 120	sylibrd 259	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → 𝐸 ∥ 𝑁))
122	61, 121	sylan2 593	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → 𝐸 ∥ 𝑁))
123		eqid 2729	. . . . 5 ⊢ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }
124	1, 3, 4, 2, 123	gexlem1 19509	. . . 4 ⊢ (𝐺 ∈ Grp → ((𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅) ∨ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }))
125	124	adantr 480	. . 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 960	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → 𝐸 ∥ 𝑁))
127	8, 126	impbid 212	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 {crab 3405 ∅c0 4296 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 · cmul 11073 < clt 11208 ≤ cle 11209 − cmin 11405 -cneg 11406 / cdiv 11835 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ℝ⁺crp 12951 ...cfz 13468 ⌊cfl 13752 mod cmo 13831 abscabs 15200 ∥ cdvds 16222 Basecbs 17179 0_gc0g 17402 Grpcgrp 18865 inv_gcminusg 18866 -_gcsg 18867 ._gcmg 18999 gExcgex 19455

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-fz 13469 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-0g 17404 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-gex 19459

This theorem is referenced by: gexdvds2 19515

W3C validator