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Theorem gexdvds 19489

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 19488	. . . . 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 3130	. . 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 4286	. . . . . . 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 2815	. . . . . . . . 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 7348	. . . . . . . . . . . 12 ⊢ (𝑦 = (abs‘𝑁) → (𝑦 · 𝑥) = ((abs‘𝑁) · 𝑥))
13	12	eqeq1d 2732	. . . . . . . . . . 11 ⊢ (𝑦 = (abs‘𝑁) → ((𝑦 · 𝑥) = 0 ↔ ((abs‘𝑁) · 𝑥) = 0 ))
14	13	ralbidv 3153	. . . . . . . . . 10 ⊢ (𝑦 = (abs‘𝑁) → (∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑋 ((abs‘𝑁) · 𝑥) = 0 ))
15	14	elrab 3645	. . . . . . . . 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 15211	. . . . . . . 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 12375	. . . . . . 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 7356	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = 𝑁) → ((abs‘𝑁) · 𝑥) = (𝑁 · 𝑥))
28	27	eqeq1d 2732	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) ∧ (abs‘𝑁) = 𝑁) → (((abs‘𝑁) · 𝑥) = 0 ↔ (𝑁 · 𝑥) = 0 ))
29		oveq1 7348	. . . . . . . . 9 ⊢ ((abs‘𝑁) = -𝑁 → ((abs‘𝑁) · 𝑥) = (-𝑁 · 𝑥))
30	29	eqeq1d 2732	. . . . . . . 8 ⊢ ((abs‘𝑁) = -𝑁 → (((abs‘𝑁) · 𝑥) = 0 ↔ (-𝑁 · 𝑥) = 0 ))
31		eqid 2730	. . . . . . . . . . . 12 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
32	1, 3, 31	mulgneg 18997	. . . . . . . . . . 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 18904	. . . . . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 0 = ((inv_g‘𝐺)‘ 0 ))
37	33, 36	eqeq12d 2746	. . . . . . . . 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 18996	. . . . . . . . . . 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 18870	. . . . . . . . . . 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 18912	. . . . . . . . 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 12464	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
47	46	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ 𝑋) → 𝑁 ∈ ℝ)
48	47	absord 15315	. . . . . . 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 3151	. . . . 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 16179	. . . . . 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 5099	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (𝐸 ∥ 𝑁 ↔ 0 ∥ 𝑁))
56		zcn 12465	. . . . . . 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 15189	. . . . 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 3641	. . . 4 ⊢ (𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } → 𝐸 ∈ ℕ)
62	46	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
63		nnrp 12894	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℝ⁺)
64		modval 13767	. . . . . . . . . . . 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 7356	. . . . . . . . 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 12481	. . . . . . . . . . . 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 12914	. . . . . . . . . . . . . 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 13694	. . . . . . . . . . . 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 12575	. . . . . . . . . 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 2730	. . . . . . . . . . 11 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
79	1, 3, 78	mulgsubdir 19019	. . . . . . . . . 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 16180	. . . . . . . . . . . . 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 19488	. . . . . . . . . . . 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 7359	. . . . . . . . . 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 18929	. . . . . . . . . . . 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 2765	. . . . . . . . 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 2769	. . . . . . . 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 3142	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 → ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ))
95		modlt 13776	. . . . . . . . 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 13787	. . . . . . . . . . 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 12435	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → (𝑁 mod 𝐸) ∈ ℝ)
100		nnre 12124	. . . . . . . . . 10 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℝ)
101	100	adantl 481	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℝ)
102	99, 101	ltnled 11252	. . . . . . . 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 19487	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝑁 mod 𝐸) ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 ((𝑁 mod 𝐸) · 𝑥) = 0 ) → 𝐸 ∈ (1...(𝑁 mod 𝐸)))
105		elfzle2 13420	. . . . . . . . . . . . 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 12375	. . . . . . . 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 16159	. . . . . 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 2730	. . . . 5 ⊢ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }
124	1, 3, 4, 2, 123	gexlem1 19484	. . . 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 1541 ∈ wcel 2110 ∀wral 3045 {crab 3393 ∅c0 4281 class class class wbr 5089 ‘cfv 6477 (class class class)co 7341 ℂcc 10996 ℝcr 10997 0cc0 10998 1c1 10999 · cmul 11003 < clt 11138 ≤ cle 11139 − cmin 11336 -cneg 11337 / cdiv 11766 ℕcn 12117 ℕ₀cn0 12373 ℤcz 12460 ℝ⁺crp 12882 ...cfz 13399 ⌊cfl 13686 mod cmo 13765 abscabs 15133 ∥ cdvds 16155 Basecbs 17112 0_gc0g 17335 Grpcgrp 18838 inv_gcminusg 18839 -_gcsg 18840 ._gcmg 18972 gExcgex 19430

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-sup 9321 df-inf 9322 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-n0 12374 df-z 12461 df-uz 12725 df-rp 12883 df-fz 13400 df-fl 13688 df-mod 13766 df-seq 13901 df-exp 13961 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-dvds 16156 df-0g 17337 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-grp 18841 df-minusg 18842 df-sbg 18843 df-mulg 18973 df-gex 19434

This theorem is referenced by: gexdvds2 19490

W3C validator