oddvds - Metamath Proof Explorer

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Theorem oddvds 19456

Description: The only multiples of 𝐴 that are equal to the identity are the multiples of the order of 𝐴. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)

**Hypotheses**
Ref	Expression
odcl.1	⊢ 𝑋 = (Base‘𝐺)
odcl.2	⊢ 𝑂 = (od‘𝐺)
odid.3	⊢ · = (._g‘𝐺)
odid.4	⊢ 0 = (0_g‘𝐺)

**Assertion**
Ref	Expression
oddvds	⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))

**Proof of Theorem oddvds**
Step	Hyp	Ref	Expression
1		simpr 485	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) ∈ ℕ)
2		simpl3 1193	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → 𝑁 ∈ ℤ)
3		dvdsval3 16205	. . . 4 ⊢ (((𝑂‘𝐴) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 mod (𝑂‘𝐴)) = 0))
4	1, 2, 3	syl2anc 584	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 mod (𝑂‘𝐴)) = 0))
5		simpl2 1192	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → 𝐴 ∈ 𝑋)
6		odcl.1	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
7		odid.4	. . . . . . 7 ⊢ 0 = (0_g‘𝐺)
8		odid.3	. . . . . . 7 ⊢ · = (._g‘𝐺)
9	6, 7, 8	mulg0 18993	. . . . . 6 ⊢ (𝐴 ∈ 𝑋 → (0 · 𝐴) = 0 )
10	5, 9	syl 17	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → (0 · 𝐴) = 0 )
11		oveq1 7418	. . . . . 6 ⊢ ((𝑁 mod (𝑂‘𝐴)) = 0 → ((𝑁 mod (𝑂‘𝐴)) · 𝐴) = (0 · 𝐴))
12	11	eqeq1d 2734	. . . . 5 ⊢ ((𝑁 mod (𝑂‘𝐴)) = 0 → (((𝑁 mod (𝑂‘𝐴)) · 𝐴) = 0 ↔ (0 · 𝐴) = 0 ))
13	10, 12	syl5ibrcom 246	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑁 mod (𝑂‘𝐴)) = 0 → ((𝑁 mod (𝑂‘𝐴)) · 𝐴) = 0 ))
14	2	zred 12670	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → 𝑁 ∈ ℝ)
15	1	nnrpd 13018	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) ∈ ℝ⁺)
16		modlt 13849	. . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ (𝑂‘𝐴) ∈ ℝ⁺) → (𝑁 mod (𝑂‘𝐴)) < (𝑂‘𝐴))
17	14, 15, 16	syl2anc 584	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑁 mod (𝑂‘𝐴)) < (𝑂‘𝐴))
18	2, 1	zmodcld 13861	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑁 mod (𝑂‘𝐴)) ∈ ℕ₀)
19	18	nn0red 12537	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑁 mod (𝑂‘𝐴)) ∈ ℝ)
20	1	nnred 12231	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) ∈ ℝ)
21	19, 20	ltnled 11365	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑁 mod (𝑂‘𝐴)) < (𝑂‘𝐴) ↔ ¬ (𝑂‘𝐴) ≤ (𝑁 mod (𝑂‘𝐴))))
22	17, 21	mpbid 231	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → ¬ (𝑂‘𝐴) ≤ (𝑁 mod (𝑂‘𝐴)))
23		odcl.2	. . . . . . . . . . . 12 ⊢ 𝑂 = (od‘𝐺)
24	6, 23, 8, 7	odlem2 19448	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝑁 mod (𝑂‘𝐴)) ∈ ℕ ∧ ((𝑁 mod (𝑂‘𝐴)) · 𝐴) = 0 ) → (𝑂‘𝐴) ∈ (1...(𝑁 mod (𝑂‘𝐴))))
25		elfzle2 13509	. . . . . . . . . . 11 ⊢ ((𝑂‘𝐴) ∈ (1...(𝑁 mod (𝑂‘𝐴))) → (𝑂‘𝐴) ≤ (𝑁 mod (𝑂‘𝐴)))
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝑁 mod (𝑂‘𝐴)) ∈ ℕ ∧ ((𝑁 mod (𝑂‘𝐴)) · 𝐴) = 0 ) → (𝑂‘𝐴) ≤ (𝑁 mod (𝑂‘𝐴)))
27	26	3com23 1126	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ ((𝑁 mod (𝑂‘𝐴)) · 𝐴) = 0 ∧ (𝑁 mod (𝑂‘𝐴)) ∈ ℕ) → (𝑂‘𝐴) ≤ (𝑁 mod (𝑂‘𝐴)))
28	27	3expia 1121	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ ((𝑁 mod (𝑂‘𝐴)) · 𝐴) = 0 ) → ((𝑁 mod (𝑂‘𝐴)) ∈ ℕ → (𝑂‘𝐴) ≤ (𝑁 mod (𝑂‘𝐴))))
29	28	con3d 152	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ ((𝑁 mod (𝑂‘𝐴)) · 𝐴) = 0 ) → (¬ (𝑂‘𝐴) ≤ (𝑁 mod (𝑂‘𝐴)) → ¬ (𝑁 mod (𝑂‘𝐴)) ∈ ℕ))
30	29	impancom 452	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ ¬ (𝑂‘𝐴) ≤ (𝑁 mod (𝑂‘𝐴))) → (((𝑁 mod (𝑂‘𝐴)) · 𝐴) = 0 → ¬ (𝑁 mod (𝑂‘𝐴)) ∈ ℕ))
31	5, 22, 30	syl2anc 584	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → (((𝑁 mod (𝑂‘𝐴)) · 𝐴) = 0 → ¬ (𝑁 mod (𝑂‘𝐴)) ∈ ℕ))
32		elnn0 12478	. . . . . . 7 ⊢ ((𝑁 mod (𝑂‘𝐴)) ∈ ℕ₀ ↔ ((𝑁 mod (𝑂‘𝐴)) ∈ ℕ ∨ (𝑁 mod (𝑂‘𝐴)) = 0))
33	18, 32	sylib 217	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑁 mod (𝑂‘𝐴)) ∈ ℕ ∨ (𝑁 mod (𝑂‘𝐴)) = 0))
34	33	ord 862	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → (¬ (𝑁 mod (𝑂‘𝐴)) ∈ ℕ → (𝑁 mod (𝑂‘𝐴)) = 0))
35	31, 34	syld 47	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → (((𝑁 mod (𝑂‘𝐴)) · 𝐴) = 0 → (𝑁 mod (𝑂‘𝐴)) = 0))
36	13, 35	impbid 211	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑁 mod (𝑂‘𝐴)) = 0 ↔ ((𝑁 mod (𝑂‘𝐴)) · 𝐴) = 0 ))
37	6, 23, 8, 7	odmod 19455	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑁 mod (𝑂‘𝐴)) · 𝐴) = (𝑁 · 𝐴))
38	37	eqeq1d 2734	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → (((𝑁 mod (𝑂‘𝐴)) · 𝐴) = 0 ↔ (𝑁 · 𝐴) = 0 ))
39	4, 36, 38	3bitrd 304	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))
40		simpr 485	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) = 0) → (𝑂‘𝐴) = 0)
41	40	breq1d 5158	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) = 0) → ((𝑂‘𝐴) ∥ 𝑁 ↔ 0 ∥ 𝑁))
42		simpl3 1193	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) = 0) → 𝑁 ∈ ℤ)
43		0dvds 16224	. . . 4 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
44	42, 43	syl 17	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) = 0) → (0 ∥ 𝑁 ↔ 𝑁 = 0))
45		simpl2 1192	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) = 0) → 𝐴 ∈ 𝑋)
46	45, 9	syl 17	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) = 0) → (0 · 𝐴) = 0 )
47		oveq1 7418	. . . . . 6 ⊢ (𝑁 = 0 → (𝑁 · 𝐴) = (0 · 𝐴))
48	47	eqeq1d 2734	. . . . 5 ⊢ (𝑁 = 0 → ((𝑁 · 𝐴) = 0 ↔ (0 · 𝐴) = 0 ))
49	46, 48	syl5ibrcom 246	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) = 0) → (𝑁 = 0 → (𝑁 · 𝐴) = 0 ))
50	6, 23, 8, 7	odnncl 19454	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≠ 0 ∧ (𝑁 · 𝐴) = 0 )) → (𝑂‘𝐴) ∈ ℕ)
51	50	nnne0d 12266	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≠ 0 ∧ (𝑁 · 𝐴) = 0 )) → (𝑂‘𝐴) ≠ 0)
52	51	expr 457	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → ((𝑁 · 𝐴) = 0 → (𝑂‘𝐴) ≠ 0))
53	52	impancom 452	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑁 · 𝐴) = 0 ) → (𝑁 ≠ 0 → (𝑂‘𝐴) ≠ 0))
54	53	necon4d 2964	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑁 · 𝐴) = 0 ) → ((𝑂‘𝐴) = 0 → 𝑁 = 0))
55	54	impancom 452	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) = 0) → ((𝑁 · 𝐴) = 0 → 𝑁 = 0))
56	49, 55	impbid 211	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) = 0) → (𝑁 = 0 ↔ (𝑁 · 𝐴) = 0 ))
57	41, 44, 56	3bitrd 304	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) = 0) → ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))
58	6, 23	odcl 19445	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
59	58	3ad2ant2 1134	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝑂‘𝐴) ∈ ℕ₀)
60		elnn0 12478	. . 3 ⊢ ((𝑂‘𝐴) ∈ ℕ₀ ↔ ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
61	59, 60	sylib 217	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
62	39, 57, 61	mpjaodan 957	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 ℝcr 11111 0cc0 11112 1c1 11113 < clt 11252 ≤ cle 11253 ℕcn 12216 ℕ₀cn0 12476 ℤcz 12562 ℝ⁺crp 12978 ...cfz 13488 mod cmo 13838 ∥ cdvds 16201 Basecbs 17148 0_gc0g 17389 Grpcgrp 18855 ._gcmg 18986 odcod 19433

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-dvds 16202 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-od 19437

This theorem is referenced by: oddvdsi 19457 odcong 19458 odeq 19459 odmulgid 19463 odbezout 19467 gexdvds2 19494 gexod 19495 gexcl3 19496 odadd1 19757 odadd2 19758 oddvdssubg 19764 pgpfac1lem3a 19987 ablsimpgfindlem2 20019 chrdvds 21299 dchrfi 26982 dchrabs 26987 dchrptlem2 26992 idomodle 42240

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