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Metamath Proof Explorer

Definition df-od

Description: Define the order of an element in a group. (Contributed by Mario Carneiro, 13-Jul-2014) (Revised by Stefan O'Rear, 4-Sep-2015) (Revised by AV, 5-Oct-2020)

Ref Expression
Assertion df-od od = ( 𝑔 ∈ V ↦ ( 𝑥 ∈ ( Base ‘ 𝑔 ) ↦ ⦋ { 𝑛 ∈ ℕ ∣ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } / 𝑖 ⦌ if ( 𝑖 = ∅ , 0 , inf ( 𝑖 , ℝ , < ) ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cod ⊢ od
1 vg ⊢ 𝑔
2 cvv ⊢ V
3 vx ⊢ 𝑥
4 cbs ⊢ Base
5 1 cv ⊢ 𝑔
6 5 4 cfv ⊢ ( Base ‘ 𝑔 )
7 vn ⊢ 𝑛
8 cn ⊢ ℕ
9 7 cv ⊢ 𝑛
10 cmg ⊢ .g
11 5 10 cfv ⊢ ( .g ‘ 𝑔 )
12 3 cv ⊢ 𝑥
13 9 12 11 co ⊢ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 )
14 c0g ⊢ 0g
15 5 14 cfv ⊢ ( 0g ‘ 𝑔 )
16 13 15 wceq ⊢ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 )
17 16 7 8 crab ⊢ { 𝑛 ∈ ℕ ∣ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) }
18 vi ⊢ 𝑖
19 18 cv ⊢ 𝑖
20 c0 ⊢ ∅
21 19 20 wceq ⊢ 𝑖 = ∅
22 cc0 ⊢ 0
23 cr ⊢ ℝ
24 clt ⊢ <
25 19 23 24 cinf ⊢ inf ( 𝑖 , ℝ , < )
26 21 22 25 cif ⊢ if ( 𝑖 = ∅ , 0 , inf ( 𝑖 , ℝ , < ) )
27 18 17 26 csb ⊢ ⦋ { 𝑛 ∈ ℕ ∣ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } / 𝑖 ⦌ if ( 𝑖 = ∅ , 0 , inf ( 𝑖 , ℝ , < ) )
28 3 6 27 cmpt ⊢ ( 𝑥 ∈ ( Base ‘ 𝑔 ) ↦ ⦋ { 𝑛 ∈ ℕ ∣ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } / 𝑖 ⦌ if ( 𝑖 = ∅ , 0 , inf ( 𝑖 , ℝ , < ) ) )
29 1 2 28 cmpt ⊢ ( 𝑔 ∈ V ↦ ( 𝑥 ∈ ( Base ‘ 𝑔 ) ↦ ⦋ { 𝑛 ∈ ℕ ∣ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } / 𝑖 ⦌ if ( 𝑖 = ∅ , 0 , inf ( 𝑖 , ℝ , < ) ) ) )
30 0 29 wceq ⊢ od = ( 𝑔 ∈ V ↦ ( 𝑥 ∈ ( Base ‘ 𝑔 ) ↦ ⦋ { 𝑛 ∈ ℕ ∣ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } / 𝑖 ⦌ if ( 𝑖 = ∅ , 0 , inf ( 𝑖 , ℝ , < ) ) ) )