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Metamath Proof Explorer

Theorem gexlem1

Description: The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 23-Apr-2016) (Proof shortened by AV, 26-Sep-2020)

Ref Expression
Hypotheses gexval.1 ⊢ 𝑋 = ( Base ‘ 𝐺 )
gexval.2 ⊢ · = ( .g ‘ 𝐺 )
gexval.3 ⊢ 0 = ( 0g ‘ 𝐺 )
gexval.4 ⊢ 𝐸 = ( gEx ‘ 𝐺 )
gexval.i ⊢ 𝐼 = { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 }
Assertion gexlem1 ( 𝐺 ∈ 𝑉 → ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) ∨ 𝐸 ∈ 𝐼 ) )

Proof

Step Hyp Ref Expression
1 gexval.1 ⊢ 𝑋 = ( Base ‘ 𝐺 )
2 gexval.2 ⊢ · = ( .g ‘ 𝐺 )
3 gexval.3 ⊢ 0 = ( 0g ‘ 𝐺 )
4 gexval.4 ⊢ 𝐸 = ( gEx ‘ 𝐺 )
5 gexval.i ⊢ 𝐼 = { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 }
6 1 2 3 4 5 gexval ⊢ ( 𝐺 ∈ 𝑉 → 𝐸 = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) )
7 eqeq2 ⊢ ( 0 = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) → ( 𝐸 = 0 ↔ 𝐸 = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) ) )
8 7 imbi1d ⊢ ( 0 = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) → ( ( 𝐸 = 0 → ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) ∨ 𝐸 ∈ 𝐼 ) ) ↔ ( 𝐸 = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) → ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) ∨ 𝐸 ∈ 𝐼 ) ) ) )
9 eqeq2 ⊢ ( inf ( 𝐼 , ℝ , < ) = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) → ( 𝐸 = inf ( 𝐼 , ℝ , < ) ↔ 𝐸 = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) ) )
10 9 imbi1d ⊢ ( inf ( 𝐼 , ℝ , < ) = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) → ( ( 𝐸 = inf ( 𝐼 , ℝ , < ) → ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) ∨ 𝐸 ∈ 𝐼 ) ) ↔ ( 𝐸 = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) → ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) ∨ 𝐸 ∈ 𝐼 ) ) ) )
11 orc ⊢ ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) → ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) ∨ 𝐸 ∈ 𝐼 ) )
12 11 expcom ⊢ ( 𝐼 = ∅ → ( 𝐸 = 0 → ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) ∨ 𝐸 ∈ 𝐼 ) ) )
13 12 adantl ⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝐼 = ∅ ) → ( 𝐸 = 0 → ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) ∨ 𝐸 ∈ 𝐼 ) ) )
14 ssrab2 ⊢ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ⊆ ℕ
15 nnuz ⊢ ℕ = ( ℤ≥ ‘ 1 )
16 15 eqcomi ⊢ ( ℤ≥ ‘ 1 ) = ℕ
17 14 5 16 3sstr4i ⊢ 𝐼 ⊆ ( ℤ≥ ‘ 1 )
18 neqne ⊢ ( ¬ 𝐼 = ∅ → 𝐼 ≠ ∅ )
19 18 adantl ⊢ ( ( 𝐺 ∈ 𝑉 ∧ ¬ 𝐼 = ∅ ) → 𝐼 ≠ ∅ )
20 infssuzcl ⊢ ( ( 𝐼 ⊆ ( ℤ≥ ‘ 1 ) ∧ 𝐼 ≠ ∅ ) → inf ( 𝐼 , ℝ , < ) ∈ 𝐼 )
21 17 19 20 sylancr ⊢ ( ( 𝐺 ∈ 𝑉 ∧ ¬ 𝐼 = ∅ ) → inf ( 𝐼 , ℝ , < ) ∈ 𝐼 )
22 eleq1a ⊢ ( inf ( 𝐼 , ℝ , < ) ∈ 𝐼 → ( 𝐸 = inf ( 𝐼 , ℝ , < ) → 𝐸 ∈ 𝐼 ) )
23 21 22 syl ⊢ ( ( 𝐺 ∈ 𝑉 ∧ ¬ 𝐼 = ∅ ) → ( 𝐸 = inf ( 𝐼 , ℝ , < ) → 𝐸 ∈ 𝐼 ) )
24 olc ⊢ ( 𝐸 ∈ 𝐼 → ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) ∨ 𝐸 ∈ 𝐼 ) )
25 23 24 syl6 ⊢ ( ( 𝐺 ∈ 𝑉 ∧ ¬ 𝐼 = ∅ ) → ( 𝐸 = inf ( 𝐼 , ℝ , < ) → ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) ∨ 𝐸 ∈ 𝐼 ) ) )
26 8 10 13 25 ifbothda ⊢ ( 𝐺 ∈ 𝑉 → ( 𝐸 = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) → ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) ∨ 𝐸 ∈ 𝐼 ) ) )
27 6 26 mpd ⊢ ( 𝐺 ∈ 𝑉 → ( ( 𝐸 = 0 ∧ 𝐼 = ∅ ) ∨ 𝐸 ∈ 𝐼 ) )