Metamath Proof Explorer


Theorem gexlem2

Description: Any positive annihilator of all the group elements is an upper bound on the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016) (Proof shortened by AV, 26-Sep-2020)

Ref Expression
Hypotheses gexcl.1 𝑋 = ( Base ‘ 𝐺 )
gexcl.2 𝐸 = ( gEx ‘ 𝐺 )
gexid.3 · = ( .g𝐺 )
gexid.4 0 = ( 0g𝐺 )
Assertion gexlem2 ( ( 𝐺𝑉𝑁 ∈ ℕ ∧ ∀ 𝑥𝑋 ( 𝑁 · 𝑥 ) = 0 ) → 𝐸 ∈ ( 1 ... 𝑁 ) )

Proof

Step Hyp Ref Expression
1 gexcl.1 𝑋 = ( Base ‘ 𝐺 )
2 gexcl.2 𝐸 = ( gEx ‘ 𝐺 )
3 gexid.3 · = ( .g𝐺 )
4 gexid.4 0 = ( 0g𝐺 )
5 oveq1 ( 𝑦 = 𝑁 → ( 𝑦 · 𝑥 ) = ( 𝑁 · 𝑥 ) )
6 5 eqeq1d ( 𝑦 = 𝑁 → ( ( 𝑦 · 𝑥 ) = 0 ↔ ( 𝑁 · 𝑥 ) = 0 ) )
7 6 ralbidv ( 𝑦 = 𝑁 → ( ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 ↔ ∀ 𝑥𝑋 ( 𝑁 · 𝑥 ) = 0 ) )
8 7 elrab ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ↔ ( 𝑁 ∈ ℕ ∧ ∀ 𝑥𝑋 ( 𝑁 · 𝑥 ) = 0 ) )
9 eqid { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } = { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 }
10 1 3 4 2 9 gexval ( 𝐺𝑉𝐸 = if ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } = ∅ , 0 , inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ) )
11 ne0i ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } → { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ≠ ∅ )
12 ifnefalse ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ≠ ∅ → if ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } = ∅ , 0 , inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ) = inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) )
13 11 12 syl ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } → if ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } = ∅ , 0 , inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ) = inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) )
14 10 13 sylan9eq ( ( 𝐺𝑉𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → 𝐸 = inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) )
15 ssrab2 { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ⊆ ℕ
16 nnuz ℕ = ( ℤ ‘ 1 )
17 15 16 sseqtri { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ⊆ ( ℤ ‘ 1 )
18 11 adantl ( ( 𝐺𝑉𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ≠ ∅ )
19 infssuzcl ( ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ⊆ ( ℤ ‘ 1 ) ∧ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ≠ ∅ ) → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } )
20 17 18 19 sylancr ( ( 𝐺𝑉𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } )
21 15 20 sselid ( ( 𝐺𝑉𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ℕ )
22 infssuzle ( ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ⊆ ( ℤ ‘ 1 ) ∧ 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ≤ 𝑁 )
23 17 22 mpan ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ≤ 𝑁 )
24 23 adantl ( ( 𝐺𝑉𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ≤ 𝑁 )
25 elrabi ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } → 𝑁 ∈ ℕ )
26 25 nnzd ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } → 𝑁 ∈ ℤ )
27 fznn ( 𝑁 ∈ ℤ → ( inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ( 1 ... 𝑁 ) ↔ ( inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ℕ ∧ inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ≤ 𝑁 ) ) )
28 26 27 syl ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } → ( inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ( 1 ... 𝑁 ) ↔ ( inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ℕ ∧ inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ≤ 𝑁 ) ) )
29 28 adantl ( ( 𝐺𝑉𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → ( inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ( 1 ... 𝑁 ) ↔ ( inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ℕ ∧ inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ≤ 𝑁 ) ) )
30 21 24 29 mpbir2and ( ( 𝐺𝑉𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ( 1 ... 𝑁 ) )
31 14 30 eqeltrd ( ( 𝐺𝑉𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → 𝐸 ∈ ( 1 ... 𝑁 ) )
32 8 31 sylan2br ( ( 𝐺𝑉 ∧ ( 𝑁 ∈ ℕ ∧ ∀ 𝑥𝑋 ( 𝑁 · 𝑥 ) = 0 ) ) → 𝐸 ∈ ( 1 ... 𝑁 ) )
33 32 3impb ( ( 𝐺𝑉𝑁 ∈ ℕ ∧ ∀ 𝑥𝑋 ( 𝑁 · 𝑥 ) = 0 ) → 𝐸 ∈ ( 1 ... 𝑁 ) )