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Metamath Proof Explorer

Theorem odhash

Description: An element of zero order generates an infinite subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015)

Ref Expression
Hypotheses odhash.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
odhash.o ⊢ 𝑂 = ( od ‘ 𝐺 )
odhash.k ⊢ 𝐾 = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) )
Assertion odhash ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) → ( ♯ ‘ ( 𝐾 ‘ { 𝐴 } ) ) = +∞ )

Proof

Step Hyp Ref Expression
1 odhash.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
2 odhash.o ⊢ 𝑂 = ( od ‘ 𝐺 )
3 odhash.k ⊢ 𝐾 = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) )
4 eqid ⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 )
5 1 4 2 3 odf1o1 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) → ( 𝑥 ∈ ℤ ↦ ( 𝑥 ( .g ‘ 𝐺 ) 𝐴 ) ) : ℤ –1-1-onto→ ( 𝐾 ‘ { 𝐴 } ) )
6 zex ⊢ ℤ ∈ V
7 6 f1oen ⊢ ( ( 𝑥 ∈ ℤ ↦ ( 𝑥 ( .g ‘ 𝐺 ) 𝐴 ) ) : ℤ –1-1-onto→ ( 𝐾 ‘ { 𝐴 } ) → ℤ ≈ ( 𝐾 ‘ { 𝐴 } ) )
8 hasheni ⊢ ( ℤ ≈ ( 𝐾 ‘ { 𝐴 } ) → ( ♯ ‘ ℤ ) = ( ♯ ‘ ( 𝐾 ‘ { 𝐴 } ) ) )
9 5 7 8 3syl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) → ( ♯ ‘ ℤ ) = ( ♯ ‘ ( 𝐾 ‘ { 𝐴 } ) ) )
10 ominf ⊢ ¬ ω ∈ Fin
11 znnen ⊢ ℤ ≈ ℕ
12 nnenom ⊢ ℕ ≈ ω
13 11 12 entri ⊢ ℤ ≈ ω
14 enfi ⊢ ( ℤ ≈ ω → ( ℤ ∈ Fin ↔ ω ∈ Fin ) )
15 13 14 ax-mp ⊢ ( ℤ ∈ Fin ↔ ω ∈ Fin )
16 10 15 mtbir ⊢ ¬ ℤ ∈ Fin
17 hashinf ⊢ ( ( ℤ ∈ V ∧ ¬ ℤ ∈ Fin ) → ( ♯ ‘ ℤ ) = +∞ )
18 6 16 17 mp2an ⊢ ( ♯ ‘ ℤ ) = +∞
19 9 18 eqtr3di ⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) → ( ♯ ‘ ( 𝐾 ‘ { 𝐴 } ) ) = +∞ )