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Metamath Proof Explorer

Theorem cyggenod

Description: An element is the generator of a finite group iff the order of the generator equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016)

Ref Expression
Hypotheses iscyg.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
iscyg.2 ⊢ · = ( .g ‘ 𝐺 )
iscyg3.e ⊢ 𝐸 = { 𝑥 ∈ 𝐵 ∣ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 }
cyggenod.o ⊢ 𝑂 = ( od ‘ 𝐺 )
Assertion cyggenod ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) → ( 𝑋 ∈ 𝐸 ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑂 ‘ 𝑋 ) = ( ♯ ‘ 𝐵 ) ) ) )

Proof

Step Hyp Ref Expression
1 iscyg.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 iscyg.2 ⊢ · = ( .g ‘ 𝐺 )
3 iscyg3.e ⊢ 𝐸 = { 𝑥 ∈ 𝐵 ∣ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 }
4 cyggenod.o ⊢ 𝑂 = ( od ‘ 𝐺 )
5 1 2 3 iscyggen ⊢ ( 𝑋 ∈ 𝐸 ↔ ( 𝑋 ∈ 𝐵 ∧ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = 𝐵 ) )
6 simplr ⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) → 𝐵 ∈ Fin )
7 simplll ⊢ ( ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑛 ∈ ℤ ) → 𝐺 ∈ Grp )
8 simpr ⊢ ( ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑛 ∈ ℤ ) → 𝑛 ∈ ℤ )
9 simplr ⊢ ( ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑛 ∈ ℤ ) → 𝑋 ∈ 𝐵 )
10 1 2 mulgcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑛 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑛 · 𝑋 ) ∈ 𝐵 )
11 7 8 9 10 syl3anc ⊢ ( ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑛 ∈ ℤ ) → ( 𝑛 · 𝑋 ) ∈ 𝐵 )
12 11 fmpttd ⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) : ℤ ⟶ 𝐵 )
13 12 frnd ⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ⊆ 𝐵 )
14 6 13 ssfid ⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ∈ Fin )
15 hashen ⊢ ( ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ) = ( ♯ ‘ 𝐵 ) ↔ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ≈ 𝐵 ) )
16 14 6 15 syl2anc ⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) → ( ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ) = ( ♯ ‘ 𝐵 ) ↔ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ≈ 𝐵 ) )
17 eqid ⊢ ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) )
18 1 4 2 17 dfod2 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑂 ‘ 𝑋 ) = if ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ∈ Fin , ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ) , 0 ) )
19 18 adantlr ⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑂 ‘ 𝑋 ) = if ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ∈ Fin , ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ) , 0 ) )
20 14 iftrued ⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) → if ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ∈ Fin , ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ) , 0 ) = ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ) )
21 19 20 eqtr2d ⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) → ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ) = ( 𝑂 ‘ 𝑋 ) )
22 21 eqeq1d ⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) → ( ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ) = ( ♯ ‘ 𝐵 ) ↔ ( 𝑂 ‘ 𝑋 ) = ( ♯ ‘ 𝐵 ) ) )
23 fisseneq ⊢ ( ( 𝐵 ∈ Fin ∧ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ⊆ 𝐵 ∧ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ≈ 𝐵 ) → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = 𝐵 )
24 23 3expia ⊢ ( ( 𝐵 ∈ Fin ∧ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ⊆ 𝐵 ) → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ≈ 𝐵 → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = 𝐵 ) )
25 enrefg ⊢ ( 𝐵 ∈ Fin → 𝐵 ≈ 𝐵 )
26 25 adantr ⊢ ( ( 𝐵 ∈ Fin ∧ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ⊆ 𝐵 ) → 𝐵 ≈ 𝐵 )
27 breq1 ⊢ ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = 𝐵 → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ≈ 𝐵 ↔ 𝐵 ≈ 𝐵 ) )
28 26 27 syl5ibrcom ⊢ ( ( 𝐵 ∈ Fin ∧ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ⊆ 𝐵 ) → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = 𝐵 → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ≈ 𝐵 ) )
29 24 28 impbid ⊢ ( ( 𝐵 ∈ Fin ∧ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ⊆ 𝐵 ) → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ≈ 𝐵 ↔ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = 𝐵 ) )
30 6 13 29 syl2anc ⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ≈ 𝐵 ↔ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = 𝐵 ) )
31 16 22 30 3bitr3rd ⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) ∧ 𝑋 ∈ 𝐵 ) → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = 𝐵 ↔ ( 𝑂 ‘ 𝑋 ) = ( ♯ ‘ 𝐵 ) ) )
32 31 pm5.32da ⊢ ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) → ( ( 𝑋 ∈ 𝐵 ∧ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = 𝐵 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑂 ‘ 𝑋 ) = ( ♯ ‘ 𝐵 ) ) ) )
33 5 32 bitrid ⊢ ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ) → ( 𝑋 ∈ 𝐸 ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑂 ‘ 𝑋 ) = ( ♯ ‘ 𝐵 ) ) ) )