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Metamath Proof Explorer

Theorem cyggic

Description: Cyclic groups are isomorphic precisely when they have the same order. (Contributed by Mario Carneiro, 21-Apr-2016)

Ref Expression
Hypotheses cygctb.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
cygctb.c ⊢ 𝐶 = ( Base ‘ 𝐻 )
Assertion cyggic ( ( 𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp ) → ( 𝐺 ≃𝑔 𝐻 ↔ 𝐵 ≈ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 cygctb.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 cygctb.c ⊢ 𝐶 = ( Base ‘ 𝐻 )
3 1 2 gicen ⊢ ( 𝐺 ≃𝑔 𝐻 → 𝐵 ≈ 𝐶 )
4 eqid ⊢ if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) = if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 )
5 eqid ⊢ ( ℤ/nℤ ‘ if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) ) = ( ℤ/nℤ ‘ if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) )
6 1 4 5 cygzn ⊢ ( 𝐺 ∈ CycGrp → 𝐺 ≃𝑔 ( ℤ/nℤ ‘ if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) ) )
7 6 ad2antrr ⊢ ( ( ( 𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp ) ∧ 𝐵 ≈ 𝐶 ) → 𝐺 ≃𝑔 ( ℤ/nℤ ‘ if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) ) )
8 enfi ⊢ ( 𝐵 ≈ 𝐶 → ( 𝐵 ∈ Fin ↔ 𝐶 ∈ Fin ) )
9 8 adantl ⊢ ( ( ( 𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp ) ∧ 𝐵 ≈ 𝐶 ) → ( 𝐵 ∈ Fin ↔ 𝐶 ∈ Fin ) )
10 hasheni ⊢ ( 𝐵 ≈ 𝐶 → ( ♯ ‘ 𝐵 ) = ( ♯ ‘ 𝐶 ) )
11 10 adantl ⊢ ( ( ( 𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp ) ∧ 𝐵 ≈ 𝐶 ) → ( ♯ ‘ 𝐵 ) = ( ♯ ‘ 𝐶 ) )
12 9 11 ifbieq1d ⊢ ( ( ( 𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp ) ∧ 𝐵 ≈ 𝐶 ) → if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) = if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) )
13 12 fveq2d ⊢ ( ( ( 𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp ) ∧ 𝐵 ≈ 𝐶 ) → ( ℤ/nℤ ‘ if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) ) = ( ℤ/nℤ ‘ if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) ) )
14 eqid ⊢ if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) = if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 )
15 eqid ⊢ ( ℤ/nℤ ‘ if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) ) = ( ℤ/nℤ ‘ if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) )
16 2 14 15 cygzn ⊢ ( 𝐻 ∈ CycGrp → 𝐻 ≃𝑔 ( ℤ/nℤ ‘ if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) ) )
17 16 ad2antlr ⊢ ( ( ( 𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp ) ∧ 𝐵 ≈ 𝐶 ) → 𝐻 ≃𝑔 ( ℤ/nℤ ‘ if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) ) )
18 gicsym ⊢ ( 𝐻 ≃𝑔 ( ℤ/nℤ ‘ if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) ) → ( ℤ/nℤ ‘ if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) ) ≃𝑔 𝐻 )
19 17 18 syl ⊢ ( ( ( 𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp ) ∧ 𝐵 ≈ 𝐶 ) → ( ℤ/nℤ ‘ if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) ) ≃𝑔 𝐻 )
20 13 19 eqbrtrd ⊢ ( ( ( 𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp ) ∧ 𝐵 ≈ 𝐶 ) → ( ℤ/nℤ ‘ if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) ) ≃𝑔 𝐻 )
21 gictr ⊢ ( ( 𝐺 ≃𝑔 ( ℤ/nℤ ‘ if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) ) ∧ ( ℤ/nℤ ‘ if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) ) ≃𝑔 𝐻 ) → 𝐺 ≃𝑔 𝐻 )
22 7 20 21 syl2anc ⊢ ( ( ( 𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp ) ∧ 𝐵 ≈ 𝐶 ) → 𝐺 ≃𝑔 𝐻 )
23 22 ex ⊢ ( ( 𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp ) → ( 𝐵 ≈ 𝐶 → 𝐺 ≃𝑔 𝐻 ) )
24 3 23 impbid2 ⊢ ( ( 𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp ) → ( 𝐺 ≃𝑔 𝐻 ↔ 𝐵 ≈ 𝐶 ) )