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Metamath Proof Explorer

Theorem qusxpid

Description: The Group quotient equivalence relation for the whole group is the cartesian product, i.e. all elements are in the same equivalence class. (Contributed by Thierry Arnoux, 16-Jan-2024)

Ref Expression
Hypothesis qustriv.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion qusxpid ( 𝐺 ∈ Grp → ( 𝐺 ~QG 𝐵 ) = ( 𝐵 × 𝐵 ) )

Proof

Step Hyp Ref Expression
1 qustriv.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 1 subgid ⊢ ( 𝐺 ∈ Grp → 𝐵 ∈ ( SubGrp ‘ 𝐺 ) )
3 eqid ⊢ ( 𝐺 ~QG 𝐵 ) = ( 𝐺 ~QG 𝐵 )
4 1 3 eqger ⊢ ( 𝐵 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐺 ~QG 𝐵 ) Er 𝐵 )
5 errel ⊢ ( ( 𝐺 ~QG 𝐵 ) Er 𝐵 → Rel ( 𝐺 ~QG 𝐵 ) )
6 2 4 5 3syl ⊢ ( 𝐺 ∈ Grp → Rel ( 𝐺 ~QG 𝐵 ) )
7 relxp ⊢ Rel ( 𝐵 × 𝐵 )
8 7 a1i ⊢ ( 𝐺 ∈ Grp → Rel ( 𝐵 × 𝐵 ) )
9 df-3an ⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) )
10 simpl ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐺 ∈ Grp )
11 eqid ⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
12 1 11 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 )
13 12 adantrr ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 )
14 simprr ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐵 )
15 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
16 1 15 grpcl ⊢ ( ( 𝐺 ∈ Grp ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 )
17 10 13 14 16 syl3anc ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 )
18 17 ex ⊢ ( 𝐺 ∈ Grp → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) )
19 18 pm4.71d ⊢ ( 𝐺 ∈ Grp → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ) )
20 9 19 bitr4id ⊢ ( 𝐺 ∈ Grp → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) )
21 ssid ⊢ 𝐵 ⊆ 𝐵
22 1 11 15 3 eqgval ⊢ ( ( 𝐺 ∈ Grp ∧ 𝐵 ⊆ 𝐵 ) → ( 𝑥 ( 𝐺 ~QG 𝐵 ) 𝑦 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ) )
23 21 22 mpan2 ⊢ ( 𝐺 ∈ Grp → ( 𝑥 ( 𝐺 ~QG 𝐵 ) 𝑦 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ) )
24 brxp ⊢ ( 𝑥 ( 𝐵 × 𝐵 ) 𝑦 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) )
25 24 a1i ⊢ ( 𝐺 ∈ Grp → ( 𝑥 ( 𝐵 × 𝐵 ) 𝑦 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) )
26 20 23 25 3bitr4d ⊢ ( 𝐺 ∈ Grp → ( 𝑥 ( 𝐺 ~QG 𝐵 ) 𝑦 ↔ 𝑥 ( 𝐵 × 𝐵 ) 𝑦 ) )
27 6 8 26 eqbrrdv ⊢ ( 𝐺 ∈ Grp → ( 𝐺 ~QG 𝐵 ) = ( 𝐵 × 𝐵 ) )