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Metamath Proof Explorer

Theorem grpoidval

Description: Lemma for grpoidcl and others. (Contributed by NM, 5-Feb-2010) (Proof shortened by Mario Carneiro, 15-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses grpoidval.1 ⊢ 𝑋 = ran 𝐺
grpoidval.2 ⊢ 𝑈 = ( GId ‘ 𝐺 )
Assertion grpoidval ( 𝐺 ∈ GrpOp → 𝑈 = ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 ) )

Proof

Step Hyp Ref Expression
1 grpoidval.1 ⊢ 𝑋 = ran 𝐺
2 grpoidval.2 ⊢ 𝑈 = ( GId ‘ 𝐺 )
3 1 gidval ⊢ ( 𝐺 ∈ GrpOp → ( GId ‘ 𝐺 ) = ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) )
4 simpl ⊢ ( ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) → ( 𝑢 𝐺 𝑥 ) = 𝑥 )
5 4 ralimi ⊢ ( ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) → ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 )
6 5 rgenw ⊢ ∀ 𝑢 ∈ 𝑋 ( ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) → ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 )
7 6 a1i ⊢ ( 𝐺 ∈ GrpOp → ∀ 𝑢 ∈ 𝑋 ( ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) → ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 ) )
8 1 grpoidinv ⊢ ( 𝐺 ∈ GrpOp → ∃ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝑋 ( ( 𝑦 𝐺 𝑥 ) = 𝑢 ∧ ( 𝑥 𝐺 𝑦 ) = 𝑢 ) ) )
9 simpl ⊢ ( ( ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝑋 ( ( 𝑦 𝐺 𝑥 ) = 𝑢 ∧ ( 𝑥 𝐺 𝑦 ) = 𝑢 ) ) → ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) )
10 9 ralimi ⊢ ( ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝑋 ( ( 𝑦 𝐺 𝑥 ) = 𝑢 ∧ ( 𝑥 𝐺 𝑦 ) = 𝑢 ) ) → ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) )
11 10 reximi ⊢ ( ∃ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝑋 ( ( 𝑦 𝐺 𝑥 ) = 𝑢 ∧ ( 𝑥 𝐺 𝑦 ) = 𝑢 ) ) → ∃ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) )
12 8 11 syl ⊢ ( 𝐺 ∈ GrpOp → ∃ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) )
13 1 grpoideu ⊢ ( 𝐺 ∈ GrpOp → ∃! 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 )
14 7 12 13 3jca ⊢ ( 𝐺 ∈ GrpOp → ( ∀ 𝑢 ∈ 𝑋 ( ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) → ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 ) ∧ ∃ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ∧ ∃! 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 ) )
15 reupick2 ⊢ ( ( ( ∀ 𝑢 ∈ 𝑋 ( ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) → ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 ) ∧ ∃ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ∧ ∃! 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 ) ∧ 𝑢 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 ↔ ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) )
16 14 15 sylan ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 ↔ ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) )
17 16 riotabidva ⊢ ( 𝐺 ∈ GrpOp → ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 ) = ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) )
18 3 17 eqtr4d ⊢ ( 𝐺 ∈ GrpOp → ( GId ‘ 𝐺 ) = ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 ) )
19 2 18 eqtrid ⊢ ( 𝐺 ∈ GrpOp → 𝑈 = ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( 𝑢 𝐺 𝑥 ) = 𝑥 ) )