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Metamath Proof Explorer

Theorem elghomlem2OLD

Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD . (Contributed by Paul Chapman, 25-Feb-2008) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypothesis elghomlem1OLD.1 ⊢ 𝑆 = { 𝑓 ∣ ( 𝑓 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝑓 ‘ 𝑥 ) 𝐻 ( 𝑓 ‘ 𝑦 ) ) = ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) ) }
Assertion elghomlem2OLD ( ( 𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ) → ( 𝐹 ∈ ( 𝐺 GrpOpHom 𝐻 ) ↔ ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 elghomlem1OLD.1 ⊢ 𝑆 = { 𝑓 ∣ ( 𝑓 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝑓 ‘ 𝑥 ) 𝐻 ( 𝑓 ‘ 𝑦 ) ) = ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) ) }
2 1 elghomlem1OLD ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ) → ( 𝐺 GrpOpHom 𝐻 ) = 𝑆 )
3 2 eleq2d ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ) → ( 𝐹 ∈ ( 𝐺 GrpOpHom 𝐻 ) ↔ 𝐹 ∈ 𝑆 ) )
4 elex ⊢ ( 𝐹 ∈ 𝑆 → 𝐹 ∈ V )
5 feq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 : ran 𝐺 ⟶ ran 𝐻 ↔ 𝐹 : ran 𝐺 ⟶ ran 𝐻 ) )
6 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
7 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
8 6 7 oveq12d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑥 ) 𝐻 ( 𝑓 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) )
9 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) )
10 8 9 eqeq12d ⊢ ( 𝑓 = 𝐹 → ( ( ( 𝑓 ‘ 𝑥 ) 𝐻 ( 𝑓 ‘ 𝑦 ) ) = ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) ↔ ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) )
11 10 2ralbidv ⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝑓 ‘ 𝑥 ) 𝐻 ( 𝑓 ‘ 𝑦 ) ) = ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) ↔ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) )
12 5 11 anbi12d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝑓 ‘ 𝑥 ) 𝐻 ( 𝑓 ‘ 𝑦 ) ) = ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) ) ↔ ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) ) )
13 12 1 elab2g ⊢ ( 𝐹 ∈ V → ( 𝐹 ∈ 𝑆 ↔ ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) ) )
14 13 biimpd ⊢ ( 𝐹 ∈ V → ( 𝐹 ∈ 𝑆 → ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) ) )
15 4 14 mpcom ⊢ ( 𝐹 ∈ 𝑆 → ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) )
16 rnexg ⊢ ( 𝐺 ∈ GrpOp → ran 𝐺 ∈ V )
17 fex ⊢ ( ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 ∧ ran 𝐺 ∈ V ) → 𝐹 ∈ V )
18 17 expcom ⊢ ( ran 𝐺 ∈ V → ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 → 𝐹 ∈ V ) )
19 16 18 syl ⊢ ( 𝐺 ∈ GrpOp → ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 → 𝐹 ∈ V ) )
20 19 adantrd ⊢ ( 𝐺 ∈ GrpOp → ( ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) → 𝐹 ∈ V ) )
21 13 biimprd ⊢ ( 𝐹 ∈ V → ( ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) → 𝐹 ∈ 𝑆 ) )
22 20 21 syli ⊢ ( 𝐺 ∈ GrpOp → ( ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) → 𝐹 ∈ 𝑆 ) )
23 15 22 impbid2 ⊢ ( 𝐺 ∈ GrpOp → ( 𝐹 ∈ 𝑆 ↔ ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) ) )
24 23 adantr ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ) → ( 𝐹 ∈ 𝑆 ↔ ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) ) )
25 3 24 bitrd ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ) → ( 𝐹 ∈ ( 𝐺 GrpOpHom 𝐻 ) ↔ ( 𝐹 : ran 𝐺 ⟶ ran 𝐻 ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝐹 ‘ 𝑥 ) 𝐻 ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) ) )