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 ) ) ) ) )