Theorem mgplem

Description: Lemma for mgpbas . (Contributed by Mario Carneiro, 5-Oct-2015)

	Ref	Expression
Hypotheses	mgpbas.1	⊢ 𝑀 = ( mulGrp ‘ 𝑅 )
	mgplem.2	⊢ 𝐸 = Slot 𝑁
	mgplem.3	⊢ 𝑁 ∈ ℕ
	mgplem.4	⊢ 𝑁 ≠ 2
Assertion	mgplem	⊢ ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ 𝑀 )

Proof

Step	Hyp	Ref	Expression
1		mgpbas.1	⊢ 𝑀 = ( mulGrp ‘ 𝑅 )
2		mgplem.2	⊢ 𝐸 = Slot 𝑁
3		mgplem.3	⊢ 𝑁 ∈ ℕ
4		mgplem.4	⊢ 𝑁 ≠ 2
5	2 3	ndxid	⊢ 𝐸 = Slot ( 𝐸 ‘ ndx )
6	2 3	ndxarg	⊢ ( 𝐸 ‘ ndx ) = 𝑁
7		plusgndx	⊢ ( +g ‘ ndx ) = 2
8	6 7	neeq12i	⊢ ( ( 𝐸 ‘ ndx ) ≠ ( +g ‘ ndx ) ↔ 𝑁 ≠ 2 )
9	4 8	mpbir	⊢ ( 𝐸 ‘ ndx ) ≠ ( +g ‘ ndx )
10	5 9	setsnid	⊢ ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ ( 𝑅 sSet ⟨ ( +g ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) )
11		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
12	1 11	mgpval	⊢ 𝑀 = ( 𝑅 sSet ⟨ ( +g ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ )
13	12	fveq2i	⊢ ( 𝐸 ‘ 𝑀 ) = ( 𝐸 ‘ ( 𝑅 sSet ⟨ ( +g ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) )
14	10 13	eqtr4i	⊢ ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ 𝑀 )