Theorem oppglem

Description: Lemma for oppgbas . (Contributed by Stefan O'Rear, 26-Aug-2015)

	Ref	Expression
Hypotheses	oppgbas.1	⊢ 𝑂 = ( oppg ‘ 𝑅 )
	oppglem.2	⊢ 𝐸 = Slot 𝑁
	oppglem.3	⊢ 𝑁 ∈ ℕ
	oppglem.4	⊢ 𝑁 ≠ 2
Assertion	oppglem	⊢ ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ 𝑂 )

Proof

Step	Hyp	Ref	Expression
1		oppgbas.1	⊢ 𝑂 = ( oppg ‘ 𝑅 )
2		oppglem.2	⊢ 𝐸 = Slot 𝑁
3		oppglem.3	⊢ 𝑁 ∈ ℕ
4		oppglem.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 ) , tpos ( +g ‘ 𝑅 ) ⟩ ) )
11		eqid	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
12	11 1	oppgval	⊢ 𝑂 = ( 𝑅 sSet ⟨ ( +g ‘ ndx ) , tpos ( +g ‘ 𝑅 ) ⟩ )
13	12	fveq2i	⊢ ( 𝐸 ‘ 𝑂 ) = ( 𝐸 ‘ ( 𝑅 sSet ⟨ ( +g ‘ ndx ) , tpos ( +g ‘ 𝑅 ) ⟩ ) )
14	10 13	eqtr4i	⊢ ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ 𝑂 )