Theorem psrneg

Description: The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014)

	Ref	Expression
Hypotheses	psrgrp.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
	psrgrp.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
	psrgrp.r	⊢ ( 𝜑 → 𝑅 ∈ Grp )
	psrneg.d	⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
	psrneg.i	⊢ 𝑁 = ( invg ‘ 𝑅 )
	psrneg.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	psrneg.m	⊢ 𝑀 = ( invg ‘ 𝑆 )
	psrneg.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	psrneg	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) = ( 𝑁 ∘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		psrgrp.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2		psrgrp.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
3		psrgrp.r	⊢ ( 𝜑 → 𝑅 ∈ Grp )
4		psrneg.d	⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
5		psrneg.i	⊢ 𝑁 = ( invg ‘ 𝑅 )
6		psrneg.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
7		psrneg.m	⊢ 𝑀 = ( invg ‘ 𝑆 )
8		psrneg.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
9		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
10		eqid	⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
11	1 2 3 4 5 6 8 9 10	psrlinv	⊢ ( 𝜑 → ( ( 𝑁 ∘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑋 ) = ( 𝐷 × { ( 0g ‘ 𝑅 ) } ) )
12		eqid	⊢ ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑆 )
13	1 2 3 4 9 12	psr0	⊢ ( 𝜑 → ( 0g ‘ 𝑆 ) = ( 𝐷 × { ( 0g ‘ 𝑅 ) } ) )
14	11 13	eqtr4d	⊢ ( 𝜑 → ( ( 𝑁 ∘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑋 ) = ( 0g ‘ 𝑆 ) )
15	1 2 3	psrgrp	⊢ ( 𝜑 → 𝑆 ∈ Grp )
16	1 2 3 4 5 6 8	psrnegcl	⊢ ( 𝜑 → ( 𝑁 ∘ 𝑋 ) ∈ 𝐵 )
17	6 10 12 7	grpinvid2	⊢ ( ( 𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑁 ∘ 𝑋 ) ∈ 𝐵 ) → ( ( 𝑀 ‘ 𝑋 ) = ( 𝑁 ∘ 𝑋 ) ↔ ( ( 𝑁 ∘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑋 ) = ( 0g ‘ 𝑆 ) ) )
18	15 8 16 17	syl3anc	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) = ( 𝑁 ∘ 𝑋 ) ↔ ( ( 𝑁 ∘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑋 ) = ( 0g ‘ 𝑆 ) ) )
19	14 18	mpbird	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) = ( 𝑁 ∘ 𝑋 ) )