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Metamath Proof Explorer

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 ) )