Metamath Proof Explorer


Theorem psrnegcl

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

Ref Expression
Hypotheses psrgrp.s 𝑆 = ( 𝐼 mPwSer 𝑅 )
psrgrp.i ( 𝜑𝐼𝑉 )
psrgrp.r ( 𝜑𝑅 ∈ Grp )
psrnegcl.d 𝐷 = { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin }
psrnegcl.i 𝑁 = ( invg𝑅 )
psrnegcl.b 𝐵 = ( Base ‘ 𝑆 )
psrnegcl.z ( 𝜑𝑋𝐵 )
Assertion psrnegcl ( 𝜑 → ( 𝑁𝑋 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 psrgrp.s 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 psrgrp.i ( 𝜑𝐼𝑉 )
3 psrgrp.r ( 𝜑𝑅 ∈ Grp )
4 psrnegcl.d 𝐷 = { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin }
5 psrnegcl.i 𝑁 = ( invg𝑅 )
6 psrnegcl.b 𝐵 = ( Base ‘ 𝑆 )
7 psrnegcl.z ( 𝜑𝑋𝐵 )
8 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
9 8 5 3 grpinvf1o ( 𝜑𝑁 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑅 ) )
10 f1of ( 𝑁 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑅 ) → 𝑁 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑅 ) )
11 9 10 syl ( 𝜑𝑁 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑅 ) )
12 1 8 4 6 7 psrelbas ( 𝜑𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
13 fco ( ( 𝑁 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑅 ) ∧ 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) → ( 𝑁𝑋 ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
14 11 12 13 syl2anc ( 𝜑 → ( 𝑁𝑋 ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
15 fvex ( Base ‘ 𝑅 ) ∈ V
16 ovex ( ℕ0m 𝐼 ) ∈ V
17 4 16 rabex2 𝐷 ∈ V
18 15 17 elmap ( ( 𝑁𝑋 ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐷 ) ↔ ( 𝑁𝑋 ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
19 14 18 sylibr ( 𝜑 → ( 𝑁𝑋 ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐷 ) )
20 1 8 4 6 2 psrbas ( 𝜑𝐵 = ( ( Base ‘ 𝑅 ) ↑m 𝐷 ) )
21 19 20 eleqtrrd ( 𝜑 → ( 𝑁𝑋 ) ∈ 𝐵 )