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Metamath Proof Explorer

Theorem psr0lid

Description: The zero element of the ring of power series is a left identity. (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 )
psr0cl.d ⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
psr0cl.o ⊢ 0 = ( 0g ‘ 𝑅 )
psr0cl.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
psr0lid.p ⊢ + = ( +g ‘ 𝑆 )
psr0lid.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion psr0lid ( 𝜑 → ( ( 𝐷 × { 0 } ) + 𝑋 ) = 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 psr0cl.d ⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
5 psr0cl.o ⊢ 0 = ( 0g ‘ 𝑅 )
6 psr0cl.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
7 psr0lid.p ⊢ + = ( +g ‘ 𝑆 )
8 psr0lid.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
9 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
10 1 2 3 4 5 6 psr0cl ⊢ ( 𝜑 → ( 𝐷 × { 0 } ) ∈ 𝐵 )
11 1 6 9 7 10 8 psradd ⊢ ( 𝜑 → ( ( 𝐷 × { 0 } ) + 𝑋 ) = ( ( 𝐷 × { 0 } ) ∘f ( +g ‘ 𝑅 ) 𝑋 ) )
12 ovex ⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
13 4 12 rabex2 ⊢ 𝐷 ∈ V
14 13 a1i ⊢ ( 𝜑 → 𝐷 ∈ V )
15 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
16 1 15 4 6 8 psrelbas ⊢ ( 𝜑 → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
17 5 fvexi ⊢ 0 ∈ V
18 17 a1i ⊢ ( 𝜑 → 0 ∈ V )
19 15 9 5 grplid ⊢ ( ( 𝑅 ∈ Grp ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 0 ( +g ‘ 𝑅 ) 𝑥 ) = 𝑥 )
20 3 19 sylan ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 0 ( +g ‘ 𝑅 ) 𝑥 ) = 𝑥 )
21 14 16 18 20 caofid0l ⊢ ( 𝜑 → ( ( 𝐷 × { 0 } ) ∘f ( +g ‘ 𝑅 ) 𝑋 ) = 𝑋 )
22 11 21 eqtrd ⊢ ( 𝜑 → ( ( 𝐷 × { 0 } ) + 𝑋 ) = 𝑋 )