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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 ∈ ℕ 0 I | f -1 ℕ ∈ Fin
psrneg.i ⊢ N = inv g ⁡ R
psrneg.b ⊢ B = Base S
psrneg.m ⊢ M = inv g ⁡ 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 ∈ ℕ 0 I | f -1 ℕ ∈ Fin
5 psrneg.i ⊢ N = inv g ⁡ R
6 psrneg.b ⊢ B = Base S
7 psrneg.m ⊢ M = inv g ⁡ S
8 psrneg.x ⊢ φ → X ∈ B
9 eqid ⊢ 0 R = 0 R
10 eqid ⊢ + S = + S
11 1 2 3 4 5 6 8 9 10 psrlinv ⊢ φ → N ∘ X + S X = D × 0 R
12 eqid ⊢ 0 S = 0 S
13 1 2 3 4 9 12 psr0 ⊢ φ → 0 S = D × 0 R
14 11 13 eqtr4d ⊢ φ → N ∘ X + S X = 0 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 + S X = 0 S
18 15 8 16 17 syl3anc ⊢ φ → M ⁡ X = N ∘ X ↔ N ∘ X + S X = 0 S
19 14 18 mpbird ⊢ φ → M ⁡ X = N ∘ X