Metamath Proof Explorer


Theorem ablinvadd

Description: The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014)

Ref Expression
Hypotheses ablinvadd.b 𝐵 = ( Base ‘ 𝐺 )
ablinvadd.p + = ( +g𝐺 )
ablinvadd.n 𝑁 = ( invg𝐺 )
Assertion ablinvadd ( ( 𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝑁𝑋 ) + ( 𝑁𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 ablinvadd.b 𝐵 = ( Base ‘ 𝐺 )
2 ablinvadd.p + = ( +g𝐺 )
3 ablinvadd.n 𝑁 = ( invg𝐺 )
4 ablgrp ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
5 1 2 3 grpinvadd ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝑁𝑌 ) + ( 𝑁𝑋 ) ) )
6 4 5 syl3an1 ( ( 𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝑁𝑌 ) + ( 𝑁𝑋 ) ) )
7 simp1 ( ( 𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵 ) → 𝐺 ∈ Abel )
8 4 3ad2ant1 ( ( 𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵 ) → 𝐺 ∈ Grp )
9 simp2 ( ( 𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵 ) → 𝑋𝐵 )
10 1 3 grpinvcl ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵 ) → ( 𝑁𝑋 ) ∈ 𝐵 )
11 8 9 10 syl2anc ( ( 𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁𝑋 ) ∈ 𝐵 )
12 simp3 ( ( 𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵 ) → 𝑌𝐵 )
13 1 3 grpinvcl ( ( 𝐺 ∈ Grp ∧ 𝑌𝐵 ) → ( 𝑁𝑌 ) ∈ 𝐵 )
14 8 12 13 syl2anc ( ( 𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁𝑌 ) ∈ 𝐵 )
15 1 2 ablcom ( ( 𝐺 ∈ Abel ∧ ( 𝑁𝑋 ) ∈ 𝐵 ∧ ( 𝑁𝑌 ) ∈ 𝐵 ) → ( ( 𝑁𝑋 ) + ( 𝑁𝑌 ) ) = ( ( 𝑁𝑌 ) + ( 𝑁𝑋 ) ) )
16 7 11 14 15 syl3anc ( ( 𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑁𝑋 ) + ( 𝑁𝑌 ) ) = ( ( 𝑁𝑌 ) + ( 𝑁𝑋 ) ) )
17 6 16 eqtr4d ( ( 𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝑁𝑋 ) + ( 𝑁𝑌 ) ) )