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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 ⊢ + = ( +g ‘ 𝐺 )
ablinvadd.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
Assertion ablinvadd ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝑁 ‘ 𝑋 ) + ( 𝑁 ‘ 𝑌 ) ) )

Proof

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