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Metamath Proof Explorer

Theorem ablnnncan

Description: Cancellation law for group subtraction. ( nnncan analog.) (Contributed by NM, 29-Feb-2008) (Revised by AV, 27-Aug-2021)

Ref Expression
Hypotheses ablnncan.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
ablnncan.m ⊢ − = ( -g ‘ 𝐺 )
ablnncan.g ⊢ ( 𝜑 → 𝐺 ∈ Abel )
ablnncan.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
ablnncan.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
ablsub32.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion ablnnncan ( 𝜑 → ( ( 𝑋 − ( 𝑌 − 𝑍 ) ) − 𝑍 ) = ( 𝑋 − 𝑌 ) )

Proof

Step Hyp Ref Expression
1 ablnncan.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 ablnncan.m ⊢ − = ( -g ‘ 𝐺 )
3 ablnncan.g ⊢ ( 𝜑 → 𝐺 ∈ Abel )
4 ablnncan.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 ablnncan.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 ablsub32.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
7 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
8 ablgrp ⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
9 3 8 syl ⊢ ( 𝜑 → 𝐺 ∈ Grp )
10 1 2 grpsubcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 − 𝑍 ) ∈ 𝐵 )
11 9 5 6 10 syl3anc ⊢ ( 𝜑 → ( 𝑌 − 𝑍 ) ∈ 𝐵 )
12 1 7 2 3 4 11 6 ablsubsub4 ⊢ ( 𝜑 → ( ( 𝑋 − ( 𝑌 − 𝑍 ) ) − 𝑍 ) = ( 𝑋 − ( ( 𝑌 − 𝑍 ) ( +g ‘ 𝐺 ) 𝑍 ) ) )
13 1 7 ablcom ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑌 − 𝑍 ) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝑌 − 𝑍 ) ( +g ‘ 𝐺 ) 𝑍 ) = ( 𝑍 ( +g ‘ 𝐺 ) ( 𝑌 − 𝑍 ) ) )
14 3 11 6 13 syl3anc ⊢ ( 𝜑 → ( ( 𝑌 − 𝑍 ) ( +g ‘ 𝐺 ) 𝑍 ) = ( 𝑍 ( +g ‘ 𝐺 ) ( 𝑌 − 𝑍 ) ) )
15 1 7 2 ablpncan3 ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑍 ( +g ‘ 𝐺 ) ( 𝑌 − 𝑍 ) ) = 𝑌 )
16 3 6 5 15 syl12anc ⊢ ( 𝜑 → ( 𝑍 ( +g ‘ 𝐺 ) ( 𝑌 − 𝑍 ) ) = 𝑌 )
17 14 16 eqtrd ⊢ ( 𝜑 → ( ( 𝑌 − 𝑍 ) ( +g ‘ 𝐺 ) 𝑍 ) = 𝑌 )
18 17 oveq2d ⊢ ( 𝜑 → ( 𝑋 − ( ( 𝑌 − 𝑍 ) ( +g ‘ 𝐺 ) 𝑍 ) ) = ( 𝑋 − 𝑌 ) )
19 12 18 eqtrd ⊢ ( 𝜑 → ( ( 𝑋 − ( 𝑌 − 𝑍 ) ) − 𝑍 ) = ( 𝑋 − 𝑌 ) )