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Metamath Proof Explorer

Theorem ablsubsub23

Description: Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007) (Revised by AV, 7-Oct-2021)

Ref Expression
Hypotheses ablsubsub23.v ⊢ 𝑉 = ( Base ‘ 𝐺 )
ablsubsub23.m ⊢ − = ( -g ‘ 𝐺 )
Assertion ablsubsub23 ( ( 𝐺 ∈ Abel ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐴 − 𝐶 ) = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ablsubsub23.v ⊢ 𝑉 = ( Base ‘ 𝐺 )
2 ablsubsub23.m ⊢ − = ( -g ‘ 𝐺 )
3 simpl ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → 𝐺 ∈ Abel )
4 simpr3 ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → 𝐶 ∈ 𝑉 )
5 simpr2 ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → 𝐵 ∈ 𝑉 )
6 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
7 1 6 ablcom ⊢ ( ( 𝐺 ∈ Abel ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( 𝐶 ( +g ‘ 𝐺 ) 𝐵 ) = ( 𝐵 ( +g ‘ 𝐺 ) 𝐶 ) )
8 3 4 5 7 syl3anc ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( 𝐶 ( +g ‘ 𝐺 ) 𝐵 ) = ( 𝐵 ( +g ‘ 𝐺 ) 𝐶 ) )
9 8 eqeq1d ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐶 ( +g ‘ 𝐺 ) 𝐵 ) = 𝐴 ↔ ( 𝐵 ( +g ‘ 𝐺 ) 𝐶 ) = 𝐴 ) )
10 ablgrp ⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
11 1 6 2 grpsubadd ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐶 ( +g ‘ 𝐺 ) 𝐵 ) = 𝐴 ) )
12 10 11 sylan ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐶 ( +g ‘ 𝐺 ) 𝐵 ) = 𝐴 ) )
13 3ancomb ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ↔ ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) )
14 13 biimpi ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) )
15 1 6 2 grpsubadd ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) → ( ( 𝐴 − 𝐶 ) = 𝐵 ↔ ( 𝐵 ( +g ‘ 𝐺 ) 𝐶 ) = 𝐴 ) )
16 10 14 15 syl2an ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐴 − 𝐶 ) = 𝐵 ↔ ( 𝐵 ( +g ‘ 𝐺 ) 𝐶 ) = 𝐴 ) )
17 9 12 16 3bitr4d ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐴 − 𝐶 ) = 𝐵 ) )