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Metamath Proof Explorer

Theorem sub31

Description: Swap the first and third terms in a double subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion sub31 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( 𝐵 − 𝐶 ) ) = ( 𝐶 − ( 𝐵 − 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 simp1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐴 ∈ ℂ )
2 simpr ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐶 ∈ ℂ )
3 simpl ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐵 ∈ ℂ )
4 2 3 subcld ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐶 − 𝐵 ) ∈ ℂ )
5 4 3adant1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐶 − 𝐵 ) ∈ ℂ )
6 1 5 addcomd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( 𝐶 − 𝐵 ) ) = ( ( 𝐶 − 𝐵 ) + 𝐴 ) )
7 subsub2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( 𝐵 − 𝐶 ) ) = ( 𝐴 + ( 𝐶 − 𝐵 ) ) )
8 simp3 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐶 ∈ ℂ )
9 simp2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐵 ∈ ℂ )
10 8 9 1 subsubd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐶 − ( 𝐵 − 𝐴 ) ) = ( ( 𝐶 − 𝐵 ) + 𝐴 ) )
11 6 7 10 3eqtr4d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( 𝐵 − 𝐶 ) ) = ( 𝐶 − ( 𝐵 − 𝐴 ) ) )