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Metamath Proof Explorer

Theorem reppncan

Description: Cancellation law for mixed addition and real subtraction. Compare ppncan . (Contributed by SN, 3-Sep-2023)

Ref Expression
Assertion reppncan ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 + 𝐶 ) + ( 𝐵 −ℝ 𝐶 ) ) = ( 𝐴 + 𝐵 ) )

Proof

Step Hyp Ref Expression
1 repnpcan ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 + 𝐵 ) −ℝ ( 𝐴 + 𝐶 ) ) = ( 𝐵 −ℝ 𝐶 ) )
2 readdcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 𝐵 ) ∈ ℝ )
3 2 3adant3 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 + 𝐵 ) ∈ ℝ )
4 readdcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 + 𝐶 ) ∈ ℝ )
5 4 3adant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 + 𝐶 ) ∈ ℝ )
6 rersubcl ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 −ℝ 𝐶 ) ∈ ℝ )
7 6 3adant1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 −ℝ 𝐶 ) ∈ ℝ )
8 3 5 7 resubaddd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( ( 𝐴 + 𝐵 ) −ℝ ( 𝐴 + 𝐶 ) ) = ( 𝐵 −ℝ 𝐶 ) ↔ ( ( 𝐴 + 𝐶 ) + ( 𝐵 −ℝ 𝐶 ) ) = ( 𝐴 + 𝐵 ) ) )
9 1 8 mpbid ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 + 𝐶 ) + ( 𝐵 −ℝ 𝐶 ) ) = ( 𝐴 + 𝐵 ) )