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Metamath Proof Explorer

Theorem div11OLD

Description: Obsolete version of div11 as of 9-Jul-2025. (Contributed by NM, 20-Apr-2006) (Proof shortened by Mario Carneiro, 27-May-2016) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion div11OLD ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 simp1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐴 ∈ ℂ )
2 simp3l ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐶 ∈ ℂ )
3 simp3r ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐶 ≠ 0 )
4 divcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( 𝐴 / 𝐶 ) ∈ ℂ )
5 1 2 3 4 syl3anc ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 / 𝐶 ) ∈ ℂ )
6 simp2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐵 ∈ ℂ )
7 divcl ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( 𝐵 / 𝐶 ) ∈ ℂ )
8 6 2 3 7 syl3anc ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐵 / 𝐶 ) ∈ ℂ )
9 5 8 2 3 mulcand ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐶 · ( 𝐴 / 𝐶 ) ) = ( 𝐶 · ( 𝐵 / 𝐶 ) ) ↔ ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) ) )
10 divcan2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( 𝐶 · ( 𝐴 / 𝐶 ) ) = 𝐴 )
11 1 2 3 10 syl3anc ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐶 · ( 𝐴 / 𝐶 ) ) = 𝐴 )
12 divcan2 ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( 𝐶 · ( 𝐵 / 𝐶 ) ) = 𝐵 )
13 6 2 3 12 syl3anc ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐶 · ( 𝐵 / 𝐶 ) ) = 𝐵 )
14 11 13 eqeq12d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐶 · ( 𝐴 / 𝐶 ) ) = ( 𝐶 · ( 𝐵 / 𝐶 ) ) ↔ 𝐴 = 𝐵 ) )
15 9 14 bitr3d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) ↔ 𝐴 = 𝐵 ) )