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Metamath Proof Explorer

Theorem divrec

Description: Relationship between division and reciprocal. Theorem I.9 of Apostol p. 18. (Contributed by NM, 2-Aug-2004) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion divrec ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 simp2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐵 ∈ ℂ )
2 simp1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐴 ∈ ℂ )
3 reccl ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 1 / 𝐵 ) ∈ ℂ )
4 3 3adant1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 1 / 𝐵 ) ∈ ℂ )
5 1 2 4 mul12d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · ( 𝐴 · ( 1 / 𝐵 ) ) ) = ( 𝐴 · ( 𝐵 · ( 1 / 𝐵 ) ) ) )
6 recid ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · ( 1 / 𝐵 ) ) = 1 )
7 6 3adant1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · ( 1 / 𝐵 ) ) = 1 )
8 7 oveq2d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 · ( 𝐵 · ( 1 / 𝐵 ) ) ) = ( 𝐴 · 1 ) )
9 2 mulridd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 · 1 ) = 𝐴 )
10 5 8 9 3eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · ( 𝐴 · ( 1 / 𝐵 ) ) ) = 𝐴 )
11 2 4 mulcld ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 · ( 1 / 𝐵 ) ) ∈ ℂ )
12 3simpc ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) )
13 divmul ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐴 · ( 1 / 𝐵 ) ) ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) ) ↔ ( 𝐵 · ( 𝐴 · ( 1 / 𝐵 ) ) ) = 𝐴 ) )
14 2 11 12 13 syl3anc ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) ) ↔ ( 𝐵 · ( 𝐴 · ( 1 / 𝐵 ) ) ) = 𝐴 ) )
15 10 14 mpbird ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) ) )