Metamath Proof Explorer
Description: Division of two ratios. Theorem I.15 of Apostol p. 18. (Contributed by NM, 22-Feb-1995)
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Ref |
Expression |
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Hypotheses |
divclz.1 |
⊢ 𝐴 ∈ ℂ |
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divclz.2 |
⊢ 𝐵 ∈ ℂ |
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divmulz.3 |
⊢ 𝐶 ∈ ℂ |
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divmuldiv.4 |
⊢ 𝐷 ∈ ℂ |
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divmuldiv.5 |
⊢ 𝐵 ≠ 0 |
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divmuldiv.6 |
⊢ 𝐷 ≠ 0 |
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divdivdiv.7 |
⊢ 𝐶 ≠ 0 |
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Assertion |
divdivdivi |
⊢ ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 𝐶 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
divclz.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
divclz.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
divmulz.3 |
⊢ 𝐶 ∈ ℂ |
4 |
|
divmuldiv.4 |
⊢ 𝐷 ∈ ℂ |
5 |
|
divmuldiv.5 |
⊢ 𝐵 ≠ 0 |
6 |
|
divmuldiv.6 |
⊢ 𝐷 ≠ 0 |
7 |
|
divdivdiv.7 |
⊢ 𝐶 ≠ 0 |
8 |
2 5
|
pm3.2i |
⊢ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) |
9 |
3 7
|
pm3.2i |
⊢ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) |
10 |
4 6
|
pm3.2i |
⊢ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) |
11 |
|
divdivdiv |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) ∧ ( ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) ) → ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 𝐶 ) ) ) |
12 |
1 8 9 10 11
|
mp4an |
⊢ ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 𝐶 ) ) |