Theorem mul4i 11374

Description: Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)

Hypotheses

Ref	Expression
mul.1	⊢ 𝐴 ∈ ℂ
mul.2	⊢ 𝐵 ∈ ℂ
mul.3	⊢ 𝐶 ∈ ℂ
mul4.4	⊢ 𝐷 ∈ ℂ

Assertion

Ref	Expression
mul4i	⊢ ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))

Proof of Theorem mul4i

Step	Hyp	Ref	Expression
1		mul.1	. 2 ⊢ 𝐴 ∈ ℂ
2		mul.2	. 2 ⊢ 𝐵 ∈ ℂ
3		mul.3	. 2 ⊢ 𝐶 ∈ ℂ
4		mul4.4	. 2 ⊢ 𝐷 ∈ ℂ
5		mul4 11345	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷)))
6	1, 2, 3, 4, 5	mp4an 703	1 ⊢ ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))

Syntax hints:   = wceq 1559   ∈ wcel 2141  (class class class)co 7391  ℂcc 11065   · cmul 11072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-mulcl 11129  ax-mulcom 11131  ax-mulass 11133

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6472  df-fv 6524  df-ov 7394

This theorem is referenced by:  faclbnd4lem1  14300  bposlem8  27343  normlem1  31270  dpmul  33051  dpmul4  33052