Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  joinlmuladdmuld Structured version   Visualization version   GIF version

 Description: Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
Hypotheses
Ref Expression
joinlmuladdmuld.4 (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
Assertion
Ref Expression
joinlmuladdmuld (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)

StepHypRef Expression
1 joinlmuladdmuld.1 . . 3 (𝜑𝐴 ∈ ℂ)
2 joinlmuladdmuld.3 . . 3 (𝜑𝐶 ∈ ℂ)
3 joinlmuladdmuld.2 . . 3 (𝜑𝐵 ∈ ℂ)
41, 2, 3adddird 9944 . 2 (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵)))
5 joinlmuladdmuld.4 . 2 (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
64, 5eqtrd 2644 1 (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  (class class class)co 6549  ℂcc 9813   + caddc 9818   · cmul 9820 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-addcl 9875  ax-mulcom 9879  ax-distr 9882 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  div4p1lem1div2  11164  fmtnorec2lem  39992  joinlmuladdmuli  42328
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