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Theorem joinlmuladdmuld 11288
Description: Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
Hypotheses
Ref Expression
joinlmuladdmuld.1 (𝜑𝐴 ∈ ℂ)
joinlmuladdmuld.2 (𝜑𝐵 ∈ ℂ)
joinlmuladdmuld.3 (𝜑𝐶 ∈ ℂ)
joinlmuladdmuld.4 (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
Assertion
Ref Expression
joinlmuladdmuld (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)

Proof of Theorem joinlmuladdmuld
StepHypRef Expression
1 joinlmuladdmuld.1 . . 3 (𝜑𝐴 ∈ ℂ)
2 joinlmuladdmuld.3 . . 3 (𝜑𝐶 ∈ ℂ)
3 joinlmuladdmuld.2 . . 3 (𝜑𝐵 ∈ ℂ)
41, 2, 3adddird 11286 . 2 (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵)))
5 joinlmuladdmuld.4 . 2 (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
64, 5eqtrd 2777 1 (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  (class class class)co 7431  cc 11153   + caddc 11158   · cmul 11160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-addcl 11215  ax-mulcom 11219  ax-distr 11222
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by:  1p1times  11432  div4p1lem1div2  12521  ltdifltdiv  13874  discr1  14278  arisum  15896  bezoutlem3  16578  bezoutlem4  16579  mbfi1fseqlem4  25753  itgmulc2  25869  tangtx  26547  binom4  26893  axcontlem8  28986  zringfrac  33582  constrrtcclem  33775  int-rightdistd  44193  fmtnorec2lem  47529  joinlmuladdmuli  49292
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