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Theorem joinlmuladdmuld 11286
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 11284 . 2 (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵)))
5 joinlmuladdmuld.4 . 2 (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
64, 5eqtrd 2775 1 (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  (class class class)co 7431  cc 11151   + caddc 11156   · cmul 11158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-addcl 11213  ax-mulcom 11217  ax-distr 11220
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  1p1times  11430  div4p1lem1div2  12519  ltdifltdiv  13871  discr1  14275  arisum  15893  bezoutlem3  16575  bezoutlem4  16576  mbfi1fseqlem4  25768  itgmulc2  25884  tangtx  26562  binom4  26908  axcontlem8  29001  zringfrac  33562  constrrtcclem  33740  int-rightdistd  44170  fmtnorec2lem  47467  joinlmuladdmuli  49004
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