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Theorem joinlmuladdmuld 11273
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 11271 . 2 (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵)))
5 joinlmuladdmuld.4 . 2 (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
64, 5eqtrd 2765 1 (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  (class class class)co 7419  cc 11138   + caddc 11143   · cmul 11145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-addcl 11200  ax-mulcom 11204  ax-distr 11207
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422
This theorem is referenced by:  1p1times  11417  div4p1lem1div2  12500  ltdifltdiv  13835  discr1  14237  arisum  15842  bezoutlem3  16520  bezoutlem4  16521  mbfi1fseqlem4  25692  itgmulc2  25807  tangtx  26485  binom4  26827  axcontlem8  28854  zringfrac  33369  int-rightdistd  43752  fmtnorec2lem  47019  joinlmuladdmuli  48392
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