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Theorem joinlmuladdmuli 42328
 Description: Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
Hypotheses
Ref Expression
joinlmuladdmuli.1 𝐴 ∈ ℂ
joinlmuladdmuli.2 𝐵 ∈ ℂ
joinlmuladdmuli.3 𝐶 ∈ ℂ
joinlmuladdmuli.4 ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷
Assertion
Ref Expression
joinlmuladdmuli ((𝐴 + 𝐶) · 𝐵) = 𝐷

Proof of Theorem joinlmuladdmuli
StepHypRef Expression
1 joinlmuladdmuli.1 . . . 4 𝐴 ∈ ℂ
21a1i 11 . . 3 (⊤ → 𝐴 ∈ ℂ)
3 joinlmuladdmuli.2 . . . 4 𝐵 ∈ ℂ
43a1i 11 . . 3 (⊤ → 𝐵 ∈ ℂ)
5 joinlmuladdmuli.3 . . . 4 𝐶 ∈ ℂ
65a1i 11 . . 3 (⊤ → 𝐶 ∈ ℂ)
7 joinlmuladdmuli.4 . . . 4 ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷
87a1i 11 . . 3 (⊤ → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
92, 4, 6, 8joinlmuladdmuld 9946 . 2 (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
109trud 1484 1 ((𝐴 + 𝐶) · 𝐵) = 𝐷
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ⊤wtru 1476   ∈ 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: (None)
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