Mathbox for David A. Wheeler < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  joinlmuladdmuli 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
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 10269 . 2 (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
109trud 1641 1 ((𝐴 + 𝐶) · 𝐵) = 𝐷
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631  ⊤wtru 1632   ∈ wcel 2145  (class class class)co 6793  ℂcc 10136   + caddc 10141   · cmul 10143 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-addcl 10198  ax-mulcom 10202  ax-distr 10205 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796 This theorem is referenced by: (None)
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