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Theorem joinlmuladdmuld 10027
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 10025 . 2 (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵)))
5 joinlmuladdmuld.4 . 2 (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
64, 5eqtrd 2655 1 (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  (class class class)co 6615  cc 9894   + caddc 9899   · cmul 9901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-addcl 9956  ax-mulcom 9960  ax-distr 9963
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618
This theorem is referenced by:  1p1times  10167  div4p1lem1div2  11247  int-rightdistd  38004  fmtnorec2lem  40783  joinlmuladdmuli  41852
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