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Theorem joinlmuladdmuli 47114
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 11140 . 2 (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
109mptru 1548 1 ((𝐴 + 𝐶) · 𝐵) = 𝐷
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wtru 1542  wcel 2106  (class class class)co 7351  cc 11007   + caddc 11012   · cmul 11014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-addcl 11069  ax-mulcom 11073  ax-distr 11076
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-iota 6445  df-fv 6501  df-ov 7354
This theorem is referenced by: (None)
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