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Theorem joinlmuladdmuli 48873
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 11319 . 2 (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
109mptru 1544 1 ((𝐴 + 𝐶) · 𝐵) = 𝐷
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wtru 1538  wcel 2108  (class class class)co 7450  cc 11184   + caddc 11189   · cmul 11191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-addcl 11246  ax-mulcom 11250  ax-distr 11253
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6527  df-fv 6583  df-ov 7453
This theorem is referenced by: (None)
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