Theorem joinlmuladdmuld 11002

Description: Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)

Hypotheses

Ref	Expression
joinlmuladdmuld.1	⊢ (𝜑 → 𝐴 ∈ ℂ)
joinlmuladdmuld.2	⊢ (𝜑 → 𝐵 ∈ ℂ)
joinlmuladdmuld.3	⊢ (𝜑 → 𝐶 ∈ ℂ)
joinlmuladdmuld.4	⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)

Assertion

Ref	Expression
joinlmuladdmuld	⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)

Proof of Theorem joinlmuladdmuld

Step	Hyp	Ref	Expression
1		joinlmuladdmuld.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		joinlmuladdmuld.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
3		joinlmuladdmuld.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ)
4	1, 2, 3	adddird 11000	. 2 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵)))
5		joinlmuladdmuld.4	. 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
6	4, 5	eqtrd 2778	1 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  (class class class)co 7275  ℂcc 10869   + caddc 10874   · cmul 10876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-addcl 10931  ax-mulcom 10935  ax-distr 10938

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  1p1times  11146  div4p1lem1div2  12228  ltdifltdiv  13554  discr1  13954  arisum  15572  bezoutlem3  16249  bezoutlem4  16250  mbfi1fseqlem4  24883  itgmulc2  24998  tangtx  25662  binom4  26000  axcontlem8  27339  int-rightdistd  41791  fmtnorec2lem  44994  joinlmuladdmuli  46477