Theorem joinlmuladdmuld 11139

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 11137	. 2 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵)))
5		joinlmuladdmuld.4	. 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
6	4, 5	eqtrd 2766	1 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  (class class class)co 7346  ℂcc 11004   + caddc 11009   · cmul 11011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-addcl 11066  ax-mulcom 11070  ax-distr 11073

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349

This theorem is referenced by:  1p1times  11284  div4p1lem1div2  12376  ltdifltdiv  13738  discr1  14146  arisum  15767  bezoutlem3  16452  bezoutlem4  16453  mbfi1fseqlem4  25646  itgmulc2  25762  tangtx  26441  binom4  26787  axcontlem8  28949  zringfrac  33519  constrrtcclem  33747  cos9thpiminplylem2  33796  int-rightdistd  44283  fmtnorec2lem  47652  joinlmuladdmuli  49884