Theorem joinlmuladdmuld 11172

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 11170	. 2 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵)))
5		joinlmuladdmuld.4	. 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
6	4, 5	eqtrd 2771	1 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  (class class class)co 7367  ℂcc 11036   + caddc 11041   · cmul 11043

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-addcl 11098  ax-mulcom 11102  ax-distr 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370

This theorem is referenced by:  1p1times  11317  div4p1lem1div2  12432  ltdifltdiv  13793  discr1  14201  arisum  15825  bezoutlem3  16510  bezoutlem4  16511  mbfi1fseqlem4  25685  itgmulc2  25801  tangtx  26469  binom4  26814  axcontlem8  29040  zringfrac  33614  constrrtcclem  33878  cos9thpiminplylem2  33927  int-rightdistd  44607  sin5tlem4  47324  fmtnorec2lem  48005  joinlmuladdmuli  50248