Theorem joinlmuladdmuld 7917

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 7915	. 2 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵)))
5		joinlmuladdmuld.4	. 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
6	4, 5	eqtrd 2197	1 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)

Syntax hints:   → wi 4   = wceq 1342   ∈ wcel 2135  (class class class)co 5836  ℂcc 7742   + caddc 7747   · cmul 7749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146  ax-addcl 7840  ax-mulcom 7845  ax-distr 7848

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-iota 5147  df-fv 5190  df-ov 5839

This theorem is referenced by:  div4p1lem1div2  9101  arisum  11425  tangtx  13300  binom4  13438