Theorem joinlmuladdmuld 8343

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 8341	. 2 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵)))
5		joinlmuladdmuld.4	. 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
6	4, 5	eqtrd 2271	1 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  (class class class)co 6075  ℂcc 8167   + caddc 8172   · cmul 8174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-addcl 8265  ax-mulcom 8270  ax-distr 8273

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078

This theorem is referenced by:  div4p1lem1div2  9538  arisum  12243  tangtx  15862  binom4  16004