Theorem joinlmuladdmuli 49884

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

Hypotheses

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

Assertion

Ref	Expression
joinlmuladdmuli	⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷

Proof of Theorem joinlmuladdmuli

Step	Hyp	Ref	Expression
1		joinlmuladdmuli.1	. . . 4 ⊢ 𝐴 ∈ ℂ
2	1	a1i 11	. . 3 ⊢ (⊤ → 𝐴 ∈ ℂ)
3		joinlmuladdmuli.2	. . . 4 ⊢ 𝐵 ∈ ℂ
4	3	a1i 11	. . 3 ⊢ (⊤ → 𝐵 ∈ ℂ)
5		joinlmuladdmuli.3	. . . 4 ⊢ 𝐶 ∈ ℂ
6	5	a1i 11	. . 3 ⊢ (⊤ → 𝐶 ∈ ℂ)
7		joinlmuladdmuli.4	. . . 4 ⊢ ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷
8	7	a1i 11	. . 3 ⊢ (⊤ → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)
9	2, 4, 6, 8	joinlmuladdmuld 11139	. 2 ⊢ (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
10	9	mptru 1548	1 ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷

Syntax hints:   = wceq 1541  ⊤wtru 1542   ∈ 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: (None)