Theorem joinlmuladdmuli 49129

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 11295	. 2 ⊢ (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
10	9	mptru 1546	1 ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷

Syntax hints:   = wceq 1539  ⊤wtru 1540   ∈ wcel 2108  (class class class)co 7438  ℂcc 11160   + caddc 11165   · cmul 11167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-addcl 11222  ax-mulcom 11226  ax-distr 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-iota 6522  df-fv 6577  df-ov 7441

This theorem is referenced by: (None)