Theorem joinlmuladdmuli 50028

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 11159	. 2 ⊢ (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
10	9	mptru 1548	1 ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷

Syntax hints:   = wceq 1541  ⊤wtru 1542   ∈ wcel 2113  (class class class)co 7358  ℂcc 11024   + caddc 11029   · cmul 11031

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 2115  ax-9 2123  ax-ext 2708  ax-addcl 11086  ax-mulcom 11090  ax-distr 11093

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 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361

This theorem is referenced by: (None)