Theorem joinlmuladdmuli 50869

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 11336	. 2 ⊢ (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
10	9	mptru 1576	1 ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷

This proof depends on syntax axioms:   = wceq 1569  ⊤wtru 1570   ∈ wcel 2144  (class class class)co 7420  ℂcc 11198   + caddc 11203   · cmul 11205

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-addcl 11260  ax-mulcom 11264  ax-distr 11267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423

This theorem is used by: (None)