Theorem joinlmuladdmuli 50627

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 11255	. 2 ⊢ (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
10	9	mptru 1577	1 ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷

This proof depends on syntax axioms:   = wceq 1570  ⊤wtru 1571   ∈ wcel 2146  (class class class)co 7419  ℂcc 11117   + caddc 11122   · cmul 11124

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-addcl 11179  ax-mulcom 11183  ax-distr 11186

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422

This theorem is used by: (None)