Theorem int-addassocd 44180

Description: AdditionAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)

Hypotheses

Ref	Expression
int-addassocd.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
int-addassocd.2	⊢ (𝜑 → 𝐶 ∈ ℝ)
int-addassocd.3	⊢ (𝜑 → 𝐷 ∈ ℝ)
int-addassocd.4	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
int-addassocd	⊢ (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷))

Proof of Theorem int-addassocd

Step	Hyp	Ref	Expression
1		int-addassocd.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2	1	recnd 11296	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		int-addassocd.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	recnd 11296	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5		int-addassocd.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ)
6	5	recnd 11296	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ)
7	2, 4, 6	addassd 11290	. 2 ⊢ (𝜑 → ((𝐴 + 𝐶) + 𝐷) = (𝐴 + (𝐶 + 𝐷)))
8		int-addassocd.4	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
9	8	oveq1d 7453	. 2 ⊢ (𝜑 → (𝐴 + (𝐶 + 𝐷)) = (𝐵 + (𝐶 + 𝐷)))
10	7, 9	eqtr2d 2778	1 ⊢ (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  (class class class)co 7438  ℝcr 11161   + caddc 11165

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-resscn 11219  ax-addass 11227

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)