Theorem int-addassocd 44206

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 11137	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		int-addassocd.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	recnd 11137	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5		int-addassocd.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ)
6	5	recnd 11137	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ)
7	2, 4, 6	addassd 11131	. 2 ⊢ (𝜑 → ((𝐴 + 𝐶) + 𝐷) = (𝐴 + (𝐶 + 𝐷)))
8		int-addassocd.4	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
9	8	oveq1d 7361	. 2 ⊢ (𝜑 → (𝐴 + (𝐶 + 𝐷)) = (𝐵 + (𝐶 + 𝐷)))
10	7, 9	eqtr2d 2767	1 ⊢ (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  (class class class)co 7346  ℝcr 11002   + caddc 11006

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 2113  ax-9 2121  ax-ext 2703  ax-resscn 11060  ax-addass 11068

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 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489  df-ov 7349

This theorem is referenced by: (None)