Theorem int-addassocd 44135

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 11220	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		int-addassocd.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	recnd 11220	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5		int-addassocd.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ)
6	5	recnd 11220	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ)
7	2, 4, 6	addassd 11214	. 2 ⊢ (𝜑 → ((𝐴 + 𝐶) + 𝐷) = (𝐴 + (𝐶 + 𝐷)))
8		int-addassocd.4	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
9	8	oveq1d 7409	. 2 ⊢ (𝜑 → (𝐴 + (𝐶 + 𝐷)) = (𝐵 + (𝐶 + 𝐷)))
10	7, 9	eqtr2d 2766	1 ⊢ (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  (class class class)co 7394  ℝcr 11085   + caddc 11089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-resscn 11143  ax-addass 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-iota 6472  df-fv 6527  df-ov 7397

This theorem is referenced by: (None)