Theorem int-addassocd 45014

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 11261	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		int-addassocd.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	recnd 11261	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5		int-addassocd.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ)
6	5	recnd 11261	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ)
7	2, 4, 6	addassd 11255	. 2 ⊢ (𝜑 → ((𝐴 + 𝐶) + 𝐷) = (𝐴 + (𝐶 + 𝐷)))
8		int-addassocd.4	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
9	8	oveq1d 7428	. 2 ⊢ (𝜑 → (𝐴 + (𝐶 + 𝐷)) = (𝐵 + (𝐶 + 𝐷)))
10	7, 9	eqtr2d 2796	1 ⊢ (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  (class class class)co 7413  ℝcr 11123   + caddc 11127

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 2147  ax-9 2155  ax-ext 2732  ax-resscn 11181  ax-addass 11189

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7416

This theorem is used by: (None)