![]() |
Mathbox for Stanislas Polu |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > int-addassocd | Structured version Visualization version GIF version |
Description: AdditionAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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) |
Ref | Expression |
---|---|
int-addassocd | ⊢ (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | int-addassocd.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | 1 | recnd 11314 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | int-addassocd.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 3 | recnd 11314 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
5 | int-addassocd.3 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
6 | 5 | recnd 11314 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
7 | 2, 4, 6 | addassd 11308 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) + 𝐷) = (𝐴 + (𝐶 + 𝐷))) |
8 | int-addassocd.4 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
9 | 8 | oveq1d 7460 | . 2 ⊢ (𝜑 → (𝐴 + (𝐶 + 𝐷)) = (𝐵 + (𝐶 + 𝐷))) |
10 | 7, 9 | eqtr2d 2775 | 1 ⊢ (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 (class class class)co 7445 ℝcr 11179 + caddc 11183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 ax-resscn 11237 ax-addass 11245 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-iota 6524 df-fv 6580 df-ov 7448 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |