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Mathbox for Stanislas Polu |
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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 11191 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | int-addassocd.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 3 | recnd 11191 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
5 | int-addassocd.3 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
6 | 5 | recnd 11191 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
7 | 2, 4, 6 | addassd 11185 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) + 𝐷) = (𝐴 + (𝐶 + 𝐷))) |
8 | int-addassocd.4 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
9 | 8 | oveq1d 7376 | . 2 ⊢ (𝜑 → (𝐴 + (𝐶 + 𝐷)) = (𝐵 + (𝐶 + 𝐷))) |
10 | 7, 9 | eqtr2d 2774 | 1 ⊢ (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 (class class class)co 7361 ℝcr 11058 + caddc 11062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-resscn 11116 ax-addass 11124 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-iota 6452 df-fv 6508 df-ov 7364 |
This theorem is referenced by: (None) |
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