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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 10669 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | int-addassocd.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 3 | recnd 10669 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
5 | int-addassocd.3 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
6 | 5 | recnd 10669 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
7 | 2, 4, 6 | addassd 10663 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) + 𝐷) = (𝐴 + (𝐶 + 𝐷))) |
8 | int-addassocd.4 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
9 | 8 | oveq1d 7171 | . 2 ⊢ (𝜑 → (𝐴 + (𝐶 + 𝐷)) = (𝐵 + (𝐶 + 𝐷))) |
10 | 7, 9 | eqtr2d 2857 | 1 ⊢ (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℝcr 10536 + caddc 10540 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-resscn 10594 ax-addass 10602 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 |
This theorem is referenced by: (None) |
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