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Mirrors > Home > ILE Home > Th. List > add42d | GIF version |
Description: Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
addd.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
add4d.4 | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
Ref | Expression |
---|---|
add42d | ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | addd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | addd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | add4d.4 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
5 | add42 7917 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))) | |
6 | 1, 2, 3, 4, 5 | syl22anc 1217 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 (class class class)co 5767 ℂcc 7611 + caddc 7616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-addcl 7709 ax-addcom 7713 ax-addass 7715 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 |
This theorem is referenced by: remullem 10636 |
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