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Mirrors > Home > ILE Home > Th. List > add42i | GIF version |
Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.) |
Ref | Expression |
---|---|
add.1 | ⊢ 𝐴 ∈ ℂ |
add.2 | ⊢ 𝐵 ∈ ℂ |
add.3 | ⊢ 𝐶 ∈ ℂ |
add4.4 | ⊢ 𝐷 ∈ ℂ |
Ref | Expression |
---|---|
add42i | ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | add.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
2 | add.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
3 | add.3 | . . 3 ⊢ 𝐶 ∈ ℂ | |
4 | add4.4 | . . 3 ⊢ 𝐷 ∈ ℂ | |
5 | 1, 2, 3, 4 | add4i 7273 | . 2 ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)) |
6 | 2, 4 | addcomi 7252 | . . 3 ⊢ (𝐵 + 𝐷) = (𝐷 + 𝐵) |
7 | 6 | oveq2i 5543 | . 2 ⊢ ((𝐴 + 𝐶) + (𝐵 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)) |
8 | 5, 7 | eqtri 2101 | 1 ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∈ wcel 1433 (class class class)co 5532 ℂcc 6979 + caddc 6984 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-addcl 7072 ax-addcom 7076 ax-addass 7078 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: (None) |
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