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Mirrors > Home > ILE Home > Th. List > add42 | GIF version |
Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.) |
Ref | Expression |
---|---|
add42 | ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | add4 7947 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))) | |
2 | addcom 7923 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (𝐵 + 𝐷) = (𝐷 + 𝐵)) | |
3 | 2 | ad2ant2l 500 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐵 + 𝐷) = (𝐷 + 𝐵)) |
4 | 3 | oveq2d 5798 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐶) + (𝐵 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))) |
5 | 1, 4 | eqtrd 2173 | 1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 (class class class)co 5782 ℂcc 7642 + caddc 7647 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-addcl 7740 ax-addcom 7744 ax-addass 7746 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 |
This theorem is referenced by: add42d 7956 |
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