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Mirrors > Home > ILE Home > Th. List > add4d | 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 |
---|---|
add4d | ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))) |
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 | add4 7172 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))) | |
6 | 1, 2, 3, 4, 5 | syl22anc 1136 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 (class class class)co 5512 ℂcc 6887 + caddc 6892 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-addcl 6980 ax-addcom 6984 ax-addass 6986 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: apadd1 7599 binom3 9366 readd 9469 imadd 9477 |
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