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Mirrors > Home > ILE Home > Th. List > muladdi | GIF version |
Description: Product of two sums. (Contributed by NM, 17-May-1999.) |
Ref | Expression |
---|---|
mulm1.1 | ⊢ 𝐴 ∈ ℂ |
mulneg.2 | ⊢ 𝐵 ∈ ℂ |
subdi.3 | ⊢ 𝐶 ∈ ℂ |
muladdi.4 | ⊢ 𝐷 ∈ ℂ |
Ref | Expression |
---|---|
muladdi | ⊢ ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) + ((𝐴 · 𝐷) + (𝐶 · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulm1.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | mulneg.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | subdi.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
4 | muladdi.4 | . 2 ⊢ 𝐷 ∈ ℂ | |
5 | muladd 8273 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) + ((𝐴 · 𝐷) + (𝐶 · 𝐵)))) | |
6 | 1, 2, 3, 4, 5 | mp4an 424 | 1 ⊢ ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) + ((𝐴 · 𝐷) + (𝐶 · 𝐵))) |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 (class class class)co 5836 ℂcc 7742 + caddc 7747 · cmul 7749 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-addcl 7840 ax-mulcl 7842 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-distr 7848 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-iota 5147 df-fv 5190 df-ov 5839 |
This theorem is referenced by: (None) |
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