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Theorem adddird 7109
Description: Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
addcld.1 (𝜑𝐴 ∈ ℂ)
addcld.2 (𝜑𝐵 ∈ ℂ)
addassd.3 (𝜑𝐶 ∈ ℂ)
Assertion
Ref Expression
adddird (𝜑 → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Proof of Theorem adddird
StepHypRef Expression
1 addcld.1 . 2 (𝜑𝐴 ∈ ℂ)
2 addcld.2 . 2 (𝜑𝐵 ∈ ℂ)
3 addassd.3 . 2 (𝜑𝐶 ∈ ℂ)
4 adddir 7075 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
51, 2, 3, 4syl3anc 1146 1 (𝜑 → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  (class class class)co 5539  cc 6944   + caddc 6949   · cmul 6951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-addcl 7037  ax-mulcom 7042  ax-distr 7045
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-iota 4894  df-fv 4937  df-ov 5542
This theorem is referenced by:  adddirp1d  7110  joinlmuladdmuld  7111  1p1times  7207  recextlem1  7705  divdirap  7747  subsq  9524  subsq2  9525  binom2  9528  binom3  9533  remullem  9698  resqrexlemover  9836  resqrexlemcalc1  9840
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