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Theorem ringdi22 33031
Description: Expand the product of two sums in a ring. (Contributed by Thierry Arnoux, 3-Jun-2025.)
Hypotheses
Ref Expression
ringdi22.1 𝐵 = (Base‘𝑅)
ringdi22.2 + = (+g𝑅)
ringdi22.3 · = (.r𝑅)
ringdi22.4 (𝜑𝑅 ∈ Ring)
ringdi22.5 (𝜑𝑋𝐵)
ringdi22.6 (𝜑𝑌𝐵)
ringdi22.7 (𝜑𝑍𝐵)
ringdi22.8 (𝜑𝑇𝐵)
Assertion
Ref Expression
ringdi22 (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))

Proof of Theorem ringdi22
StepHypRef Expression
1 ringdi22.1 . . 3 𝐵 = (Base‘𝑅)
2 ringdi22.2 . . 3 + = (+g𝑅)
3 ringdi22.3 . . 3 · = (.r𝑅)
4 ringdi22.4 . . 3 (𝜑𝑅 ∈ Ring)
54ringgrpd 20194 . . . 4 (𝜑𝑅 ∈ Grp)
6 ringdi22.5 . . . 4 (𝜑𝑋𝐵)
7 ringdi22.6 . . . 4 (𝜑𝑌𝐵)
81, 2, 5, 6, 7grpcld 18912 . . 3 (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
9 ringdi22.7 . . 3 (𝜑𝑍𝐵)
10 ringdi22.8 . . 3 (𝜑𝑇𝐵)
111, 2, 3, 4, 8, 9, 10ringdid 33029 . 2 (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)))
121, 2, 3, 4, 6, 7, 9ringdird 33030 . . 3 (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
131, 2, 3, 4, 6, 7, 10ringdird 33030 . . 3 (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇)))
1412, 13oveq12d 7437 . 2 (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
1511, 14eqtrd 2765 1 (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6549  (class class class)co 7419  Basecbs 17183  +gcplusg 17236  .rcmulr 17237  Ringcrg 20185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18901  df-ring 20187
This theorem is referenced by:  ssdifidlprm  33270
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