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Theorem ringdi22 33235
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 20239 . . . 4 (𝜑𝑅 ∈ Grp)
6 ringdi22.5 . . . 4 (𝜑𝑋𝐵)
7 ringdi22.6 . . . 4 (𝜑𝑌𝐵)
81, 2, 5, 6, 7grpcld 18965 . . 3 (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
9 ringdi22.7 . . 3 (𝜑𝑍𝐵)
10 ringdi22.8 . . 3 (𝜑𝑇𝐵)
111, 2, 3, 4, 8, 9, 10ringdid 20260 . 2 (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)))
121, 2, 3, 4, 6, 7, 9ringdird 20261 . . 3 (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
131, 2, 3, 4, 6, 7, 10ringdird 20261 . . 3 (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇)))
1412, 13oveq12d 7449 . 2 (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
1511, 14eqtrd 2777 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 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  Ringcrg 20230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-ring 20232
This theorem is referenced by:  ssdifidlprm  33486
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