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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringdi22 | Structured version Visualization version GIF version |
Description: Expand the product of two sums in a ring. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
Ref | Expression |
---|---|
ringdi22.1 | ⊢ 𝐵 = (Base‘𝑅) |
ringdi22.2 | ⊢ + = (+g‘𝑅) |
ringdi22.3 | ⊢ · = (.r‘𝑅) |
ringdi22.4 | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ringdi22.5 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ringdi22.6 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ringdi22.7 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
ringdi22.8 | ⊢ (𝜑 → 𝑇 ∈ 𝐵) |
Ref | Expression |
---|---|
ringdi22 | ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringdi22.1 | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | ringdi22.2 | . . 3 ⊢ + = (+g‘𝑅) | |
3 | ringdi22.3 | . . 3 ⊢ · = (.r‘𝑅) | |
4 | ringdi22.4 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
5 | 4 | ringgrpd 20194 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) |
6 | ringdi22.5 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | ringdi22.6 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
8 | 1, 2, 5, 6, 7 | grpcld 18912 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
9 | ringdi22.7 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
10 | ringdi22.8 | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐵) | |
11 | 1, 2, 3, 4, 8, 9, 10 | ringdid 33029 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇))) |
12 | 1, 2, 3, 4, 6, 7, 9 | ringdird 33030 | . . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))) |
13 | 1, 2, 3, 4, 6, 7, 10 | ringdird 33030 | . . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇))) |
14 | 12, 13 | oveq12d 7437 | . 2 ⊢ (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇)))) |
15 | 11, 14 | eqtrd 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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