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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 20259 | . . . 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 18977 | . . 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 33218 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇))) |
12 | 1, 2, 3, 4, 6, 7, 9 | ringdird 33219 | . . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))) |
13 | 1, 2, 3, 4, 6, 7, 10 | ringdird 33219 | . . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇))) |
14 | 12, 13 | oveq12d 7448 | . 2 ⊢ (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇)))) |
15 | 11, 14 | eqtrd 2774 | 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 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 .rcmulr 17298 Ringcrg 20250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-12 2174 ax-ext 2705 ax-nul 5311 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-ov 7433 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-ring 20252 |
This theorem is referenced by: ssdifidlprm 33465 |
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