| Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
| 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 20239 | . . . 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 18965 | . . 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 20260 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇))) |
| 12 | 1, 2, 3, 4, 6, 7, 9 | ringdird 20261 | . . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))) |
| 13 | 1, 2, 3, 4, 6, 7, 10 | ringdird 20261 | . . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇))) |
| 14 | 12, 13 | oveq12d 7449 | . 2 ⊢ (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇)))) |
| 15 | 11, 14 | eqtrd 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 |
| Copyright terms: Public domain | W3C validator |