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Mirrors > Home > MPE Home > Th. List > 2idlval | Structured version Visualization version GIF version |
Description: Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
2idlval.i | ⊢ 𝐼 = (LIdeal‘𝑅) |
2idlval.o | ⊢ 𝑂 = (oppr‘𝑅) |
2idlval.j | ⊢ 𝐽 = (LIdeal‘𝑂) |
2idlval.t | ⊢ 𝑇 = (2Ideal‘𝑅) |
Ref | Expression |
---|---|
2idlval | ⊢ 𝑇 = (𝐼 ∩ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2idlval.t | . 2 ⊢ 𝑇 = (2Ideal‘𝑅) | |
2 | fveq2 6354 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅)) | |
3 | 2idlval.i | . . . . . 6 ⊢ 𝐼 = (LIdeal‘𝑅) | |
4 | 2, 3 | syl6eqr 2813 | . . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼) |
5 | fveq2 6354 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅)) | |
6 | 2idlval.o | . . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅) | |
7 | 5, 6 | syl6eqr 2813 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑂) |
8 | 7 | fveq2d 6358 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = (LIdeal‘𝑂)) |
9 | 2idlval.j | . . . . . 6 ⊢ 𝐽 = (LIdeal‘𝑂) | |
10 | 8, 9 | syl6eqr 2813 | . . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = 𝐽) |
11 | 4, 10 | ineq12d 3959 | . . . 4 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))) = (𝐼 ∩ 𝐽)) |
12 | df-2idl 19455 | . . . 4 ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟)))) | |
13 | fvex 6364 | . . . . . 6 ⊢ (LIdeal‘𝑅) ∈ V | |
14 | 3, 13 | eqeltri 2836 | . . . . 5 ⊢ 𝐼 ∈ V |
15 | 14 | inex1 4952 | . . . 4 ⊢ (𝐼 ∩ 𝐽) ∈ V |
16 | 11, 12, 15 | fvmpt 6446 | . . 3 ⊢ (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)) |
17 | fvprc 6348 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = ∅) | |
18 | inss1 3977 | . . . . 5 ⊢ (𝐼 ∩ 𝐽) ⊆ 𝐼 | |
19 | fvprc 6348 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (LIdeal‘𝑅) = ∅) | |
20 | 3, 19 | syl5eq 2807 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐼 = ∅) |
21 | sseq0 4119 | . . . . 5 ⊢ (((𝐼 ∩ 𝐽) ⊆ 𝐼 ∧ 𝐼 = ∅) → (𝐼 ∩ 𝐽) = ∅) | |
22 | 18, 20, 21 | sylancr 698 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝐼 ∩ 𝐽) = ∅) |
23 | 17, 22 | eqtr4d 2798 | . . 3 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)) |
24 | 16, 23 | pm2.61i 176 | . 2 ⊢ (2Ideal‘𝑅) = (𝐼 ∩ 𝐽) |
25 | 1, 24 | eqtri 2783 | 1 ⊢ 𝑇 = (𝐼 ∩ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1632 ∈ wcel 2140 Vcvv 3341 ∩ cin 3715 ⊆ wss 3716 ∅c0 4059 ‘cfv 6050 opprcoppr 18843 LIdealclidl 19393 2Idealc2idl 19454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3343 df-sbc 3578 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-nul 4060 df-if 4232 df-sn 4323 df-pr 4325 df-op 4329 df-uni 4590 df-br 4806 df-opab 4866 df-mpt 4883 df-id 5175 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-iota 6013 df-fun 6052 df-fv 6058 df-2idl 19455 |
This theorem is referenced by: 2idlcpbl 19457 qus1 19458 qusrhm 19460 crng2idl 19462 |
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