Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6103 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅)) | |
3 | 2idlval.i | . . . . . 6 ⊢ 𝐼 = (LIdeal‘𝑅) | |
4 | 2, 3 | syl6eqr 2662 | . . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼) |
5 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅)) | |
6 | 2idlval.o | . . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅) | |
7 | 5, 6 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑂) |
8 | 7 | fveq2d 6107 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = (LIdeal‘𝑂)) |
9 | 2idlval.j | . . . . . 6 ⊢ 𝐽 = (LIdeal‘𝑂) | |
10 | 8, 9 | syl6eqr 2662 | . . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = 𝐽) |
11 | 4, 10 | ineq12d 3777 | . . . 4 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))) = (𝐼 ∩ 𝐽)) |
12 | df-2idl 19053 | . . . 4 ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟)))) | |
13 | fvex 6113 | . . . . . 6 ⊢ (LIdeal‘𝑅) ∈ V | |
14 | 3, 13 | eqeltri 2684 | . . . . 5 ⊢ 𝐼 ∈ V |
15 | 14 | inex1 4727 | . . . 4 ⊢ (𝐼 ∩ 𝐽) ∈ V |
16 | 11, 12, 15 | fvmpt 6191 | . . 3 ⊢ (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)) |
17 | fvprc 6097 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = ∅) | |
18 | inss1 3795 | . . . . 5 ⊢ (𝐼 ∩ 𝐽) ⊆ 𝐼 | |
19 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (LIdeal‘𝑅) = ∅) | |
20 | 3, 19 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐼 = ∅) |
21 | sseq0 3927 | . . . . 5 ⊢ (((𝐼 ∩ 𝐽) ⊆ 𝐼 ∧ 𝐼 = ∅) → (𝐼 ∩ 𝐽) = ∅) | |
22 | 18, 20, 21 | sylancr 694 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝐼 ∩ 𝐽) = ∅) |
23 | 17, 22 | eqtr4d 2647 | . . 3 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)) |
24 | 16, 23 | pm2.61i 175 | . 2 ⊢ (2Ideal‘𝑅) = (𝐼 ∩ 𝐽) |
25 | 1, 24 | eqtri 2632 | 1 ⊢ 𝑇 = (𝐼 ∩ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ‘cfv 5804 opprcoppr 18445 LIdealclidl 18991 2Idealc2idl 19052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-2idl 19053 |
This theorem is referenced by: 2idlcpbl 19055 qus1 19056 qusrhm 19058 crng2idl 19060 |
Copyright terms: Public domain | W3C validator |