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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 6907 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅)) | |
3 | 2idlval.i | . . . . . 6 ⊢ 𝐼 = (LIdeal‘𝑅) | |
4 | 2, 3 | eqtr4di 2793 | . . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼) |
5 | fveq2 6907 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅)) | |
6 | 2idlval.o | . . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅) | |
7 | 5, 6 | eqtr4di 2793 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑂) |
8 | 7 | fveq2d 6911 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = (LIdeal‘𝑂)) |
9 | 2idlval.j | . . . . . 6 ⊢ 𝐽 = (LIdeal‘𝑂) | |
10 | 8, 9 | eqtr4di 2793 | . . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = 𝐽) |
11 | 4, 10 | ineq12d 4229 | . . . 4 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))) = (𝐼 ∩ 𝐽)) |
12 | df-2idl 21278 | . . . 4 ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟)))) | |
13 | 3 | fvexi 6921 | . . . . 5 ⊢ 𝐼 ∈ V |
14 | 13 | inex1 5323 | . . . 4 ⊢ (𝐼 ∩ 𝐽) ∈ V |
15 | 11, 12, 14 | fvmpt 7016 | . . 3 ⊢ (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)) |
16 | fvprc 6899 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = ∅) | |
17 | inss1 4245 | . . . . 5 ⊢ (𝐼 ∩ 𝐽) ⊆ 𝐼 | |
18 | fvprc 6899 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (LIdeal‘𝑅) = ∅) | |
19 | 3, 18 | eqtrid 2787 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐼 = ∅) |
20 | sseq0 4409 | . . . . 5 ⊢ (((𝐼 ∩ 𝐽) ⊆ 𝐼 ∧ 𝐼 = ∅) → (𝐼 ∩ 𝐽) = ∅) | |
21 | 17, 19, 20 | sylancr 587 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝐼 ∩ 𝐽) = ∅) |
22 | 16, 21 | eqtr4d 2778 | . . 3 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)) |
23 | 15, 22 | pm2.61i 182 | . 2 ⊢ (2Ideal‘𝑅) = (𝐼 ∩ 𝐽) |
24 | 1, 23 | eqtri 2763 | 1 ⊢ 𝑇 = (𝐼 ∩ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ‘cfv 6563 opprcoppr 20350 LIdealclidl 21234 2Idealc2idl 21277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-2idl 21278 |
This theorem is referenced by: 2idlelb 21281 2idllidld 21282 2idlridld 21283 2idl0 21288 2idl1 21289 qus1 21302 qusrhm 21304 crng2idl 21309 oppr2idl 33494 qsdrngilem 33502 qsdrngi 33503 |
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