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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 6714 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅)) | |
3 | 2idlval.i | . . . . . 6 ⊢ 𝐼 = (LIdeal‘𝑅) | |
4 | 2, 3 | eqtr4di 2796 | . . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼) |
5 | fveq2 6714 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅)) | |
6 | 2idlval.o | . . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅) | |
7 | 5, 6 | eqtr4di 2796 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑂) |
8 | 7 | fveq2d 6718 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = (LIdeal‘𝑂)) |
9 | 2idlval.j | . . . . . 6 ⊢ 𝐽 = (LIdeal‘𝑂) | |
10 | 8, 9 | eqtr4di 2796 | . . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = 𝐽) |
11 | 4, 10 | ineq12d 4125 | . . . 4 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))) = (𝐼 ∩ 𝐽)) |
12 | df-2idl 20267 | . . . 4 ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟)))) | |
13 | 3 | fvexi 6728 | . . . . 5 ⊢ 𝐼 ∈ V |
14 | 13 | inex1 5207 | . . . 4 ⊢ (𝐼 ∩ 𝐽) ∈ V |
15 | 11, 12, 14 | fvmpt 6815 | . . 3 ⊢ (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)) |
16 | fvprc 6706 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = ∅) | |
17 | inss1 4140 | . . . . 5 ⊢ (𝐼 ∩ 𝐽) ⊆ 𝐼 | |
18 | fvprc 6706 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (LIdeal‘𝑅) = ∅) | |
19 | 3, 18 | syl5eq 2790 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐼 = ∅) |
20 | sseq0 4311 | . . . . 5 ⊢ (((𝐼 ∩ 𝐽) ⊆ 𝐼 ∧ 𝐼 = ∅) → (𝐼 ∩ 𝐽) = ∅) | |
21 | 17, 19, 20 | sylancr 590 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝐼 ∩ 𝐽) = ∅) |
22 | 16, 21 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)) |
23 | 15, 22 | pm2.61i 185 | . 2 ⊢ (2Ideal‘𝑅) = (𝐼 ∩ 𝐽) |
24 | 1, 23 | eqtri 2765 | 1 ⊢ 𝑇 = (𝐼 ∩ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2110 Vcvv 3405 ∩ cin 3862 ⊆ wss 3863 ∅c0 4234 ‘cfv 6377 opprcoppr 19637 LIdealclidl 20204 2Idealc2idl 20266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pr 5319 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6335 df-fun 6379 df-fv 6385 df-2idl 20267 |
This theorem is referenced by: 2idlcpbl 20269 qus1 20270 qusrhm 20272 crng2idl 20274 |
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