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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 6906 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅)) | |
| 3 | 2idlval.i | . . . . . 6 ⊢ 𝐼 = (LIdeal‘𝑅) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼) |
| 5 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅)) | |
| 6 | 2idlval.o | . . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅) | |
| 7 | 5, 6 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑂) |
| 8 | 7 | fveq2d 6910 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = (LIdeal‘𝑂)) |
| 9 | 2idlval.j | . . . . . 6 ⊢ 𝐽 = (LIdeal‘𝑂) | |
| 10 | 8, 9 | eqtr4di 2795 | . . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = 𝐽) |
| 11 | 4, 10 | ineq12d 4221 | . . . 4 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))) = (𝐼 ∩ 𝐽)) |
| 12 | df-2idl 21260 | . . . 4 ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟)))) | |
| 13 | 3 | fvexi 6920 | . . . . 5 ⊢ 𝐼 ∈ V |
| 14 | 13 | inex1 5317 | . . . 4 ⊢ (𝐼 ∩ 𝐽) ∈ V |
| 15 | 11, 12, 14 | fvmpt 7016 | . . 3 ⊢ (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)) |
| 16 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = ∅) | |
| 17 | inss1 4237 | . . . . 5 ⊢ (𝐼 ∩ 𝐽) ⊆ 𝐼 | |
| 18 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (LIdeal‘𝑅) = ∅) | |
| 19 | 3, 18 | eqtrid 2789 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐼 = ∅) |
| 20 | sseq0 4403 | . . . . 5 ⊢ (((𝐼 ∩ 𝐽) ⊆ 𝐼 ∧ 𝐼 = ∅) → (𝐼 ∩ 𝐽) = ∅) | |
| 21 | 17, 19, 20 | sylancr 587 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝐼 ∩ 𝐽) = ∅) |
| 22 | 16, 21 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)) |
| 23 | 15, 22 | pm2.61i 182 | . 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 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ‘cfv 6561 opprcoppr 20333 LIdealclidl 21216 2Idealc2idl 21259 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 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-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-2idl 21260 |
| This theorem is referenced by: 2idlelb 21263 2idllidld 21264 2idlridld 21265 2idl0 21270 2idl1 21271 qus1 21284 qusrhm 21286 crng2idl 21291 oppr2idl 33514 qsdrngilem 33522 qsdrngi 33523 |
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