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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 6541 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅)) | |
3 | 2idlval.i | . . . . . 6 ⊢ 𝐼 = (LIdeal‘𝑅) | |
4 | 2, 3 | syl6eqr 2848 | . . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼) |
5 | fveq2 6541 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅)) | |
6 | 2idlval.o | . . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅) | |
7 | 5, 6 | syl6eqr 2848 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑂) |
8 | 7 | fveq2d 6545 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = (LIdeal‘𝑂)) |
9 | 2idlval.j | . . . . . 6 ⊢ 𝐽 = (LIdeal‘𝑂) | |
10 | 8, 9 | syl6eqr 2848 | . . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = 𝐽) |
11 | 4, 10 | ineq12d 4112 | . . . 4 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))) = (𝐼 ∩ 𝐽)) |
12 | df-2idl 19694 | . . . 4 ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟)))) | |
13 | 3 | fvexi 6555 | . . . . 5 ⊢ 𝐼 ∈ V |
14 | 13 | inex1 5115 | . . . 4 ⊢ (𝐼 ∩ 𝐽) ∈ V |
15 | 11, 12, 14 | fvmpt 6638 | . . 3 ⊢ (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)) |
16 | fvprc 6534 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = ∅) | |
17 | inss1 4127 | . . . . 5 ⊢ (𝐼 ∩ 𝐽) ⊆ 𝐼 | |
18 | fvprc 6534 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (LIdeal‘𝑅) = ∅) | |
19 | 3, 18 | syl5eq 2842 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐼 = ∅) |
20 | sseq0 4275 | . . . . 5 ⊢ (((𝐼 ∩ 𝐽) ⊆ 𝐼 ∧ 𝐼 = ∅) → (𝐼 ∩ 𝐽) = ∅) | |
21 | 17, 19, 20 | sylancr 587 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝐼 ∩ 𝐽) = ∅) |
22 | 16, 21 | eqtr4d 2833 | . . 3 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)) |
23 | 15, 22 | pm2.61i 183 | . 2 ⊢ (2Ideal‘𝑅) = (𝐼 ∩ 𝐽) |
24 | 1, 23 | eqtri 2818 | 1 ⊢ 𝑇 = (𝐼 ∩ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1522 ∈ wcel 2080 Vcvv 3436 ∩ cin 3860 ⊆ wss 3861 ∅c0 4213 ‘cfv 6228 opprcoppr 19062 LIdealclidl 19632 2Idealc2idl 19693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-sbc 3708 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-br 4965 df-opab 5027 df-mpt 5044 df-id 5351 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-iota 6192 df-fun 6230 df-fv 6236 df-2idl 19694 |
This theorem is referenced by: 2idlcpbl 19696 qus1 19697 qusrhm 19699 crng2idl 19701 |
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