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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 | β’ πΌ = (LIdealβπ ) |
2idlval.o | β’ π = (opprβπ ) |
2idlval.j | β’ π½ = (LIdealβπ) |
2idlval.t | β’ π = (2Idealβπ ) |
Ref | Expression |
---|---|
2idlval | β’ π = (πΌ β© π½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2idlval.t | . 2 β’ π = (2Idealβπ ) | |
2 | fveq2 6884 | . . . . . 6 β’ (π = π β (LIdealβπ) = (LIdealβπ )) | |
3 | 2idlval.i | . . . . . 6 β’ πΌ = (LIdealβπ ) | |
4 | 2, 3 | eqtr4di 2784 | . . . . 5 β’ (π = π β (LIdealβπ) = πΌ) |
5 | fveq2 6884 | . . . . . . . 8 β’ (π = π β (opprβπ) = (opprβπ )) | |
6 | 2idlval.o | . . . . . . . 8 β’ π = (opprβπ ) | |
7 | 5, 6 | eqtr4di 2784 | . . . . . . 7 β’ (π = π β (opprβπ) = π) |
8 | 7 | fveq2d 6888 | . . . . . 6 β’ (π = π β (LIdealβ(opprβπ)) = (LIdealβπ)) |
9 | 2idlval.j | . . . . . 6 β’ π½ = (LIdealβπ) | |
10 | 8, 9 | eqtr4di 2784 | . . . . 5 β’ (π = π β (LIdealβ(opprβπ)) = π½) |
11 | 4, 10 | ineq12d 4208 | . . . 4 β’ (π = π β ((LIdealβπ) β© (LIdealβ(opprβπ))) = (πΌ β© π½)) |
12 | df-2idl 21104 | . . . 4 β’ 2Ideal = (π β V β¦ ((LIdealβπ) β© (LIdealβ(opprβπ)))) | |
13 | 3 | fvexi 6898 | . . . . 5 β’ πΌ β V |
14 | 13 | inex1 5310 | . . . 4 β’ (πΌ β© π½) β V |
15 | 11, 12, 14 | fvmpt 6991 | . . 3 β’ (π β V β (2Idealβπ ) = (πΌ β© π½)) |
16 | fvprc 6876 | . . . 4 β’ (Β¬ π β V β (2Idealβπ ) = β ) | |
17 | inss1 4223 | . . . . 5 β’ (πΌ β© π½) β πΌ | |
18 | fvprc 6876 | . . . . . 6 β’ (Β¬ π β V β (LIdealβπ ) = β ) | |
19 | 3, 18 | eqtrid 2778 | . . . . 5 β’ (Β¬ π β V β πΌ = β ) |
20 | sseq0 4394 | . . . . 5 β’ (((πΌ β© π½) β πΌ β§ πΌ = β ) β (πΌ β© π½) = β ) | |
21 | 17, 19, 20 | sylancr 586 | . . . 4 β’ (Β¬ π β V β (πΌ β© π½) = β ) |
22 | 16, 21 | eqtr4d 2769 | . . 3 β’ (Β¬ π β V β (2Idealβπ ) = (πΌ β© π½)) |
23 | 15, 22 | pm2.61i 182 | . 2 β’ (2Idealβπ ) = (πΌ β© π½) |
24 | 1, 23 | eqtri 2754 | 1 β’ π = (πΌ β© π½) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1533 β wcel 2098 Vcvv 3468 β© cin 3942 β wss 3943 β c0 4317 βcfv 6536 opprcoppr 20232 LIdealclidl 21062 2Idealc2idl 21103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-2idl 21104 |
This theorem is referenced by: 2idlelb 21107 2idllidld 21108 2idlridld 21109 2idl0 21114 2idl1 21115 qus1 21128 qusrhm 21130 crng2idl 21133 oppr2idl 33105 qsdrngilem 33113 qsdrngi 33114 |
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