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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 6902 | . . . . . 6 β’ (π = π β (LIdealβπ) = (LIdealβπ )) | |
3 | 2idlval.i | . . . . . 6 β’ πΌ = (LIdealβπ ) | |
4 | 2, 3 | eqtr4di 2786 | . . . . 5 β’ (π = π β (LIdealβπ) = πΌ) |
5 | fveq2 6902 | . . . . . . . 8 β’ (π = π β (opprβπ) = (opprβπ )) | |
6 | 2idlval.o | . . . . . . . 8 β’ π = (opprβπ ) | |
7 | 5, 6 | eqtr4di 2786 | . . . . . . 7 β’ (π = π β (opprβπ) = π) |
8 | 7 | fveq2d 6906 | . . . . . 6 β’ (π = π β (LIdealβ(opprβπ)) = (LIdealβπ)) |
9 | 2idlval.j | . . . . . 6 β’ π½ = (LIdealβπ) | |
10 | 8, 9 | eqtr4di 2786 | . . . . 5 β’ (π = π β (LIdealβ(opprβπ)) = π½) |
11 | 4, 10 | ineq12d 4215 | . . . 4 β’ (π = π β ((LIdealβπ) β© (LIdealβ(opprβπ))) = (πΌ β© π½)) |
12 | df-2idl 21151 | . . . 4 β’ 2Ideal = (π β V β¦ ((LIdealβπ) β© (LIdealβ(opprβπ)))) | |
13 | 3 | fvexi 6916 | . . . . 5 β’ πΌ β V |
14 | 13 | inex1 5321 | . . . 4 β’ (πΌ β© π½) β V |
15 | 11, 12, 14 | fvmpt 7010 | . . 3 β’ (π β V β (2Idealβπ ) = (πΌ β© π½)) |
16 | fvprc 6894 | . . . 4 β’ (Β¬ π β V β (2Idealβπ ) = β ) | |
17 | inss1 4231 | . . . . 5 β’ (πΌ β© π½) β πΌ | |
18 | fvprc 6894 | . . . . . 6 β’ (Β¬ π β V β (LIdealβπ ) = β ) | |
19 | 3, 18 | eqtrid 2780 | . . . . 5 β’ (Β¬ π β V β πΌ = β ) |
20 | sseq0 4403 | . . . . 5 β’ (((πΌ β© π½) β πΌ β§ πΌ = β ) β (πΌ β© π½) = β ) | |
21 | 17, 19, 20 | sylancr 585 | . . . 4 β’ (Β¬ π β V β (πΌ β© π½) = β ) |
22 | 16, 21 | eqtr4d 2771 | . . 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 2756 | 1 β’ π = (πΌ β© π½) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1533 β wcel 2098 Vcvv 3473 β© cin 3948 β wss 3949 β c0 4326 βcfv 6553 opprcoppr 20279 LIdealclidl 21109 2Idealc2idl 21150 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-2idl 21151 |
This theorem is referenced by: 2idlelb 21154 2idllidld 21155 2idlridld 21156 2idl0 21161 2idl1 21162 qus1 21175 qusrhm 21177 crng2idl 21180 oppr2idl 33222 qsdrngilem 33230 qsdrngi 33231 |
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