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Mirrors > Home > MPE Home > Th. List > 2idlelb | Structured version Visualization version GIF version |
Description: Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21171. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.) |
Ref | Expression |
---|---|
2idlel.i | β’ πΌ = (LIdealβπ ) |
2idlel.o | β’ π = (opprβπ ) |
2idlel.j | β’ π½ = (LIdealβπ) |
2idlel.t | β’ π = (2Idealβπ ) |
Ref | Expression |
---|---|
2idlelb | β’ (π β π β (π β πΌ β§ π β π½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2idlel.i | . . 3 β’ πΌ = (LIdealβπ ) | |
2 | 2idlel.o | . . 3 β’ π = (opprβπ ) | |
3 | 2idlel.j | . . 3 β’ π½ = (LIdealβπ) | |
4 | 2idlel.t | . . 3 β’ π = (2Idealβπ ) | |
5 | 1, 2, 3, 4 | 2idlval 21150 | . 2 β’ π = (πΌ β© π½) |
6 | 5 | elin2 4197 | 1 β’ (π β π β (π β πΌ β§ π β π½)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6551 opprcoppr 20277 LIdealclidl 21107 2Idealc2idl 21148 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
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 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-iota 6503 df-fun 6553 df-fv 6559 df-2idl 21149 |
This theorem is referenced by: df2idl2rng 21155 2idlelbas 21163 rng2idlsubgsubrng 21167 2idlcpblrng 21170 2idlcpbl 21171 |
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