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Mirrors > Home > MPE Home > Th. List > irrednu | Structured version Visualization version GIF version |
Description: An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
irredn0.i | β’ πΌ = (Irredβπ ) |
irrednu.u | β’ π = (Unitβπ ) |
Ref | Expression |
---|---|
irrednu | β’ (π β πΌ β Β¬ π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
2 | irrednu.u | . . 3 β’ π = (Unitβπ ) | |
3 | irredn0.i | . . 3 β’ πΌ = (Irredβπ ) | |
4 | eqid 2727 | . . 3 β’ (.rβπ ) = (.rβπ ) | |
5 | 1, 2, 3, 4 | isirred2 20365 | . 2 β’ (π β πΌ β (π β (Baseβπ ) β§ Β¬ π β π β§ βπ₯ β (Baseβπ )βπ¦ β (Baseβπ )((π₯(.rβπ )π¦) = π β (π₯ β π β¨ π¦ β π)))) |
6 | 5 | simp2bi 1143 | 1 β’ (π β πΌ β Β¬ π β π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β¨ wo 845 = wceq 1533 β wcel 2098 βwral 3057 βcfv 6551 (class class class)co 7424 Basecbs 17185 .rcmulr 17239 Unitcui 20299 Irredcir 20300 |
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-sbc 3777 df-csb 3893 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-ov 7427 df-irred 20303 |
This theorem is referenced by: irredn1 20370 mxidlirred 33203 irredminply 33389 |
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