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Mirrors > Home > MPE Home > Th. List > irredcl | Structured version Visualization version GIF version |
Description: An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
irredn0.i | β’ πΌ = (Irredβπ ) |
irredcl.b | β’ π΅ = (Baseβπ ) |
Ref | Expression |
---|---|
irredcl | β’ (π β πΌ β π β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | irredcl.b | . . 3 β’ π΅ = (Baseβπ ) | |
2 | eqid 2732 | . . 3 β’ (Unitβπ ) = (Unitβπ ) | |
3 | irredn0.i | . . 3 β’ πΌ = (Irredβπ ) | |
4 | eqid 2732 | . . 3 β’ (.rβπ ) = (.rβπ ) | |
5 | 1, 2, 3, 4 | isirred2 20234 | . 2 β’ (π β πΌ β (π β π΅ β§ Β¬ π β (Unitβπ ) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯(.rβπ )π¦) = π β (π₯ β (Unitβπ ) β¨ π¦ β (Unitβπ ))))) |
6 | 5 | simp1bi 1145 | 1 β’ (π β πΌ β π β π΅) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β¨ wo 845 = wceq 1541 β wcel 2106 βwral 3061 βcfv 6543 (class class class)co 7408 Basecbs 17143 .rcmulr 17197 Unitcui 20168 Irredcir 20169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7411 df-irred 20172 |
This theorem is referenced by: irredrmul 20240 irredneg 20243 prmirred 21043 |
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