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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 2737 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
2 | irrednu.u | . . 3 β’ π = (Unitβπ ) | |
3 | irredn0.i | . . 3 β’ πΌ = (Irredβπ ) | |
4 | eqid 2737 | . . 3 β’ (.rβπ ) = (.rβπ ) | |
5 | 1, 2, 3, 4 | isirred2 20131 | . 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 1147 | 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 846 = wceq 1542 β wcel 2107 βwral 3065 βcfv 6497 (class class class)co 7358 Basecbs 17084 .rcmulr 17135 Unitcui 20069 Irredcir 20070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-irred 20073 |
This theorem is referenced by: irredn1 20136 |
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