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Mirrors > Home > MPE Home > Th. List > uc1pn0 | Structured version Visualization version GIF version |
Description: Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
uc1pn0.p | β’ π = (Poly1βπ ) |
uc1pn0.z | β’ 0 = (0gβπ) |
uc1pn0.c | β’ πΆ = (Unic1pβπ ) |
Ref | Expression |
---|---|
uc1pn0 | β’ (πΉ β πΆ β πΉ β 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uc1pn0.p | . . 3 β’ π = (Poly1βπ ) | |
2 | eqid 2730 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
3 | uc1pn0.z | . . 3 β’ 0 = (0gβπ) | |
4 | eqid 2730 | . . 3 β’ ( deg1 βπ ) = ( deg1 βπ ) | |
5 | uc1pn0.c | . . 3 β’ πΆ = (Unic1pβπ ) | |
6 | eqid 2730 | . . 3 β’ (Unitβπ ) = (Unitβπ ) | |
7 | 1, 2, 3, 4, 5, 6 | isuc1p 25892 | . 2 β’ (πΉ β πΆ β (πΉ β (Baseβπ) β§ πΉ β 0 β§ ((coe1βπΉ)β(( deg1 βπ )βπΉ)) β (Unitβπ ))) |
8 | 7 | simp2bi 1144 | 1 β’ (πΉ β πΆ β πΉ β 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 β wne 2938 βcfv 6544 Basecbs 17150 0gc0g 17391 Unitcui 20248 Poly1cpl1 21922 coe1cco1 21923 deg1 cdg1 25803 Unic1pcuc1p 25878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-1cn 11172 ax-addcl 11174 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7416 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-nn 12219 df-slot 17121 df-ndx 17133 df-base 17151 df-uc1p 25883 |
This theorem is referenced by: uc1pdeg 25899 q1peqb 25906 r1pid2 32952 |
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