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Mirrors > Home > MPE Home > Th. List > mon1pid | Structured version Visualization version GIF version |
Description: Monicity and degree of the unit polynomial. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Ref | Expression |
---|---|
mon1pid.p | β’ π = (Poly1βπ ) |
mon1pid.o | β’ 1 = (1rβπ) |
mon1pid.m | β’ π = (Monic1pβπ ) |
mon1pid.d | β’ π· = ( deg1 βπ ) |
Ref | Expression |
---|---|
mon1pid | β’ (π β NzRing β ( 1 β π β§ (π·β 1 ) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mon1pid.p | . . . . 5 β’ π = (Poly1βπ ) | |
2 | 1 | ply1nz 26007 | . . . 4 β’ (π β NzRing β π β NzRing) |
3 | nzrring 20415 | . . . 4 β’ (π β NzRing β π β Ring) | |
4 | eqid 2726 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
5 | mon1pid.o | . . . . 5 β’ 1 = (1rβπ) | |
6 | 4, 5 | ringidcl 20162 | . . . 4 β’ (π β Ring β 1 β (Baseβπ)) |
7 | 2, 3, 6 | 3syl 18 | . . 3 β’ (π β NzRing β 1 β (Baseβπ)) |
8 | eqid 2726 | . . . . 5 β’ (0gβπ) = (0gβπ) | |
9 | 5, 8 | nzrnz 20414 | . . . 4 β’ (π β NzRing β 1 β (0gβπ)) |
10 | 2, 9 | syl 17 | . . 3 β’ (π β NzRing β 1 β (0gβπ)) |
11 | nzrring 20415 | . . . . . . . 8 β’ (π β NzRing β π β Ring) | |
12 | eqid 2726 | . . . . . . . . 9 β’ (algScβπ) = (algScβπ) | |
13 | eqid 2726 | . . . . . . . . 9 β’ (1rβπ ) = (1rβπ ) | |
14 | 1, 12, 13, 5 | ply1scl1 22162 | . . . . . . . 8 β’ (π β Ring β ((algScβπ)β(1rβπ )) = 1 ) |
15 | 11, 14 | syl 17 | . . . . . . 7 β’ (π β NzRing β ((algScβπ)β(1rβπ )) = 1 ) |
16 | 15 | fveq2d 6888 | . . . . . 6 β’ (π β NzRing β (coe1β((algScβπ)β(1rβπ ))) = (coe1β 1 )) |
17 | eqid 2726 | . . . . . . . 8 β’ (Baseβπ ) = (Baseβπ ) | |
18 | 17, 13 | ringidcl 20162 | . . . . . . 7 β’ (π β Ring β (1rβπ ) β (Baseβπ )) |
19 | eqid 2726 | . . . . . . . 8 β’ (0gβπ ) = (0gβπ ) | |
20 | 1, 12, 17, 19 | coe1scl 22156 | . . . . . . 7 β’ ((π β Ring β§ (1rβπ ) β (Baseβπ )) β (coe1β((algScβπ)β(1rβπ ))) = (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))) |
21 | 11, 18, 20 | syl2anc2 584 | . . . . . 6 β’ (π β NzRing β (coe1β((algScβπ)β(1rβπ ))) = (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))) |
22 | 16, 21 | eqtr3d 2768 | . . . . 5 β’ (π β NzRing β (coe1β 1 ) = (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))) |
23 | 15 | fveq2d 6888 | . . . . . 6 β’ (π β NzRing β (π·β((algScβπ)β(1rβπ ))) = (π·β 1 )) |
24 | 11, 18 | syl 17 | . . . . . . 7 β’ (π β NzRing β (1rβπ ) β (Baseβπ )) |
25 | 13, 19 | nzrnz 20414 | . . . . . . 7 β’ (π β NzRing β (1rβπ ) β (0gβπ )) |
26 | mon1pid.d | . . . . . . . 8 β’ π· = ( deg1 βπ ) | |
27 | 26, 1, 17, 12, 19 | deg1scl 25999 | . . . . . . 7 β’ ((π β Ring β§ (1rβπ ) β (Baseβπ ) β§ (1rβπ ) β (0gβπ )) β (π·β((algScβπ)β(1rβπ ))) = 0) |
28 | 11, 24, 25, 27 | syl3anc 1368 | . . . . . 6 β’ (π β NzRing β (π·β((algScβπ)β(1rβπ ))) = 0) |
29 | 23, 28 | eqtr3d 2768 | . . . . 5 β’ (π β NzRing β (π·β 1 ) = 0) |
30 | 22, 29 | fveq12d 6891 | . . . 4 β’ (π β NzRing β ((coe1β 1 )β(π·β 1 )) = ((π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))β0)) |
31 | 0nn0 12488 | . . . . 5 β’ 0 β β0 | |
32 | iftrue 4529 | . . . . . 6 β’ (π₯ = 0 β if(π₯ = 0, (1rβπ ), (0gβπ )) = (1rβπ )) | |
33 | eqid 2726 | . . . . . 6 β’ (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ ))) = (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ ))) | |
34 | fvex 6897 | . . . . . 6 β’ (1rβπ ) β V | |
35 | 32, 33, 34 | fvmpt 6991 | . . . . 5 β’ (0 β β0 β ((π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))β0) = (1rβπ )) |
36 | 31, 35 | ax-mp 5 | . . . 4 β’ ((π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))β0) = (1rβπ ) |
37 | 30, 36 | eqtrdi 2782 | . . 3 β’ (π β NzRing β ((coe1β 1 )β(π·β 1 )) = (1rβπ )) |
38 | mon1pid.m | . . . 4 β’ π = (Monic1pβπ ) | |
39 | 1, 4, 8, 26, 38, 13 | ismon1p 26028 | . . 3 β’ ( 1 β π β ( 1 β (Baseβπ) β§ 1 β (0gβπ) β§ ((coe1β 1 )β(π·β 1 )) = (1rβπ ))) |
40 | 7, 10, 37, 39 | syl3anbrc 1340 | . 2 β’ (π β NzRing β 1 β π) |
41 | 40, 29 | jca 511 | 1 β’ (π β NzRing β ( 1 β π β§ (π·β 1 ) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2934 ifcif 4523 β¦ cmpt 5224 βcfv 6536 0cc0 11109 β0cn0 12473 Basecbs 17150 0gc0g 17391 1rcur 20083 Ringcrg 20135 NzRingcnzr 20411 algSccascl 21742 Poly1cpl1 22046 coe1cco1 22047 deg1 cdg1 25937 Monic1pcmn1 26011 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14293 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-hom 17227 df-cco 17228 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-submnd 18711 df-grp 18863 df-minusg 18864 df-sbg 18865 df-mulg 18993 df-subg 19047 df-ghm 19136 df-cntz 19230 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-cring 20138 df-nzr 20412 df-subrng 20443 df-subrg 20468 df-lmod 20705 df-lss 20776 df-cnfld 21236 df-ascl 21745 df-psr 21798 df-mvr 21799 df-mpl 21800 df-opsr 21802 df-psr1 22049 df-vr1 22050 df-ply1 22051 df-coe1 22052 df-mdeg 25938 df-deg1 25939 df-mon1 26016 |
This theorem is referenced by: ply1unit 33163 mon1psubm 42506 deg1mhm 42507 |
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