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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 26077 | . . . 4 β’ (π β NzRing β π β NzRing) |
3 | nzrring 20462 | . . . 4 β’ (π β NzRing β π β Ring) | |
4 | eqid 2728 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
5 | mon1pid.o | . . . . 5 β’ 1 = (1rβπ) | |
6 | 4, 5 | ringidcl 20209 | . . . 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 2728 | . . . . 5 β’ (0gβπ) = (0gβπ) | |
9 | 5, 8 | nzrnz 20461 | . . . 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 20462 | . . . . . . . 8 β’ (π β NzRing β π β Ring) | |
12 | eqid 2728 | . . . . . . . . 9 β’ (algScβπ) = (algScβπ) | |
13 | eqid 2728 | . . . . . . . . 9 β’ (1rβπ ) = (1rβπ ) | |
14 | 1, 12, 13, 5 | ply1scl1 22219 | . . . . . . . 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 6906 | . . . . . 6 β’ (π β NzRing β (coe1β((algScβπ)β(1rβπ ))) = (coe1β 1 )) |
17 | eqid 2728 | . . . . . . . 8 β’ (Baseβπ ) = (Baseβπ ) | |
18 | 17, 13 | ringidcl 20209 | . . . . . . 7 β’ (π β Ring β (1rβπ ) β (Baseβπ )) |
19 | eqid 2728 | . . . . . . . 8 β’ (0gβπ ) = (0gβπ ) | |
20 | 1, 12, 17, 19 | coe1scl 22213 | . . . . . . 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 583 | . . . . . 6 β’ (π β NzRing β (coe1β((algScβπ)β(1rβπ ))) = (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))) |
22 | 16, 21 | eqtr3d 2770 | . . . . 5 β’ (π β NzRing β (coe1β 1 ) = (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))) |
23 | 15 | fveq2d 6906 | . . . . . 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 20461 | . . . . . . 7 β’ (π β NzRing β (1rβπ ) β (0gβπ )) |
26 | mon1pid.d | . . . . . . . 8 β’ π· = ( deg1 βπ ) | |
27 | 26, 1, 17, 12, 19 | deg1scl 26069 | . . . . . . 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 2770 | . . . . 5 β’ (π β NzRing β (π·β 1 ) = 0) |
30 | 22, 29 | fveq12d 6909 | . . . 4 β’ (π β NzRing β ((coe1β 1 )β(π·β 1 )) = ((π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))β0)) |
31 | 0nn0 12525 | . . . . 5 β’ 0 β β0 | |
32 | iftrue 4538 | . . . . . 6 β’ (π₯ = 0 β if(π₯ = 0, (1rβπ ), (0gβπ )) = (1rβπ )) | |
33 | eqid 2728 | . . . . . 6 β’ (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ ))) = (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ ))) | |
34 | fvex 6915 | . . . . . 6 β’ (1rβπ ) β V | |
35 | 32, 33, 34 | fvmpt 7010 | . . . . 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 2784 | . . 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 26098 | . . 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 510 | 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 394 = wceq 1533 β wcel 2098 β wne 2937 ifcif 4532 β¦ cmpt 5235 βcfv 6553 0cc0 11146 β0cn0 12510 Basecbs 17187 0gc0g 17428 1rcur 20128 Ringcrg 20180 NzRingcnzr 20458 algSccascl 21793 Poly1cpl1 22103 coe1cco1 22104 deg1 cdg1 26007 Monic1pcmn1 26081 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-nzr 20459 df-subrng 20490 df-subrg 20515 df-lmod 20752 df-lss 20823 df-cnfld 21287 df-ascl 21796 df-psr 21849 df-mvr 21850 df-mpl 21851 df-opsr 21853 df-psr1 22106 df-vr1 22107 df-ply1 22108 df-coe1 22109 df-mdeg 26008 df-deg1 26009 df-mon1 26086 |
This theorem is referenced by: ply1unit 33293 mon1psubm 42658 deg1mhm 42659 |
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