![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ressdeg1 | Structured version Visualization version GIF version |
Description: The degree of a univariate polynomial in a structure restriction. (Contributed by Thierry Arnoux, 20-Jan-2025.) |
Ref | Expression |
---|---|
ressdeg1.h | β’ π» = (π βΎs π) |
ressdeg1.d | β’ π· = ( deg1 βπ ) |
ressdeg1.u | β’ π = (Poly1βπ») |
ressdeg1.b | β’ π΅ = (Baseβπ) |
ressdeg1.p | β’ (π β π β π΅) |
ressdeg1.t | β’ (π β π β (SubRingβπ )) |
Ref | Expression |
---|---|
ressdeg1 | β’ (π β (π·βπ) = (( deg1 βπ»)βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressdeg1.t | . . . . 5 β’ (π β π β (SubRingβπ )) | |
2 | ressdeg1.h | . . . . . 6 β’ π» = (π βΎs π) | |
3 | eqid 2728 | . . . . . 6 β’ (0gβπ ) = (0gβπ ) | |
4 | 2, 3 | subrg0 20518 | . . . . 5 β’ (π β (SubRingβπ ) β (0gβπ ) = (0gβπ»)) |
5 | 1, 4 | syl 17 | . . . 4 β’ (π β (0gβπ ) = (0gβπ»)) |
6 | 5 | oveq2d 7436 | . . 3 β’ (π β ((coe1βπ) supp (0gβπ )) = ((coe1βπ) supp (0gβπ»))) |
7 | 6 | supeq1d 9470 | . 2 β’ (π β sup(((coe1βπ) supp (0gβπ )), β*, < ) = sup(((coe1βπ) supp (0gβπ»)), β*, < )) |
8 | ressdeg1.p | . . . . 5 β’ (π β π β π΅) | |
9 | eqid 2728 | . . . . . 6 β’ (Poly1βπ ) = (Poly1βπ ) | |
10 | ressdeg1.u | . . . . . 6 β’ π = (Poly1βπ») | |
11 | ressdeg1.b | . . . . . 6 β’ π΅ = (Baseβπ) | |
12 | eqid 2728 | . . . . . 6 β’ (PwSer1βπ») = (PwSer1βπ») | |
13 | eqid 2728 | . . . . . 6 β’ (Baseβ(PwSer1βπ»)) = (Baseβ(PwSer1βπ»)) | |
14 | eqid 2728 | . . . . . 6 β’ (Baseβ(Poly1βπ )) = (Baseβ(Poly1βπ )) | |
15 | 9, 2, 10, 11, 1, 12, 13, 14 | ressply1bas2 22146 | . . . . 5 β’ (π β π΅ = ((Baseβ(PwSer1βπ»)) β© (Baseβ(Poly1βπ )))) |
16 | 8, 15 | eleqtrd 2831 | . . . 4 β’ (π β π β ((Baseβ(PwSer1βπ»)) β© (Baseβ(Poly1βπ )))) |
17 | 16 | elin2d 4199 | . . 3 β’ (π β π β (Baseβ(Poly1βπ ))) |
18 | ressdeg1.d | . . . 4 β’ π· = ( deg1 βπ ) | |
19 | eqid 2728 | . . . 4 β’ (coe1βπ) = (coe1βπ) | |
20 | 18, 9, 14, 3, 19 | deg1val 26045 | . . 3 β’ (π β (Baseβ(Poly1βπ )) β (π·βπ) = sup(((coe1βπ) supp (0gβπ )), β*, < )) |
21 | 17, 20 | syl 17 | . 2 β’ (π β (π·βπ) = sup(((coe1βπ) supp (0gβπ )), β*, < )) |
22 | eqid 2728 | . . . 4 β’ ( deg1 βπ») = ( deg1 βπ») | |
23 | eqid 2728 | . . . 4 β’ (0gβπ») = (0gβπ») | |
24 | 22, 10, 11, 23, 19 | deg1val 26045 | . . 3 β’ (π β π΅ β (( deg1 βπ»)βπ) = sup(((coe1βπ) supp (0gβπ»)), β*, < )) |
25 | 8, 24 | syl 17 | . 2 β’ (π β (( deg1 βπ»)βπ) = sup(((coe1βπ) supp (0gβπ»)), β*, < )) |
26 | 7, 21, 25 | 3eqtr4d 2778 | 1 β’ (π β (π·βπ) = (( deg1 βπ»)βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β© cin 3946 βcfv 6548 (class class class)co 7420 supp csupp 8165 supcsup 9464 β*cxr 11278 < clt 11279 Basecbs 17180 βΎs cress 17209 0gc0g 17421 SubRingcsubrg 20506 PwSer1cps1 22094 Poly1cpl1 22096 coe1cco1 22097 deg1 cdg1 26000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-addf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-ofr 7686 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-subrng 20483 df-subrg 20508 df-cnfld 21280 df-psr 21842 df-mpl 21844 df-opsr 21846 df-psr1 22099 df-ply1 22101 df-coe1 22102 df-mdeg 26001 df-deg1 26002 |
This theorem is referenced by: ressply1mon1p 33253 algextdeglem7 33391 algextdeglem8 33392 |
Copyright terms: Public domain | W3C validator |