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Mathbox for Thierry Arnoux |
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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 2733 | . . . . . 6 β’ (0gβπ ) = (0gβπ ) | |
4 | 2, 3 | subrg0 20271 | . . . . 5 β’ (π β (SubRingβπ ) β (0gβπ ) = (0gβπ»)) |
5 | 1, 4 | syl 17 | . . . 4 β’ (π β (0gβπ ) = (0gβπ»)) |
6 | 5 | oveq2d 7377 | . . 3 β’ (π β ((coe1βπ) supp (0gβπ )) = ((coe1βπ) supp (0gβπ»))) |
7 | 6 | supeq1d 9390 | . 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 2733 | . . . . . 6 β’ (Poly1βπ ) = (Poly1βπ ) | |
10 | ressdeg1.u | . . . . . 6 β’ π = (Poly1βπ») | |
11 | ressdeg1.b | . . . . . 6 β’ π΅ = (Baseβπ) | |
12 | eqid 2733 | . . . . . 6 β’ (PwSer1βπ») = (PwSer1βπ») | |
13 | eqid 2733 | . . . . . 6 β’ (Baseβ(PwSer1βπ»)) = (Baseβ(PwSer1βπ»)) | |
14 | eqid 2733 | . . . . . 6 β’ (Baseβ(Poly1βπ )) = (Baseβ(Poly1βπ )) | |
15 | 9, 2, 10, 11, 1, 12, 13, 14 | ressply1bas2 21622 | . . . . 5 β’ (π β π΅ = ((Baseβ(PwSer1βπ»)) β© (Baseβ(Poly1βπ )))) |
16 | 8, 15 | eleqtrd 2836 | . . . 4 β’ (π β π β ((Baseβ(PwSer1βπ»)) β© (Baseβ(Poly1βπ )))) |
17 | 16 | elin2d 4163 | . . 3 β’ (π β π β (Baseβ(Poly1βπ ))) |
18 | ressdeg1.d | . . . 4 β’ π· = ( deg1 βπ ) | |
19 | eqid 2733 | . . . 4 β’ (coe1βπ) = (coe1βπ) | |
20 | 18, 9, 14, 3, 19 | deg1val 25484 | . . 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 2733 | . . . 4 β’ ( deg1 βπ») = ( deg1 βπ») | |
23 | eqid 2733 | . . . 4 β’ (0gβπ») = (0gβπ») | |
24 | 22, 10, 11, 23, 19 | deg1val 25484 | . . 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 2783 | 1 β’ (π β (π·βπ) = (( deg1 βπ»)βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β© cin 3913 βcfv 6500 (class class class)co 7361 supp csupp 8096 supcsup 9384 β*cxr 11196 < clt 11197 Basecbs 17091 βΎs cress 17120 0gc0g 17329 SubRingcsubrg 20260 PwSer1cps1 21569 Poly1cpl1 21571 coe1cco1 21572 deg1 cdg1 25439 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-ofr 7622 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-pm 8774 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-sup 9386 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-0g 17331 df-gsum 17332 df-prds 17337 df-pws 17339 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-submnd 18610 df-grp 18759 df-minusg 18760 df-mulg 18881 df-subg 18933 df-ghm 19014 df-cntz 19105 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-subrg 20262 df-cnfld 20820 df-psr 21334 df-mpl 21336 df-opsr 21338 df-psr1 21574 df-ply1 21576 df-coe1 21577 df-mdeg 25440 df-deg1 25441 |
This theorem is referenced by: ressply1mon1p 32334 |
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