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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 2724 | . . . . . 6 β’ (0gβπ ) = (0gβπ ) | |
4 | 2, 3 | subrg0 20477 | . . . . 5 β’ (π β (SubRingβπ ) β (0gβπ ) = (0gβπ»)) |
5 | 1, 4 | syl 17 | . . . 4 β’ (π β (0gβπ ) = (0gβπ»)) |
6 | 5 | oveq2d 7418 | . . 3 β’ (π β ((coe1βπ) supp (0gβπ )) = ((coe1βπ) supp (0gβπ»))) |
7 | 6 | supeq1d 9438 | . 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 2724 | . . . . . 6 β’ (Poly1βπ ) = (Poly1βπ ) | |
10 | ressdeg1.u | . . . . . 6 β’ π = (Poly1βπ») | |
11 | ressdeg1.b | . . . . . 6 β’ π΅ = (Baseβπ) | |
12 | eqid 2724 | . . . . . 6 β’ (PwSer1βπ») = (PwSer1βπ») | |
13 | eqid 2724 | . . . . . 6 β’ (Baseβ(PwSer1βπ»)) = (Baseβ(PwSer1βπ»)) | |
14 | eqid 2724 | . . . . . 6 β’ (Baseβ(Poly1βπ )) = (Baseβ(Poly1βπ )) | |
15 | 9, 2, 10, 11, 1, 12, 13, 14 | ressply1bas2 22090 | . . . . 5 β’ (π β π΅ = ((Baseβ(PwSer1βπ»)) β© (Baseβ(Poly1βπ )))) |
16 | 8, 15 | eleqtrd 2827 | . . . 4 β’ (π β π β ((Baseβ(PwSer1βπ»)) β© (Baseβ(Poly1βπ )))) |
17 | 16 | elin2d 4192 | . . 3 β’ (π β π β (Baseβ(Poly1βπ ))) |
18 | ressdeg1.d | . . . 4 β’ π· = ( deg1 βπ ) | |
19 | eqid 2724 | . . . 4 β’ (coe1βπ) = (coe1βπ) | |
20 | 18, 9, 14, 3, 19 | deg1val 25976 | . . 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 2724 | . . . 4 β’ ( deg1 βπ») = ( deg1 βπ») | |
23 | eqid 2724 | . . . 4 β’ (0gβπ») = (0gβπ») | |
24 | 22, 10, 11, 23, 19 | deg1val 25976 | . . 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 2774 | 1 β’ (π β (π·βπ) = (( deg1 βπ»)βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β© cin 3940 βcfv 6534 (class class class)co 7402 supp csupp 8141 supcsup 9432 β*cxr 11246 < clt 11247 Basecbs 17149 βΎs cress 17178 0gc0g 17390 SubRingcsubrg 20465 PwSer1cps1 22038 Poly1cpl1 22040 coe1cco1 22041 deg1 cdg1 25931 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-ofr 7665 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18709 df-submnd 18710 df-grp 18862 df-minusg 18863 df-mulg 18992 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-subrng 20442 df-subrg 20467 df-cnfld 21235 df-psr 21792 df-mpl 21794 df-opsr 21796 df-psr1 22043 df-ply1 22045 df-coe1 22046 df-mdeg 25932 df-deg1 25933 |
This theorem is referenced by: ressply1mon1p 33142 algextdeglem7 33290 algextdeglem8 33291 |
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