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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > evl1at0 | Structured version Visualization version GIF version |
Description: Polynomial evaluation for the 0 scalar. (Contributed by AV, 10-Aug-2019.) |
Ref | Expression |
---|---|
evl1at0.o | β’ π = (eval1βπ ) |
evl1at0.p | β’ π = (Poly1βπ ) |
evl1at0.0 | β’ 0 = (0gβπ ) |
evl1at0.z | β’ π = (0gβπ) |
Ref | Expression |
---|---|
evl1at0 | β’ (π β CRing β ((πβπ)β 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 20142 | . . . . . 6 β’ (π β CRing β π β Ring) | |
2 | evl1at0.p | . . . . . . 7 β’ π = (Poly1βπ ) | |
3 | eqid 2724 | . . . . . . 7 β’ (algScβπ) = (algScβπ) | |
4 | evl1at0.0 | . . . . . . 7 β’ 0 = (0gβπ ) | |
5 | evl1at0.z | . . . . . . 7 β’ π = (0gβπ) | |
6 | 2, 3, 4, 5 | ply1scl0 22133 | . . . . . 6 β’ (π β Ring β ((algScβπ)β 0 ) = π) |
7 | 1, 6 | syl 17 | . . . . 5 β’ (π β CRing β ((algScβπ)β 0 ) = π) |
8 | 7 | eqcomd 2730 | . . . 4 β’ (π β CRing β π = ((algScβπ)β 0 )) |
9 | 8 | fveq2d 6886 | . . 3 β’ (π β CRing β (πβπ) = (πβ((algScβπ)β 0 ))) |
10 | 9 | fveq1d 6884 | . 2 β’ (π β CRing β ((πβπ)β 0 ) = ((πβ((algScβπ)β 0 ))β 0 )) |
11 | evl1at0.o | . . . 4 β’ π = (eval1βπ ) | |
12 | eqid 2724 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
13 | eqid 2724 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
14 | id 22 | . . . 4 β’ (π β CRing β π β CRing) | |
15 | ringgrp 20135 | . . . . 5 β’ (π β Ring β π β Grp) | |
16 | 12, 4 | grpidcl 18887 | . . . . 5 β’ (π β Grp β 0 β (Baseβπ )) |
17 | 1, 15, 16 | 3syl 18 | . . . 4 β’ (π β CRing β 0 β (Baseβπ )) |
18 | 11, 2, 12, 3, 13, 14, 17, 17 | evl1scad 22178 | . . 3 β’ (π β CRing β (((algScβπ)β 0 ) β (Baseβπ) β§ ((πβ((algScβπ)β 0 ))β 0 ) = 0 )) |
19 | 18 | simprd 495 | . 2 β’ (π β CRing β ((πβ((algScβπ)β 0 ))β 0 ) = 0 ) |
20 | 10, 19 | eqtrd 2764 | 1 β’ (π β CRing β ((πβπ)β 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6534 Basecbs 17145 0gc0g 17386 Grpcgrp 18855 Ringcrg 20130 CRingccrg 20131 algSccascl 21717 Poly1cpl1 22021 eval1ce1 22157 |
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 |
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 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-fz 13483 df-fzo 13626 df-seq 13965 df-hash 14289 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-ip 17216 df-tset 17217 df-ple 17218 df-ds 17220 df-hom 17222 df-cco 17223 df-0g 17388 df-gsum 17389 df-prds 17394 df-pws 17396 df-mre 17531 df-mrc 17532 df-acs 17534 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18988 df-subg 19042 df-ghm 19131 df-cntz 19225 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-srg 20084 df-ring 20132 df-cring 20133 df-rhm 20366 df-subrng 20438 df-subrg 20463 df-lmod 20700 df-lss 20771 df-lsp 20811 df-assa 21718 df-asp 21719 df-ascl 21720 df-psr 21773 df-mvr 21774 df-mpl 21775 df-opsr 21777 df-evls 21947 df-evl 21948 df-psr1 22024 df-ply1 22026 df-evl1 22159 |
This theorem is referenced by: (None) |
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