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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 | ⊢ 𝑂 = (eval1‘𝑅) |
evl1at0.p | ⊢ 𝑃 = (Poly1‘𝑅) |
evl1at0.0 | ⊢ 0 = (0g‘𝑅) |
evl1at0.z | ⊢ 𝑍 = (0g‘𝑃) |
Ref | Expression |
---|---|
evl1at0 | ⊢ (𝑅 ∈ CRing → ((𝑂‘𝑍)‘ 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 19805 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | evl1at0.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | eqid 2738 | . . . . . . 7 ⊢ (algSc‘𝑃) = (algSc‘𝑃) | |
4 | evl1at0.0 | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
5 | evl1at0.z | . . . . . . 7 ⊢ 𝑍 = (0g‘𝑃) | |
6 | 2, 3, 4, 5 | ply1scl0 21471 | . . . . . 6 ⊢ (𝑅 ∈ Ring → ((algSc‘𝑃)‘ 0 ) = 𝑍) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ CRing → ((algSc‘𝑃)‘ 0 ) = 𝑍) |
8 | 7 | eqcomd 2744 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑍 = ((algSc‘𝑃)‘ 0 )) |
9 | 8 | fveq2d 6770 | . . 3 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑍) = (𝑂‘((algSc‘𝑃)‘ 0 ))) |
10 | 9 | fveq1d 6768 | . 2 ⊢ (𝑅 ∈ CRing → ((𝑂‘𝑍)‘ 0 ) = ((𝑂‘((algSc‘𝑃)‘ 0 ))‘ 0 )) |
11 | evl1at0.o | . . . 4 ⊢ 𝑂 = (eval1‘𝑅) | |
12 | eqid 2738 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
13 | eqid 2738 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
14 | id 22 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing) | |
15 | ringgrp 19798 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
16 | 12, 4 | grpidcl 18617 | . . . . 5 ⊢ (𝑅 ∈ Grp → 0 ∈ (Base‘𝑅)) |
17 | 1, 15, 16 | 3syl 18 | . . . 4 ⊢ (𝑅 ∈ CRing → 0 ∈ (Base‘𝑅)) |
18 | 11, 2, 12, 3, 13, 14, 17, 17 | evl1scad 21511 | . . 3 ⊢ (𝑅 ∈ CRing → (((algSc‘𝑃)‘ 0 ) ∈ (Base‘𝑃) ∧ ((𝑂‘((algSc‘𝑃)‘ 0 ))‘ 0 ) = 0 )) |
19 | 18 | simprd 496 | . 2 ⊢ (𝑅 ∈ CRing → ((𝑂‘((algSc‘𝑃)‘ 0 ))‘ 0 ) = 0 ) |
20 | 10, 19 | eqtrd 2778 | 1 ⊢ (𝑅 ∈ CRing → ((𝑂‘𝑍)‘ 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ‘cfv 6426 Basecbs 16922 0gc0g 17160 Grpcgrp 18587 Ringcrg 19793 CRingccrg 19794 algSccascl 21069 Poly1cpl1 21358 eval1ce1 21490 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-ofr 7524 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-pm 8605 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-sup 9188 df-oi 9256 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-fz 13250 df-fzo 13393 df-seq 13732 df-hash 14055 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-hom 16996 df-cco 16997 df-0g 17162 df-gsum 17163 df-prds 17168 df-pws 17170 df-mre 17305 df-mrc 17306 df-acs 17308 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-mhm 18440 df-submnd 18441 df-grp 18590 df-minusg 18591 df-sbg 18592 df-mulg 18711 df-subg 18762 df-ghm 18842 df-cntz 18933 df-cmn 19398 df-abl 19399 df-mgp 19731 df-ur 19748 df-srg 19752 df-ring 19795 df-cring 19796 df-rnghom 19969 df-subrg 20032 df-lmod 20135 df-lss 20204 df-lsp 20244 df-assa 21070 df-asp 21071 df-ascl 21072 df-psr 21122 df-mvr 21123 df-mpl 21124 df-opsr 21126 df-evls 21292 df-evl 21293 df-psr1 21361 df-ply1 21363 df-evl1 21492 |
This theorem is referenced by: (None) |
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