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Mirrors > Home > MPE Home > Th. List > Mathboxes > evl1at1 | Structured version Visualization version GIF version |
Description: Polynomial evaluation for the 1 scalar. (Contributed by AV, 10-Aug-2019.) |
Ref | Expression |
---|---|
evl1at0.o | ⊢ 𝑂 = (eval1‘𝑅) |
evl1at0.p | ⊢ 𝑃 = (Poly1‘𝑅) |
evl1at1.1 | ⊢ 1 = (1r‘𝑅) |
evl1at1.i | ⊢ 𝐼 = (1r‘𝑃) |
Ref | Expression |
---|---|
evl1at1 | ⊢ (𝑅 ∈ CRing → ((𝑂‘𝐼)‘ 1 ) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 19031 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | evl1at0.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | eqid 2779 | . . . . . . 7 ⊢ (algSc‘𝑃) = (algSc‘𝑃) | |
4 | evl1at1.1 | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
5 | evl1at1.i | . . . . . . 7 ⊢ 𝐼 = (1r‘𝑃) | |
6 | 2, 3, 4, 5 | ply1scl1 20163 | . . . . . 6 ⊢ (𝑅 ∈ Ring → ((algSc‘𝑃)‘ 1 ) = 𝐼) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ CRing → ((algSc‘𝑃)‘ 1 ) = 𝐼) |
8 | 7 | eqcomd 2785 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝐼 = ((algSc‘𝑃)‘ 1 )) |
9 | 8 | fveq2d 6503 | . . 3 ⊢ (𝑅 ∈ CRing → (𝑂‘𝐼) = (𝑂‘((algSc‘𝑃)‘ 1 ))) |
10 | 9 | fveq1d 6501 | . 2 ⊢ (𝑅 ∈ CRing → ((𝑂‘𝐼)‘ 1 ) = ((𝑂‘((algSc‘𝑃)‘ 1 ))‘ 1 )) |
11 | evl1at0.o | . . . 4 ⊢ 𝑂 = (eval1‘𝑅) | |
12 | eqid 2779 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
13 | eqid 2779 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
14 | id 22 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing) | |
15 | 12, 4 | ringidcl 19041 | . . . . 5 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
16 | 1, 15 | syl 17 | . . . 4 ⊢ (𝑅 ∈ CRing → 1 ∈ (Base‘𝑅)) |
17 | 11, 2, 12, 3, 13, 14, 16, 16 | evl1scad 20200 | . . 3 ⊢ (𝑅 ∈ CRing → (((algSc‘𝑃)‘ 1 ) ∈ (Base‘𝑃) ∧ ((𝑂‘((algSc‘𝑃)‘ 1 ))‘ 1 ) = 1 )) |
18 | 17 | simprd 488 | . 2 ⊢ (𝑅 ∈ CRing → ((𝑂‘((algSc‘𝑃)‘ 1 ))‘ 1 ) = 1 ) |
19 | 10, 18 | eqtrd 2815 | 1 ⊢ (𝑅 ∈ CRing → ((𝑂‘𝐼)‘ 1 ) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ‘cfv 6188 Basecbs 16339 1rcur 18974 Ringcrg 19020 CRingccrg 19021 algSccascl 19805 Poly1cpl1 20048 eval1ce1 20180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-ofr 7228 df-om 7397 df-1st 7501 df-2nd 7502 df-supp 7634 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fsupp 8629 df-sup 8701 df-oi 8769 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-fz 12709 df-fzo 12850 df-seq 13185 df-hash 13506 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-hom 16445 df-cco 16446 df-0g 16571 df-gsum 16572 df-prds 16577 df-pws 16579 df-mre 16715 df-mrc 16716 df-acs 16718 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-mhm 17803 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-mulg 18012 df-subg 18060 df-ghm 18127 df-cntz 18218 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-srg 18979 df-ring 19022 df-cring 19023 df-rnghom 19190 df-subrg 19256 df-lmod 19358 df-lss 19426 df-lsp 19466 df-assa 19806 df-asp 19807 df-ascl 19808 df-psr 19850 df-mvr 19851 df-mpl 19852 df-opsr 19854 df-evls 19999 df-evl 20000 df-psr1 20051 df-ply1 20053 df-evl1 20182 |
This theorem is referenced by: (None) |
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