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Mirrors > Home > MPE Home > Th. List > evls1varsrng | Structured version Visualization version GIF version |
Description: The evaluation of the variable of univariate polynomials over subring yields the same result as evaluated as variable of the polynomials over the ring itself. (Contributed by AV, 12-Sep-2019.) |
Ref | Expression |
---|---|
evls1varsrng.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
evls1varsrng.o | ⊢ 𝑂 = (eval1‘𝑆) |
evls1varsrng.v | ⊢ 𝑉 = (var1‘𝑈) |
evls1varsrng.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evls1varsrng.b | ⊢ 𝐵 = (Base‘𝑆) |
evls1varsrng.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evls1varsrng.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
Ref | Expression |
---|---|
evls1varsrng | ⊢ (𝜑 → (𝑄‘𝑉) = (𝑂‘𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1varsrng.q | . . 3 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
2 | evls1varsrng.v | . . 3 ⊢ 𝑉 = (var1‘𝑈) | |
3 | evls1varsrng.u | . . 3 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
4 | evls1varsrng.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
5 | evls1varsrng.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
6 | evls1varsrng.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
7 | 1, 2, 3, 4, 5, 6 | evls1var 20069 | . 2 ⊢ (𝜑 → (𝑄‘𝑉) = ( I ↾ 𝐵)) |
8 | evls1varsrng.o | . . . . . 6 ⊢ 𝑂 = (eval1‘𝑆) | |
9 | 8, 4 | evl1fval1 20062 | . . . . 5 ⊢ 𝑂 = (𝑆 evalSub1 𝐵) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑂 = (𝑆 evalSub1 𝐵)) |
11 | 10 | fveq1d 6439 | . . 3 ⊢ (𝜑 → (𝑂‘𝑉) = ((𝑆 evalSub1 𝐵)‘𝑉)) |
12 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑉 = (var1‘𝑈)) |
13 | eqid 2825 | . . . . . 6 ⊢ (var1‘𝑆) = (var1‘𝑆) | |
14 | 13, 6, 3 | subrgvr1 19998 | . . . . 5 ⊢ (𝜑 → (var1‘𝑆) = (var1‘𝑈)) |
15 | 4 | ressid 16305 | . . . . . . . 8 ⊢ (𝑆 ∈ CRing → (𝑆 ↾s 𝐵) = 𝑆) |
16 | 5, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ↾s 𝐵) = 𝑆) |
17 | 16 | eqcomd 2831 | . . . . . 6 ⊢ (𝜑 → 𝑆 = (𝑆 ↾s 𝐵)) |
18 | 17 | fveq2d 6441 | . . . . 5 ⊢ (𝜑 → (var1‘𝑆) = (var1‘(𝑆 ↾s 𝐵))) |
19 | 12, 14, 18 | 3eqtr2d 2867 | . . . 4 ⊢ (𝜑 → 𝑉 = (var1‘(𝑆 ↾s 𝐵))) |
20 | 19 | fveq2d 6441 | . . 3 ⊢ (𝜑 → ((𝑆 evalSub1 𝐵)‘𝑉) = ((𝑆 evalSub1 𝐵)‘(var1‘(𝑆 ↾s 𝐵)))) |
21 | eqid 2825 | . . . 4 ⊢ (𝑆 evalSub1 𝐵) = (𝑆 evalSub1 𝐵) | |
22 | eqid 2825 | . . . 4 ⊢ (var1‘(𝑆 ↾s 𝐵)) = (var1‘(𝑆 ↾s 𝐵)) | |
23 | eqid 2825 | . . . 4 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
24 | crngring 18919 | . . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
25 | 4 | subrgid 19145 | . . . . 5 ⊢ (𝑆 ∈ Ring → 𝐵 ∈ (SubRing‘𝑆)) |
26 | 5, 24, 25 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑆)) |
27 | 21, 22, 23, 4, 5, 26 | evls1var 20069 | . . 3 ⊢ (𝜑 → ((𝑆 evalSub1 𝐵)‘(var1‘(𝑆 ↾s 𝐵))) = ( I ↾ 𝐵)) |
28 | 11, 20, 27 | 3eqtrrd 2866 | . 2 ⊢ (𝜑 → ( I ↾ 𝐵) = (𝑂‘𝑉)) |
29 | 7, 28 | eqtrd 2861 | 1 ⊢ (𝜑 → (𝑄‘𝑉) = (𝑂‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 I cid 5251 ↾ cres 5348 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 ↾s cress 16230 Ringcrg 18908 CRingccrg 18909 SubRingcsubrg 19139 var1cv1 19913 evalSub1 ces1 20045 eval1ce1 20046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-ofr 7163 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-fzo 12768 df-seq 13103 df-hash 13418 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-hom 16336 df-cco 16337 df-0g 16462 df-gsum 16463 df-prds 16468 df-pws 16470 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-ghm 18016 df-cntz 18107 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-srg 18867 df-ring 18910 df-cring 18911 df-rnghom 19078 df-subrg 19141 df-lmod 19228 df-lss 19296 df-lsp 19338 df-assa 19680 df-asp 19681 df-ascl 19682 df-psr 19724 df-mvr 19725 df-mpl 19726 df-opsr 19728 df-evls 19873 df-evl 19874 df-psr1 19917 df-vr1 19918 df-ply1 19919 df-evls1 20047 df-evl1 20048 |
This theorem is referenced by: (None) |
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