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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 20429 | . 2 ⊢ (𝜑 → (𝑄‘𝑉) = ( I ↾ 𝐵)) |
8 | evls1varsrng.o | . . . . . 6 ⊢ 𝑂 = (eval1‘𝑆) | |
9 | 8, 4 | evl1fval1 20422 | . . . . 5 ⊢ 𝑂 = (𝑆 evalSub1 𝐵) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑂 = (𝑆 evalSub1 𝐵)) |
11 | 10 | fveq1d 6665 | . . 3 ⊢ (𝜑 → (𝑂‘𝑉) = ((𝑆 evalSub1 𝐵)‘𝑉)) |
12 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑉 = (var1‘𝑈)) |
13 | eqid 2818 | . . . . . 6 ⊢ (var1‘𝑆) = (var1‘𝑆) | |
14 | 13, 6, 3 | subrgvr1 20357 | . . . . 5 ⊢ (𝜑 → (var1‘𝑆) = (var1‘𝑈)) |
15 | 4 | ressid 16547 | . . . . . . . 8 ⊢ (𝑆 ∈ CRing → (𝑆 ↾s 𝐵) = 𝑆) |
16 | 5, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ↾s 𝐵) = 𝑆) |
17 | 16 | eqcomd 2824 | . . . . . 6 ⊢ (𝜑 → 𝑆 = (𝑆 ↾s 𝐵)) |
18 | 17 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → (var1‘𝑆) = (var1‘(𝑆 ↾s 𝐵))) |
19 | 12, 14, 18 | 3eqtr2d 2859 | . . . 4 ⊢ (𝜑 → 𝑉 = (var1‘(𝑆 ↾s 𝐵))) |
20 | 19 | fveq2d 6667 | . . 3 ⊢ (𝜑 → ((𝑆 evalSub1 𝐵)‘𝑉) = ((𝑆 evalSub1 𝐵)‘(var1‘(𝑆 ↾s 𝐵)))) |
21 | eqid 2818 | . . . 4 ⊢ (𝑆 evalSub1 𝐵) = (𝑆 evalSub1 𝐵) | |
22 | eqid 2818 | . . . 4 ⊢ (var1‘(𝑆 ↾s 𝐵)) = (var1‘(𝑆 ↾s 𝐵)) | |
23 | eqid 2818 | . . . 4 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
24 | crngring 19237 | . . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
25 | 4 | subrgid 19466 | . . . . 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 20429 | . . 3 ⊢ (𝜑 → ((𝑆 evalSub1 𝐵)‘(var1‘(𝑆 ↾s 𝐵))) = ( I ↾ 𝐵)) |
28 | 11, 20, 27 | 3eqtrrd 2858 | . 2 ⊢ (𝜑 → ( I ↾ 𝐵) = (𝑂‘𝑉)) |
29 | 7, 28 | eqtrd 2853 | 1 ⊢ (𝜑 → (𝑄‘𝑉) = (𝑂‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 I cid 5452 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 ↾s cress 16472 Ringcrg 19226 CRingccrg 19227 SubRingcsubrg 19460 var1cv1 20272 evalSub1 ces1 20404 eval1ce1 20405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-ofr 7399 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-0g 16703 df-gsum 16704 df-prds 16709 df-pws 16711 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-srg 19185 df-ring 19228 df-cring 19229 df-rnghom 19396 df-subrg 19462 df-lmod 19565 df-lss 19633 df-lsp 19673 df-assa 20013 df-asp 20014 df-ascl 20015 df-psr 20064 df-mvr 20065 df-mpl 20066 df-opsr 20068 df-evls 20214 df-evl 20215 df-psr1 20276 df-vr1 20277 df-ply1 20278 df-evls1 20406 df-evl1 20407 |
This theorem is referenced by: (None) |
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