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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 22287 | . 2 ⊢ (𝜑 → (𝑄‘𝑉) = ( I ↾ 𝐵)) |
8 | evls1varsrng.o | . . . . . 6 ⊢ 𝑂 = (eval1‘𝑆) | |
9 | 8, 4 | evl1fval1 22280 | . . . . 5 ⊢ 𝑂 = (𝑆 evalSub1 𝐵) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑂 = (𝑆 evalSub1 𝐵)) |
11 | 10 | fveq1d 6898 | . . 3 ⊢ (𝜑 → (𝑂‘𝑉) = ((𝑆 evalSub1 𝐵)‘𝑉)) |
12 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑉 = (var1‘𝑈)) |
13 | eqid 2725 | . . . . . 6 ⊢ (var1‘𝑆) = (var1‘𝑆) | |
14 | 13, 6, 3 | subrgvr1 22210 | . . . . 5 ⊢ (𝜑 → (var1‘𝑆) = (var1‘𝑈)) |
15 | 4 | ressid 17233 | . . . . . . . 8 ⊢ (𝑆 ∈ CRing → (𝑆 ↾s 𝐵) = 𝑆) |
16 | 5, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ↾s 𝐵) = 𝑆) |
17 | 16 | eqcomd 2731 | . . . . . 6 ⊢ (𝜑 → 𝑆 = (𝑆 ↾s 𝐵)) |
18 | 17 | fveq2d 6900 | . . . . 5 ⊢ (𝜑 → (var1‘𝑆) = (var1‘(𝑆 ↾s 𝐵))) |
19 | 12, 14, 18 | 3eqtr2d 2771 | . . . 4 ⊢ (𝜑 → 𝑉 = (var1‘(𝑆 ↾s 𝐵))) |
20 | 19 | fveq2d 6900 | . . 3 ⊢ (𝜑 → ((𝑆 evalSub1 𝐵)‘𝑉) = ((𝑆 evalSub1 𝐵)‘(var1‘(𝑆 ↾s 𝐵)))) |
21 | eqid 2725 | . . . 4 ⊢ (𝑆 evalSub1 𝐵) = (𝑆 evalSub1 𝐵) | |
22 | eqid 2725 | . . . 4 ⊢ (var1‘(𝑆 ↾s 𝐵)) = (var1‘(𝑆 ↾s 𝐵)) | |
23 | eqid 2725 | . . . 4 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
24 | crngring 20202 | . . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
25 | 4 | subrgid 20529 | . . . . 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 22287 | . . 3 ⊢ (𝜑 → ((𝑆 evalSub1 𝐵)‘(var1‘(𝑆 ↾s 𝐵))) = ( I ↾ 𝐵)) |
28 | 11, 20, 27 | 3eqtrrd 2770 | . 2 ⊢ (𝜑 → ( I ↾ 𝐵) = (𝑂‘𝑉)) |
29 | 7, 28 | eqtrd 2765 | 1 ⊢ (𝜑 → (𝑄‘𝑉) = (𝑂‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 I cid 5575 ↾ cres 5680 ‘cfv 6549 (class class class)co 7419 Basecbs 17188 ↾s cress 17217 Ringcrg 20190 CRingccrg 20191 SubRingcsubrg 20523 var1cv1 22123 evalSub1 ces1 22262 eval1ce1 22263 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-ofr 7686 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9393 df-sup 9472 df-oi 9540 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14008 df-hash 14331 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-sca 17257 df-vsca 17258 df-ip 17259 df-tset 17260 df-ple 17261 df-ds 17263 df-hom 17265 df-cco 17266 df-0g 17431 df-gsum 17432 df-prds 17437 df-pws 17439 df-mre 17574 df-mrc 17575 df-acs 17577 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18748 df-submnd 18749 df-grp 18906 df-minusg 18907 df-sbg 18908 df-mulg 19037 df-subg 19091 df-ghm 19181 df-cntz 19285 df-cmn 19754 df-abl 19755 df-mgp 20092 df-rng 20110 df-ur 20139 df-srg 20144 df-ring 20192 df-cring 20193 df-rhm 20428 df-subrng 20500 df-subrg 20525 df-lmod 20762 df-lss 20833 df-lsp 20873 df-assa 21809 df-asp 21810 df-ascl 21811 df-psr 21864 df-mvr 21865 df-mpl 21866 df-opsr 21868 df-evls 22045 df-evl 22046 df-psr1 22127 df-vr1 22128 df-ply1 22129 df-evls1 22264 df-evl1 22265 |
This theorem is referenced by: (None) |
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