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Mirrors > Home > MPE Home > Th. List > evl1var | Structured version Visualization version GIF version |
Description: Polynomial evaluation maps the variable to the identity function. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
evl1var.q | ⊢ 𝑂 = (eval1‘𝑅) |
evl1var.v | ⊢ 𝑋 = (var1‘𝑅) |
evl1var.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
evl1var | ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = ( I ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 18911 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | evl1var.v | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
3 | eqid 2824 | . . . . 5 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
4 | eqid 2824 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
5 | 2, 3, 4 | vr1cl 19946 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘(Poly1‘𝑅))) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ (Base‘(Poly1‘𝑅))) |
7 | evl1var.q | . . . 4 ⊢ 𝑂 = (eval1‘𝑅) | |
8 | eqid 2824 | . . . 4 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
9 | evl1var.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
10 | eqid 2824 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
11 | eqid 2824 | . . . . 5 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
12 | 3, 11, 4 | ply1bas 19924 | . . . 4 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1o mPoly 𝑅)) |
13 | 7, 8, 9, 10, 12 | evl1val 20052 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘(Poly1‘𝑅))) → (𝑂‘𝑋) = (((1o eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
14 | 6, 13 | mpdan 680 | . 2 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = (((1o eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
15 | df1o2 7838 | . . . . 5 ⊢ 1o = {∅} | |
16 | 9 | fvexi 6446 | . . . . 5 ⊢ 𝐵 ∈ V |
17 | 0ex 5013 | . . . . 5 ⊢ ∅ ∈ V | |
18 | eqid 2824 | . . . . 5 ⊢ (𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅)) = (𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅)) | |
19 | 15, 16, 17, 18 | mapsncnv 8170 | . . . 4 ⊢ ◡(𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅)) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) |
20 | 19 | coeq2i 5514 | . . 3 ⊢ (((1o eval 𝑅)‘𝑋) ∘ ◡(𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅))) = (((1o eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) |
21 | 9 | ressid 16297 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅) |
22 | 21 | oveq2d 6920 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (1o mVar (𝑅 ↾s 𝐵)) = (1o mVar 𝑅)) |
23 | 22 | fveq1d 6434 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → ((1o mVar (𝑅 ↾s 𝐵))‘∅) = ((1o mVar 𝑅)‘∅)) |
24 | 2 | vr1val 19921 | . . . . . . 7 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) |
25 | 23, 24 | syl6eqr 2878 | . . . . . 6 ⊢ (𝑅 ∈ CRing → ((1o mVar (𝑅 ↾s 𝐵))‘∅) = 𝑋) |
26 | 25 | fveq2d 6436 | . . . . 5 ⊢ (𝑅 ∈ CRing → ((1o eval 𝑅)‘((1o mVar (𝑅 ↾s 𝐵))‘∅)) = ((1o eval 𝑅)‘𝑋)) |
27 | 8, 9 | evlval 19883 | . . . . . 6 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵) |
28 | eqid 2824 | . . . . . 6 ⊢ (1o mVar (𝑅 ↾s 𝐵)) = (1o mVar (𝑅 ↾s 𝐵)) | |
29 | eqid 2824 | . . . . . 6 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵) | |
30 | 1on 7832 | . . . . . . 7 ⊢ 1o ∈ On | |
31 | 30 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 1o ∈ On) |
32 | id 22 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing) | |
33 | 9 | subrgid 19137 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) |
34 | 1, 33 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐵 ∈ (SubRing‘𝑅)) |
35 | 0lt1o 7850 | . . . . . . 7 ⊢ ∅ ∈ 1o | |
36 | 35 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → ∅ ∈ 1o) |
37 | 27, 28, 29, 9, 31, 32, 34, 36 | evlsvar 19882 | . . . . 5 ⊢ (𝑅 ∈ CRing → ((1o eval 𝑅)‘((1o mVar (𝑅 ↾s 𝐵))‘∅)) = (𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅))) |
38 | 26, 37 | eqtr3d 2862 | . . . 4 ⊢ (𝑅 ∈ CRing → ((1o eval 𝑅)‘𝑋) = (𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅))) |
39 | 38 | coeq1d 5515 | . . 3 ⊢ (𝑅 ∈ CRing → (((1o eval 𝑅)‘𝑋) ∘ ◡(𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅))) = ((𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅)) ∘ ◡(𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅)))) |
40 | 20, 39 | syl5eqr 2874 | . 2 ⊢ (𝑅 ∈ CRing → (((1o eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = ((𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅)) ∘ ◡(𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅)))) |
41 | 15, 16, 17, 18 | mapsnf1o2 8171 | . . 3 ⊢ (𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅)):(𝐵 ↑𝑚 1o)–1-1-onto→𝐵 |
42 | f1ococnv2 6403 | . . 3 ⊢ ((𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅)):(𝐵 ↑𝑚 1o)–1-1-onto→𝐵 → ((𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅)) ∘ ◡(𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅))) = ( I ↾ 𝐵)) | |
43 | 41, 42 | mp1i 13 | . 2 ⊢ (𝑅 ∈ CRing → ((𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅)) ∘ ◡(𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ (𝑧‘∅))) = ( I ↾ 𝐵)) |
44 | 14, 40, 43 | 3eqtrd 2864 | 1 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∅c0 4143 {csn 4396 ↦ cmpt 4951 I cid 5248 × cxp 5339 ◡ccnv 5340 ↾ cres 5343 ∘ ccom 5345 Oncon0 5962 –1-1-onto→wf1o 6121 ‘cfv 6122 (class class class)co 6904 1oc1o 7818 ↑𝑚 cmap 8121 Basecbs 16221 ↾s cress 16222 Ringcrg 18900 CRingccrg 18901 SubRingcsubrg 19131 mVar cmvr 19712 mPoly cmpl 19713 eval cevl 19864 PwSer1cps1 19904 var1cv1 19905 Poly1cpl1 19906 eval1ce1 20038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-iin 4742 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-ofr 7157 df-om 7326 df-1st 7427 df-2nd 7428 df-supp 7559 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-2o 7826 df-oadd 7829 df-er 8008 df-map 8123 df-pm 8124 df-ixp 8175 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-fsupp 8544 df-sup 8616 df-oi 8683 df-card 9077 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-7 11418 df-8 11419 df-9 11420 df-n0 11618 df-z 11704 df-dec 11821 df-uz 11968 df-fz 12619 df-fzo 12760 df-seq 13095 df-hash 13410 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-sca 16320 df-vsca 16321 df-ip 16322 df-tset 16323 df-ple 16324 df-ds 16326 df-hom 16328 df-cco 16329 df-0g 16454 df-gsum 16455 df-prds 16460 df-pws 16462 df-mre 16598 df-mrc 16599 df-acs 16601 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-mhm 17687 df-submnd 17688 df-grp 17778 df-minusg 17779 df-sbg 17780 df-mulg 17894 df-subg 17941 df-ghm 18008 df-cntz 18099 df-cmn 18547 df-abl 18548 df-mgp 18843 df-ur 18855 df-srg 18859 df-ring 18902 df-cring 18903 df-rnghom 19070 df-subrg 19133 df-lmod 19220 df-lss 19288 df-lsp 19330 df-assa 19672 df-asp 19673 df-ascl 19674 df-psr 19716 df-mvr 19717 df-mpl 19718 df-opsr 19720 df-evls 19865 df-evl 19866 df-psr1 19909 df-vr1 19910 df-ply1 19911 df-evl1 20040 |
This theorem is referenced by: evl1vard 20060 evls1var 20061 pf1id 20070 fta1blem 24326 |
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