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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 18558 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | evl1var.v | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
3 | eqid 2622 | . . . . 5 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
4 | eqid 2622 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
5 | 2, 3, 4 | vr1cl 19587 | . . . 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 2622 | . . . 4 ⊢ (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅) | |
9 | evl1var.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
10 | eqid 2622 | . . . 4 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅) | |
11 | eqid 2622 | . . . . 5 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
12 | 3, 11, 4 | ply1bas 19565 | . . . 4 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1𝑜 mPoly 𝑅)) |
13 | 7, 8, 9, 10, 12 | evl1val 19693 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘(Poly1‘𝑅))) → (𝑂‘𝑋) = (((1𝑜 eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) |
14 | 6, 13 | mpdan 702 | . 2 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = (((1𝑜 eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) |
15 | df1o2 7572 | . . . . 5 ⊢ 1𝑜 = {∅} | |
16 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
17 | 9, 16 | eqeltri 2697 | . . . . 5 ⊢ 𝐵 ∈ V |
18 | 0ex 4790 | . . . . 5 ⊢ ∅ ∈ V | |
19 | eqid 2622 | . . . . 5 ⊢ (𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)) = (𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)) | |
20 | 15, 17, 18, 19 | mapsncnv 7904 | . . . 4 ⊢ ◡(𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)) = (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})) |
21 | 20 | coeq2i 5282 | . . 3 ⊢ (((1𝑜 eval 𝑅)‘𝑋) ∘ ◡(𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅))) = (((1𝑜 eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))) |
22 | 9 | ressid 15935 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅) |
23 | 22 | oveq2d 6666 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (1𝑜 mVar (𝑅 ↾s 𝐵)) = (1𝑜 mVar 𝑅)) |
24 | 23 | fveq1d 6193 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → ((1𝑜 mVar (𝑅 ↾s 𝐵))‘∅) = ((1𝑜 mVar 𝑅)‘∅)) |
25 | 2 | vr1val 19562 | . . . . . . 7 ⊢ 𝑋 = ((1𝑜 mVar 𝑅)‘∅) |
26 | 24, 25 | syl6eqr 2674 | . . . . . 6 ⊢ (𝑅 ∈ CRing → ((1𝑜 mVar (𝑅 ↾s 𝐵))‘∅) = 𝑋) |
27 | 26 | fveq2d 6195 | . . . . 5 ⊢ (𝑅 ∈ CRing → ((1𝑜 eval 𝑅)‘((1𝑜 mVar (𝑅 ↾s 𝐵))‘∅)) = ((1𝑜 eval 𝑅)‘𝑋)) |
28 | 8, 9 | evlval 19524 | . . . . . 6 ⊢ (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘𝐵) |
29 | eqid 2622 | . . . . . 6 ⊢ (1𝑜 mVar (𝑅 ↾s 𝐵)) = (1𝑜 mVar (𝑅 ↾s 𝐵)) | |
30 | eqid 2622 | . . . . . 6 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵) | |
31 | 1on 7567 | . . . . . . 7 ⊢ 1𝑜 ∈ On | |
32 | 31 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 1𝑜 ∈ On) |
33 | id 22 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing) | |
34 | 9 | subrgid 18782 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) |
35 | 1, 34 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐵 ∈ (SubRing‘𝑅)) |
36 | 0lt1o 7584 | . . . . . . 7 ⊢ ∅ ∈ 1𝑜 | |
37 | 36 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → ∅ ∈ 1𝑜) |
38 | 28, 29, 30, 9, 32, 33, 35, 37 | evlsvar 19523 | . . . . 5 ⊢ (𝑅 ∈ CRing → ((1𝑜 eval 𝑅)‘((1𝑜 mVar (𝑅 ↾s 𝐵))‘∅)) = (𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅))) |
39 | 27, 38 | eqtr3d 2658 | . . . 4 ⊢ (𝑅 ∈ CRing → ((1𝑜 eval 𝑅)‘𝑋) = (𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅))) |
40 | 39 | coeq1d 5283 | . . 3 ⊢ (𝑅 ∈ CRing → (((1𝑜 eval 𝑅)‘𝑋) ∘ ◡(𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅))) = ((𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)) ∘ ◡(𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)))) |
41 | 21, 40 | syl5eqr 2670 | . 2 ⊢ (𝑅 ∈ CRing → (((1𝑜 eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))) = ((𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)) ∘ ◡(𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)))) |
42 | 15, 17, 18, 19 | mapsnf1o2 7905 | . . 3 ⊢ (𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)):(𝐵 ↑𝑚 1𝑜)–1-1-onto→𝐵 |
43 | f1ococnv2 6163 | . . 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)) | |
44 | 42, 43 | mp1i 13 | . 2 ⊢ (𝑅 ∈ CRing → ((𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)) ∘ ◡(𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅))) = ( I ↾ 𝐵)) |
45 | 14, 41, 44 | 3eqtrd 2660 | 1 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 {csn 4177 ↦ cmpt 4729 I cid 5023 × cxp 5112 ◡ccnv 5113 ↾ cres 5116 ∘ ccom 5118 Oncon0 5723 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 ↑𝑚 cmap 7857 Basecbs 15857 ↾s cress 15858 Ringcrg 18547 CRingccrg 18548 SubRingcsubrg 18776 mVar cmvr 19352 mPoly cmpl 19353 eval cevl 19505 PwSer1cps1 19545 var1cv1 19546 Poly1cpl1 19547 eval1ce1 19679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-srg 18506 df-ring 18549 df-cring 18550 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-assa 19312 df-asp 19313 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-evls 19506 df-evl 19507 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-evl1 19681 |
This theorem is referenced by: evl1vard 19701 evls1var 19702 pf1id 19711 fta1blem 23928 |
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