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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 | β’ π = (eval1βπ ) |
evl1var.v | β’ π = (var1βπ ) |
evl1var.b | β’ π΅ = (Baseβπ ) |
Ref | Expression |
---|---|
evl1var | β’ (π β CRing β (πβπ) = ( I βΎ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 20192 | . . . 4 β’ (π β CRing β π β Ring) | |
2 | evl1var.v | . . . . 5 β’ π = (var1βπ ) | |
3 | eqid 2728 | . . . . 5 β’ (Poly1βπ ) = (Poly1βπ ) | |
4 | eqid 2728 | . . . . 5 β’ (Baseβ(Poly1βπ )) = (Baseβ(Poly1βπ )) | |
5 | 2, 3, 4 | vr1cl 22143 | . . . 4 β’ (π β Ring β π β (Baseβ(Poly1βπ ))) |
6 | 1, 5 | syl 17 | . . 3 β’ (π β CRing β π β (Baseβ(Poly1βπ ))) |
7 | evl1var.q | . . . 4 β’ π = (eval1βπ ) | |
8 | eqid 2728 | . . . 4 β’ (1o eval π ) = (1o eval π ) | |
9 | evl1var.b | . . . 4 β’ π΅ = (Baseβπ ) | |
10 | eqid 2728 | . . . 4 β’ (1o mPoly π ) = (1o mPoly π ) | |
11 | eqid 2728 | . . . . 5 β’ (PwSer1βπ ) = (PwSer1βπ ) | |
12 | 3, 11, 4 | ply1bas 22121 | . . . 4 β’ (Baseβ(Poly1βπ )) = (Baseβ(1o mPoly π )) |
13 | 7, 8, 9, 10, 12 | evl1val 22255 | . . 3 β’ ((π β CRing β§ π β (Baseβ(Poly1βπ ))) β (πβπ) = (((1o eval π )βπ) β (π¦ β π΅ β¦ (1o Γ {π¦})))) |
14 | 6, 13 | mpdan 685 | . 2 β’ (π β CRing β (πβπ) = (((1o eval π )βπ) β (π¦ β π΅ β¦ (1o Γ {π¦})))) |
15 | df1o2 8500 | . . . . 5 β’ 1o = {β } | |
16 | 9 | fvexi 6916 | . . . . 5 β’ π΅ β V |
17 | 0ex 5311 | . . . . 5 β’ β β V | |
18 | eqid 2728 | . . . . 5 β’ (π§ β (π΅ βm 1o) β¦ (π§ββ )) = (π§ β (π΅ βm 1o) β¦ (π§ββ )) | |
19 | 15, 16, 17, 18 | mapsncnv 8918 | . . . 4 β’ β‘(π§ β (π΅ βm 1o) β¦ (π§ββ )) = (π¦ β π΅ β¦ (1o Γ {π¦})) |
20 | 19 | coeq2i 5867 | . . 3 β’ (((1o eval π )βπ) β β‘(π§ β (π΅ βm 1o) β¦ (π§ββ ))) = (((1o eval π )βπ) β (π¦ β π΅ β¦ (1o Γ {π¦}))) |
21 | 9 | ressid 17232 | . . . . . . . . 9 β’ (π β CRing β (π βΎs π΅) = π ) |
22 | 21 | oveq2d 7442 | . . . . . . . 8 β’ (π β CRing β (1o mVar (π βΎs π΅)) = (1o mVar π )) |
23 | 22 | fveq1d 6904 | . . . . . . 7 β’ (π β CRing β ((1o mVar (π βΎs π΅))ββ ) = ((1o mVar π )ββ )) |
24 | 2 | vr1val 22118 | . . . . . . 7 β’ π = ((1o mVar π )ββ ) |
25 | 23, 24 | eqtr4di 2786 | . . . . . 6 β’ (π β CRing β ((1o mVar (π βΎs π΅))ββ ) = π) |
26 | 25 | fveq2d 6906 | . . . . 5 β’ (π β CRing β ((1o eval π )β((1o mVar (π βΎs π΅))ββ )) = ((1o eval π )βπ)) |
27 | 8, 9 | evlval 22048 | . . . . . 6 β’ (1o eval π ) = ((1o evalSub π )βπ΅) |
28 | eqid 2728 | . . . . . 6 β’ (1o mVar (π βΎs π΅)) = (1o mVar (π βΎs π΅)) | |
29 | eqid 2728 | . . . . . 6 β’ (π βΎs π΅) = (π βΎs π΅) | |
30 | 1on 8505 | . . . . . . 7 β’ 1o β On | |
31 | 30 | a1i 11 | . . . . . 6 β’ (π β CRing β 1o β On) |
32 | id 22 | . . . . . 6 β’ (π β CRing β π β CRing) | |
33 | 9 | subrgid 20519 | . . . . . . 7 β’ (π β Ring β π΅ β (SubRingβπ )) |
34 | 1, 33 | syl 17 | . . . . . 6 β’ (π β CRing β π΅ β (SubRingβπ )) |
35 | 0lt1o 8531 | . . . . . . 7 β’ β β 1o | |
36 | 35 | a1i 11 | . . . . . 6 β’ (π β CRing β β β 1o) |
37 | 27, 28, 29, 9, 31, 32, 34, 36 | evlsvar 22043 | . . . . 5 β’ (π β CRing β ((1o eval π )β((1o mVar (π βΎs π΅))ββ )) = (π§ β (π΅ βm 1o) β¦ (π§ββ ))) |
38 | 26, 37 | eqtr3d 2770 | . . . 4 β’ (π β CRing β ((1o eval π )βπ) = (π§ β (π΅ βm 1o) β¦ (π§ββ ))) |
39 | 38 | coeq1d 5868 | . . 3 β’ (π β CRing β (((1o eval π )βπ) β β‘(π§ β (π΅ βm 1o) β¦ (π§ββ ))) = ((π§ β (π΅ βm 1o) β¦ (π§ββ )) β β‘(π§ β (π΅ βm 1o) β¦ (π§ββ )))) |
40 | 20, 39 | eqtr3id 2782 | . 2 β’ (π β CRing β (((1o eval π )βπ) β (π¦ β π΅ β¦ (1o Γ {π¦}))) = ((π§ β (π΅ βm 1o) β¦ (π§ββ )) β β‘(π§ β (π΅ βm 1o) β¦ (π§ββ )))) |
41 | 15, 16, 17, 18 | mapsnf1o2 8919 | . . 3 β’ (π§ β (π΅ βm 1o) β¦ (π§ββ )):(π΅ βm 1o)β1-1-ontoβπ΅ |
42 | f1ococnv2 6871 | . . 3 β’ ((π§ β (π΅ βm 1o) β¦ (π§ββ )):(π΅ βm 1o)β1-1-ontoβπ΅ β ((π§ β (π΅ βm 1o) β¦ (π§ββ )) β β‘(π§ β (π΅ βm 1o) β¦ (π§ββ ))) = ( I βΎ π΅)) | |
43 | 41, 42 | mp1i 13 | . 2 β’ (π β CRing β ((π§ β (π΅ βm 1o) β¦ (π§ββ )) β β‘(π§ β (π΅ βm 1o) β¦ (π§ββ ))) = ( I βΎ π΅)) |
44 | 14, 40, 43 | 3eqtrd 2772 | 1 β’ (π β CRing β (πβπ) = ( I βΎ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β c0 4326 {csn 4632 β¦ cmpt 5235 I cid 5579 Γ cxp 5680 β‘ccnv 5681 βΎ cres 5684 β ccom 5686 Oncon0 6374 β1-1-ontoβwf1o 6552 βcfv 6553 (class class class)co 7426 1oc1o 8486 βm cmap 8851 Basecbs 17187 βΎs cress 17216 Ringcrg 20180 CRingccrg 20181 SubRingcsubrg 20513 mVar cmvr 21845 mPoly cmpl 21846 eval cevl 22024 PwSer1cps1 22101 var1cv1 22102 Poly1cpl1 22103 eval1ce1 22240 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 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 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-rhm 20418 df-subrng 20490 df-subrg 20515 df-lmod 20752 df-lss 20823 df-lsp 20863 df-assa 21794 df-asp 21795 df-ascl 21796 df-psr 21849 df-mvr 21850 df-mpl 21851 df-opsr 21853 df-evls 22025 df-evl 22026 df-psr1 22106 df-vr1 22107 df-ply1 22108 df-evl1 22242 |
This theorem is referenced by: evl1vard 22263 evls1var 22264 pf1id 22273 fta1blem 26125 |
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