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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 20190 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | evl1var.v | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
3 | eqid 2727 | . . . . 5 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
4 | eqid 2727 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
5 | 2, 3, 4 | vr1cl 22141 | . . . 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 2727 | . . . 4 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
9 | evl1var.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
10 | eqid 2727 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
11 | eqid 2727 | . . . . 5 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
12 | 3, 11, 4 | ply1bas 22119 | . . . 4 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1o mPoly 𝑅)) |
13 | 7, 8, 9, 10, 12 | evl1val 22253 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘(Poly1‘𝑅))) → (𝑂‘𝑋) = (((1o eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
14 | 6, 13 | mpdan 685 | . 2 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = (((1o eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
15 | df1o2 8498 | . . . . 5 ⊢ 1o = {∅} | |
16 | 9 | fvexi 6914 | . . . . 5 ⊢ 𝐵 ∈ V |
17 | 0ex 5309 | . . . . 5 ⊢ ∅ ∈ V | |
18 | eqid 2727 | . . . . 5 ⊢ (𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅)) = (𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅)) | |
19 | 15, 16, 17, 18 | mapsncnv 8916 | . . . 4 ⊢ ◡(𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅)) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) |
20 | 19 | coeq2i 5865 | . . 3 ⊢ (((1o eval 𝑅)‘𝑋) ∘ ◡(𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅))) = (((1o eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) |
21 | 9 | ressid 17230 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅) |
22 | 21 | oveq2d 7440 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (1o mVar (𝑅 ↾s 𝐵)) = (1o mVar 𝑅)) |
23 | 22 | fveq1d 6902 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → ((1o mVar (𝑅 ↾s 𝐵))‘∅) = ((1o mVar 𝑅)‘∅)) |
24 | 2 | vr1val 22116 | . . . . . . 7 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) |
25 | 23, 24 | eqtr4di 2785 | . . . . . 6 ⊢ (𝑅 ∈ CRing → ((1o mVar (𝑅 ↾s 𝐵))‘∅) = 𝑋) |
26 | 25 | fveq2d 6904 | . . . . 5 ⊢ (𝑅 ∈ CRing → ((1o eval 𝑅)‘((1o mVar (𝑅 ↾s 𝐵))‘∅)) = ((1o eval 𝑅)‘𝑋)) |
27 | 8, 9 | evlval 22046 | . . . . . 6 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵) |
28 | eqid 2727 | . . . . . 6 ⊢ (1o mVar (𝑅 ↾s 𝐵)) = (1o mVar (𝑅 ↾s 𝐵)) | |
29 | eqid 2727 | . . . . . 6 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵) | |
30 | 1on 8503 | . . . . . . 7 ⊢ 1o ∈ On | |
31 | 30 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 1o ∈ On) |
32 | id 22 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing) | |
33 | 9 | subrgid 20517 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) |
34 | 1, 33 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐵 ∈ (SubRing‘𝑅)) |
35 | 0lt1o 8529 | . . . . . . 7 ⊢ ∅ ∈ 1o | |
36 | 35 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → ∅ ∈ 1o) |
37 | 27, 28, 29, 9, 31, 32, 34, 36 | evlsvar 22041 | . . . . 5 ⊢ (𝑅 ∈ CRing → ((1o eval 𝑅)‘((1o mVar (𝑅 ↾s 𝐵))‘∅)) = (𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅))) |
38 | 26, 37 | eqtr3d 2769 | . . . 4 ⊢ (𝑅 ∈ CRing → ((1o eval 𝑅)‘𝑋) = (𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅))) |
39 | 38 | coeq1d 5866 | . . 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 | eqtr3id 2781 | . 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 8917 | . . 3 ⊢ (𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅)):(𝐵 ↑m 1o)–1-1-onto→𝐵 |
42 | f1ococnv2 6869 | . . 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 2771 | 1 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∅c0 4324 {csn 4630 ↦ cmpt 5233 I cid 5577 × cxp 5678 ◡ccnv 5679 ↾ cres 5682 ∘ ccom 5684 Oncon0 6372 –1-1-onto→wf1o 6550 ‘cfv 6551 (class class class)co 7424 1oc1o 8484 ↑m cmap 8849 Basecbs 17185 ↾s cress 17214 Ringcrg 20178 CRingccrg 20179 SubRingcsubrg 20511 mVar cmvr 21843 mPoly cmpl 21844 eval cevl 22022 PwSer1cps1 22099 var1cv1 22100 Poly1cpl1 22101 eval1ce1 22238 |
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 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
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 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-ofr 7690 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-map 8851 df-pm 8852 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-sup 9471 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-fz 13523 df-fzo 13666 df-seq 14005 df-hash 14328 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-hom 17262 df-cco 17263 df-0g 17428 df-gsum 17429 df-prds 17434 df-pws 17436 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18745 df-submnd 18746 df-grp 18898 df-minusg 18899 df-sbg 18900 df-mulg 19029 df-subg 19083 df-ghm 19173 df-cntz 19273 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-srg 20132 df-ring 20180 df-cring 20181 df-rhm 20416 df-subrng 20488 df-subrg 20513 df-lmod 20750 df-lss 20821 df-lsp 20861 df-assa 21792 df-asp 21793 df-ascl 21794 df-psr 21847 df-mvr 21848 df-mpl 21849 df-opsr 21851 df-evls 22023 df-evl 22024 df-psr1 22104 df-vr1 22105 df-ply1 22106 df-evl1 22240 |
This theorem is referenced by: evl1vard 22261 evls1var 22262 pf1id 22271 fta1blem 26123 |
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