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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 19429 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | evl1var.v | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
3 | eqid 2738 | . . . . 5 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
4 | eqid 2738 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
5 | 2, 3, 4 | vr1cl 20993 | . . . 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 2738 | . . . 4 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
9 | evl1var.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
10 | eqid 2738 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
11 | eqid 2738 | . . . . 5 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
12 | 3, 11, 4 | ply1bas 20971 | . . . 4 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1o mPoly 𝑅)) |
13 | 7, 8, 9, 10, 12 | evl1val 21100 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘(Poly1‘𝑅))) → (𝑂‘𝑋) = (((1o eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
14 | 6, 13 | mpdan 687 | . 2 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = (((1o eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
15 | df1o2 8144 | . . . . 5 ⊢ 1o = {∅} | |
16 | 9 | fvexi 6689 | . . . . 5 ⊢ 𝐵 ∈ V |
17 | 0ex 5176 | . . . . 5 ⊢ ∅ ∈ V | |
18 | eqid 2738 | . . . . 5 ⊢ (𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅)) = (𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅)) | |
19 | 15, 16, 17, 18 | mapsncnv 8504 | . . . 4 ⊢ ◡(𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅)) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) |
20 | 19 | coeq2i 5704 | . . 3 ⊢ (((1o eval 𝑅)‘𝑋) ∘ ◡(𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅))) = (((1o eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) |
21 | 9 | ressid 16663 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅) |
22 | 21 | oveq2d 7187 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (1o mVar (𝑅 ↾s 𝐵)) = (1o mVar 𝑅)) |
23 | 22 | fveq1d 6677 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → ((1o mVar (𝑅 ↾s 𝐵))‘∅) = ((1o mVar 𝑅)‘∅)) |
24 | 2 | vr1val 20968 | . . . . . . 7 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) |
25 | 23, 24 | eqtr4di 2791 | . . . . . 6 ⊢ (𝑅 ∈ CRing → ((1o mVar (𝑅 ↾s 𝐵))‘∅) = 𝑋) |
26 | 25 | fveq2d 6679 | . . . . 5 ⊢ (𝑅 ∈ CRing → ((1o eval 𝑅)‘((1o mVar (𝑅 ↾s 𝐵))‘∅)) = ((1o eval 𝑅)‘𝑋)) |
27 | 8, 9 | evlval 20910 | . . . . . 6 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵) |
28 | eqid 2738 | . . . . . 6 ⊢ (1o mVar (𝑅 ↾s 𝐵)) = (1o mVar (𝑅 ↾s 𝐵)) | |
29 | eqid 2738 | . . . . . 6 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵) | |
30 | 1on 8139 | . . . . . . 7 ⊢ 1o ∈ On | |
31 | 30 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 1o ∈ On) |
32 | id 22 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing) | |
33 | 9 | subrgid 19657 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) |
34 | 1, 33 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐵 ∈ (SubRing‘𝑅)) |
35 | 0lt1o 8161 | . . . . . . 7 ⊢ ∅ ∈ 1o | |
36 | 35 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → ∅ ∈ 1o) |
37 | 27, 28, 29, 9, 31, 32, 34, 36 | evlsvar 20905 | . . . . 5 ⊢ (𝑅 ∈ CRing → ((1o eval 𝑅)‘((1o mVar (𝑅 ↾s 𝐵))‘∅)) = (𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅))) |
38 | 26, 37 | eqtr3d 2775 | . . . 4 ⊢ (𝑅 ∈ CRing → ((1o eval 𝑅)‘𝑋) = (𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅))) |
39 | 38 | coeq1d 5705 | . . 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 2787 | . 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 8505 | . . 3 ⊢ (𝑧 ∈ (𝐵 ↑m 1o) ↦ (𝑧‘∅)):(𝐵 ↑m 1o)–1-1-onto→𝐵 |
42 | f1ococnv2 6645 | . . 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 2777 | 1 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ∅c0 4212 {csn 4517 ↦ cmpt 5111 I cid 5429 × cxp 5524 ◡ccnv 5525 ↾ cres 5528 ∘ ccom 5530 Oncon0 6173 –1-1-onto→wf1o 6339 ‘cfv 6340 (class class class)co 7171 1oc1o 8125 ↑m cmap 8438 Basecbs 16587 ↾s cress 16588 Ringcrg 19417 CRingccrg 19418 SubRingcsubrg 19651 mVar cmvr 20719 mPoly cmpl 20720 eval cevl 20886 PwSer1cps1 20951 var1cv1 20952 Poly1cpl1 20953 eval1ce1 21085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-of 7426 df-ofr 7427 df-om 7601 df-1st 7715 df-2nd 7716 df-supp 7858 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-1o 8132 df-er 8321 df-map 8440 df-pm 8441 df-ixp 8509 df-en 8557 df-dom 8558 df-sdom 8559 df-fin 8560 df-fsupp 8908 df-sup 8980 df-oi 9048 df-card 9442 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-nn 11718 df-2 11780 df-3 11781 df-4 11782 df-5 11783 df-6 11784 df-7 11785 df-8 11786 df-9 11787 df-n0 11978 df-z 12064 df-dec 12181 df-uz 12326 df-fz 12983 df-fzo 13126 df-seq 13462 df-hash 13784 df-struct 16589 df-ndx 16590 df-slot 16591 df-base 16593 df-sets 16594 df-ress 16595 df-plusg 16682 df-mulr 16683 df-sca 16685 df-vsca 16686 df-ip 16687 df-tset 16688 df-ple 16689 df-ds 16691 df-hom 16693 df-cco 16694 df-0g 16819 df-gsum 16820 df-prds 16825 df-pws 16827 df-mre 16961 df-mrc 16962 df-acs 16964 df-mgm 17969 df-sgrp 18018 df-mnd 18029 df-mhm 18073 df-submnd 18074 df-grp 18223 df-minusg 18224 df-sbg 18225 df-mulg 18344 df-subg 18395 df-ghm 18475 df-cntz 18566 df-cmn 19027 df-abl 19028 df-mgp 19360 df-ur 19372 df-srg 19376 df-ring 19419 df-cring 19420 df-rnghom 19590 df-subrg 19653 df-lmod 19756 df-lss 19824 df-lsp 19864 df-assa 20670 df-asp 20671 df-ascl 20672 df-psr 20723 df-mvr 20724 df-mpl 20725 df-opsr 20727 df-evls 20887 df-evl 20888 df-psr1 20956 df-vr1 20957 df-ply1 20958 df-evl1 21087 |
This theorem is referenced by: evl1vard 21108 evls1var 21109 pf1id 21118 fta1blem 24921 |
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