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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 18381 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | evl1var.v | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
3 | eqid 2610 | . . . . 5 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
4 | eqid 2610 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
5 | 2, 3, 4 | vr1cl 19408 | . . . 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 2610 | . . . 4 ⊢ (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅) | |
9 | evl1var.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
10 | eqid 2610 | . . . 4 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅) | |
11 | eqid 2610 | . . . . 5 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
12 | 3, 11, 4 | ply1bas 19386 | . . . 4 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1𝑜 mPoly 𝑅)) |
13 | 7, 8, 9, 10, 12 | evl1val 19514 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘(Poly1‘𝑅))) → (𝑂‘𝑋) = (((1𝑜 eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) |
14 | 6, 13 | mpdan 699 | . 2 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = (((1𝑜 eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) |
15 | df1o2 7459 | . . . . 5 ⊢ 1𝑜 = {∅} | |
16 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
17 | 9, 16 | eqeltri 2684 | . . . . 5 ⊢ 𝐵 ∈ V |
18 | 0ex 4718 | . . . . 5 ⊢ ∅ ∈ V | |
19 | eqid 2610 | . . . . 5 ⊢ (𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)) = (𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)) | |
20 | 15, 17, 18, 19 | mapsncnv 7790 | . . . 4 ⊢ ◡(𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)) = (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})) |
21 | 20 | coeq2i 5204 | . . 3 ⊢ (((1𝑜 eval 𝑅)‘𝑋) ∘ ◡(𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅))) = (((1𝑜 eval 𝑅)‘𝑋) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))) |
22 | 9 | ressid 15762 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅) |
23 | 22 | oveq2d 6565 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (1𝑜 mVar (𝑅 ↾s 𝐵)) = (1𝑜 mVar 𝑅)) |
24 | 23 | fveq1d 6105 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → ((1𝑜 mVar (𝑅 ↾s 𝐵))‘∅) = ((1𝑜 mVar 𝑅)‘∅)) |
25 | 2 | vr1val 19383 | . . . . . . 7 ⊢ 𝑋 = ((1𝑜 mVar 𝑅)‘∅) |
26 | 24, 25 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑅 ∈ CRing → ((1𝑜 mVar (𝑅 ↾s 𝐵))‘∅) = 𝑋) |
27 | 26 | fveq2d 6107 | . . . . 5 ⊢ (𝑅 ∈ CRing → ((1𝑜 eval 𝑅)‘((1𝑜 mVar (𝑅 ↾s 𝐵))‘∅)) = ((1𝑜 eval 𝑅)‘𝑋)) |
28 | 8, 9 | evlval 19345 | . . . . . 6 ⊢ (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘𝐵) |
29 | eqid 2610 | . . . . . 6 ⊢ (1𝑜 mVar (𝑅 ↾s 𝐵)) = (1𝑜 mVar (𝑅 ↾s 𝐵)) | |
30 | eqid 2610 | . . . . . 6 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵) | |
31 | 1on 7454 | . . . . . . 7 ⊢ 1𝑜 ∈ On | |
32 | 31 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 1𝑜 ∈ On) |
33 | id 22 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing) | |
34 | 9 | subrgid 18605 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) |
35 | 1, 34 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐵 ∈ (SubRing‘𝑅)) |
36 | 0lt1o 7471 | . . . . . . 7 ⊢ ∅ ∈ 1𝑜 | |
37 | 36 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → ∅ ∈ 1𝑜) |
38 | 28, 29, 30, 9, 32, 33, 35, 37 | evlsvar 19344 | . . . . 5 ⊢ (𝑅 ∈ CRing → ((1𝑜 eval 𝑅)‘((1𝑜 mVar (𝑅 ↾s 𝐵))‘∅)) = (𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅))) |
39 | 27, 38 | eqtr3d 2646 | . . . 4 ⊢ (𝑅 ∈ CRing → ((1𝑜 eval 𝑅)‘𝑋) = (𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅))) |
40 | 39 | coeq1d 5205 | . . 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 2658 | . 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 7791 | . . 3 ⊢ (𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑧‘∅)):(𝐵 ↑𝑚 1𝑜)–1-1-onto→𝐵 |
43 | f1ococnv2 6076 | . . 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 2648 | 1 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {csn 4125 ↦ cmpt 4643 I cid 4948 × cxp 5036 ◡ccnv 5037 ↾ cres 5040 ∘ ccom 5042 Oncon0 5640 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 ↑𝑚 cmap 7744 Basecbs 15695 ↾s cress 15696 Ringcrg 18370 CRingccrg 18371 SubRingcsubrg 18599 mVar cmvr 19173 mPoly cmpl 19174 eval cevl 19326 PwSer1cps1 19366 var1cv1 19367 Poly1cpl1 19368 eval1ce1 19500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-srg 18329 df-ring 18372 df-cring 18373 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-assa 19133 df-asp 19134 df-ascl 19135 df-psr 19177 df-mvr 19178 df-mpl 19179 df-opsr 19181 df-evls 19327 df-evl 19328 df-psr1 19371 df-vr1 19372 df-ply1 19373 df-evl1 19502 |
This theorem is referenced by: evl1vard 19522 evls1var 19523 pf1id 19532 fta1blem 23732 |
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