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Mirrors > Home > MPE Home > Th. List > evls1var | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation for subrings maps the variable to the identity function. (Contributed by AV, 13-Sep-2019.) |
Ref | Expression |
---|---|
evls1var.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
evls1var.x | ⊢ 𝑋 = (var1‘𝑈) |
evls1var.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evls1var.b | ⊢ 𝐵 = (Base‘𝑆) |
evls1var.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evls1var.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
Ref | Expression |
---|---|
evls1var | ⊢ (𝜑 → (𝑄‘𝑋) = ( I ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . 5 ⊢ (var1‘𝑆) = (var1‘𝑆) | |
2 | evls1var.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
3 | evls1var.u | . . . . 5 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
4 | 1, 2, 3 | subrgvr1 20357 | . . . 4 ⊢ (𝜑 → (var1‘𝑆) = (var1‘𝑈)) |
5 | evls1var.x | . . . 4 ⊢ 𝑋 = (var1‘𝑈) | |
6 | 4, 5 | syl6reqr 2872 | . . 3 ⊢ (𝜑 → 𝑋 = (var1‘𝑆)) |
7 | 6 | fveq2d 6667 | . 2 ⊢ (𝜑 → (𝑄‘𝑋) = (𝑄‘(var1‘𝑆))) |
8 | eqid 2818 | . . . . . 6 ⊢ ((1o evalSub 𝑆)‘𝑅) = ((1o evalSub 𝑆)‘𝑅) | |
9 | eqid 2818 | . . . . . 6 ⊢ (1o eval 𝑆) = (1o eval 𝑆) | |
10 | eqid 2818 | . . . . . 6 ⊢ (1o mVar 𝑈) = (1o mVar 𝑈) | |
11 | evls1var.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑆) | |
12 | 1on 8098 | . . . . . . 7 ⊢ 1o ∈ On | |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1o ∈ On) |
14 | evls1var.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
15 | 0lt1o 8118 | . . . . . . 7 ⊢ ∅ ∈ 1o | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ∅ ∈ 1o) |
17 | 8, 9, 10, 3, 11, 13, 14, 2, 16 | evlsvarsrng 20240 | . . . . 5 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘((1o mVar 𝑈)‘∅)) = ((1o eval 𝑆)‘((1o mVar 𝑈)‘∅))) |
18 | 1 | vr1val 20288 | . . . . . . 7 ⊢ (var1‘𝑆) = ((1o mVar 𝑆)‘∅) |
19 | eqid 2818 | . . . . . . . . 9 ⊢ (1o mVar 𝑆) = (1o mVar 𝑆) | |
20 | 19, 13, 2, 3 | subrgmvr 20170 | . . . . . . . 8 ⊢ (𝜑 → (1o mVar 𝑆) = (1o mVar 𝑈)) |
21 | 20 | fveq1d 6665 | . . . . . . 7 ⊢ (𝜑 → ((1o mVar 𝑆)‘∅) = ((1o mVar 𝑈)‘∅)) |
22 | 18, 21 | syl5eq 2865 | . . . . . 6 ⊢ (𝜑 → (var1‘𝑆) = ((1o mVar 𝑈)‘∅)) |
23 | 22 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) = (((1o evalSub 𝑆)‘𝑅)‘((1o mVar 𝑈)‘∅))) |
24 | 22 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → ((1o eval 𝑆)‘(var1‘𝑆)) = ((1o eval 𝑆)‘((1o mVar 𝑈)‘∅))) |
25 | 17, 23, 24 | 3eqtr4d 2863 | . . . 4 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) = ((1o eval 𝑆)‘(var1‘𝑆))) |
26 | 25 | coeq1d 5725 | . . 3 ⊢ (𝜑 → ((((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (((1o eval 𝑆)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
27 | eqid 2818 | . . . . 5 ⊢ (Poly1‘𝑈) = (Poly1‘𝑈) | |
28 | eqid 2818 | . . . . . . 7 ⊢ (Poly1‘(𝑆 ↾s 𝑅)) = (Poly1‘(𝑆 ↾s 𝑅)) | |
29 | eqid 2818 | . . . . . . 7 ⊢ (PwSer1‘(𝑆 ↾s 𝑅)) = (PwSer1‘(𝑆 ↾s 𝑅)) | |
30 | 3 | fveq2i 6666 | . . . . . . . 8 ⊢ (Poly1‘𝑈) = (Poly1‘(𝑆 ↾s 𝑅)) |
31 | 30 | fveq2i 6666 | . . . . . . 7 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(Poly1‘(𝑆 ↾s 𝑅))) |
32 | 28, 29, 31 | ply1bas 20291 | . . . . . 6 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(1o mPoly (𝑆 ↾s 𝑅))) |
33 | 32 | eqcomi 2827 | . . . . 5 ⊢ (Base‘(1o mPoly (𝑆 ↾s 𝑅))) = (Base‘(Poly1‘𝑈)) |
34 | 1, 2, 3, 27, 33 | subrgvr1cl 20358 | . . . 4 ⊢ (𝜑 → (var1‘𝑆) ∈ (Base‘(1o mPoly (𝑆 ↾s 𝑅)))) |
35 | evls1var.q | . . . . 5 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
36 | eqid 2818 | . . . . 5 ⊢ (1o evalSub 𝑆) = (1o evalSub 𝑆) | |
37 | eqid 2818 | . . . . 5 ⊢ (1o mPoly (𝑆 ↾s 𝑅)) = (1o mPoly (𝑆 ↾s 𝑅)) | |
38 | eqid 2818 | . . . . 5 ⊢ (Base‘(1o mPoly (𝑆 ↾s 𝑅))) = (Base‘(1o mPoly (𝑆 ↾s 𝑅))) | |
39 | 35, 36, 11, 37, 38 | evls1val 20411 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ (var1‘𝑆) ∈ (Base‘(1o mPoly (𝑆 ↾s 𝑅)))) → (𝑄‘(var1‘𝑆)) = ((((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
40 | 14, 2, 34, 39 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝑄‘(var1‘𝑆)) = ((((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
41 | crngring 19237 | . . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
42 | eqid 2818 | . . . . . 6 ⊢ (Poly1‘𝑆) = (Poly1‘𝑆) | |
43 | eqid 2818 | . . . . . . . 8 ⊢ (PwSer1‘𝑆) = (PwSer1‘𝑆) | |
44 | eqid 2818 | . . . . . . . 8 ⊢ (Base‘(Poly1‘𝑆)) = (Base‘(Poly1‘𝑆)) | |
45 | 42, 43, 44 | ply1bas 20291 | . . . . . . 7 ⊢ (Base‘(Poly1‘𝑆)) = (Base‘(1o mPoly 𝑆)) |
46 | 45 | eqcomi 2827 | . . . . . 6 ⊢ (Base‘(1o mPoly 𝑆)) = (Base‘(Poly1‘𝑆)) |
47 | 1, 42, 46 | vr1cl 20313 | . . . . 5 ⊢ (𝑆 ∈ Ring → (var1‘𝑆) ∈ (Base‘(1o mPoly 𝑆))) |
48 | 14, 41, 47 | 3syl 18 | . . . 4 ⊢ (𝜑 → (var1‘𝑆) ∈ (Base‘(1o mPoly 𝑆))) |
49 | eqid 2818 | . . . . 5 ⊢ (eval1‘𝑆) = (eval1‘𝑆) | |
50 | eqid 2818 | . . . . 5 ⊢ (1o mPoly 𝑆) = (1o mPoly 𝑆) | |
51 | eqid 2818 | . . . . 5 ⊢ (Base‘(1o mPoly 𝑆)) = (Base‘(1o mPoly 𝑆)) | |
52 | 49, 9, 11, 50, 51 | evl1val 20420 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ (var1‘𝑆) ∈ (Base‘(1o mPoly 𝑆))) → ((eval1‘𝑆)‘(var1‘𝑆)) = (((1o eval 𝑆)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
53 | 14, 48, 52 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((eval1‘𝑆)‘(var1‘𝑆)) = (((1o eval 𝑆)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
54 | 26, 40, 53 | 3eqtr4d 2863 | . 2 ⊢ (𝜑 → (𝑄‘(var1‘𝑆)) = ((eval1‘𝑆)‘(var1‘𝑆))) |
55 | 49, 1, 11 | evl1var 20427 | . . 3 ⊢ (𝑆 ∈ CRing → ((eval1‘𝑆)‘(var1‘𝑆)) = ( I ↾ 𝐵)) |
56 | 14, 55 | syl 17 | . 2 ⊢ (𝜑 → ((eval1‘𝑆)‘(var1‘𝑆)) = ( I ↾ 𝐵)) |
57 | 7, 54, 56 | 3eqtrd 2857 | 1 ⊢ (𝜑 → (𝑄‘𝑋) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∅c0 4288 {csn 4557 ↦ cmpt 5137 I cid 5452 × cxp 5546 ↾ cres 5550 ∘ ccom 5552 Oncon0 6184 ‘cfv 6348 (class class class)co 7145 1oc1o 8084 Basecbs 16471 ↾s cress 16472 Ringcrg 19226 CRingccrg 19227 SubRingcsubrg 19460 mVar cmvr 20060 mPoly cmpl 20061 evalSub ces 20212 eval cevl 20213 PwSer1cps1 20271 var1cv1 20272 Poly1cpl1 20273 evalSub1 ces1 20404 eval1ce1 20405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-ofr 7399 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-0g 16703 df-gsum 16704 df-prds 16709 df-pws 16711 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-srg 19185 df-ring 19228 df-cring 19229 df-rnghom 19396 df-subrg 19462 df-lmod 19565 df-lss 19633 df-lsp 19673 df-assa 20013 df-asp 20014 df-ascl 20015 df-psr 20064 df-mvr 20065 df-mpl 20066 df-opsr 20068 df-evls 20214 df-evl 20215 df-psr1 20276 df-vr1 20277 df-ply1 20278 df-evls1 20406 df-evl1 20407 |
This theorem is referenced by: evls1varsrng 20431 |
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