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Mirrors > Home > MPE Home > Th. List > coe1pwmul | Structured version Visualization version GIF version |
Description: Coefficient vector of a polynomial multiplied on the left by a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
Ref | Expression |
---|---|
coe1pwmul.z | ⊢ 0 = (0g‘𝑅) |
coe1pwmul.p | ⊢ 𝑃 = (Poly1‘𝑅) |
coe1pwmul.x | ⊢ 𝑋 = (var1‘𝑅) |
coe1pwmul.n | ⊢ 𝑁 = (mulGrp‘𝑃) |
coe1pwmul.e | ⊢ ↑ = (.g‘𝑁) |
coe1pwmul.b | ⊢ 𝐵 = (Base‘𝑃) |
coe1pwmul.t | ⊢ · = (.r‘𝑃) |
coe1pwmul.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
coe1pwmul.a | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
coe1pwmul.d | ⊢ (𝜑 → 𝐷 ∈ ℕ0) |
Ref | Expression |
---|---|
coe1pwmul | ⊢ (𝜑 → (coe1‘((𝐷 ↑ 𝑋) · 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coe1pwmul.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
2 | eqid 2725 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | coe1pwmul.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | coe1pwmul.x | . . 3 ⊢ 𝑋 = (var1‘𝑅) | |
5 | eqid 2725 | . . 3 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
6 | coe1pwmul.n | . . 3 ⊢ 𝑁 = (mulGrp‘𝑃) | |
7 | coe1pwmul.e | . . 3 ⊢ ↑ = (.g‘𝑁) | |
8 | coe1pwmul.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
9 | coe1pwmul.t | . . 3 ⊢ · = (.r‘𝑃) | |
10 | eqid 2725 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | coe1pwmul.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
12 | coe1pwmul.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
13 | eqid 2725 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
14 | 2, 13 | ringidcl 20231 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
15 | 12, 14 | syl 17 | . . 3 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅)) |
16 | coe1pwmul.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℕ0) | |
17 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16 | coe1tmmul 22238 | . 2 ⊢ (𝜑 → (coe1‘(((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) · 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((1r‘𝑅)(.r‘𝑅)((coe1‘𝐴)‘(𝑥 − 𝐷))), 0 ))) |
18 | 3 | ply1sca 22212 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
19 | 12, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
20 | 19 | fveq2d 6900 | . . . . 5 ⊢ (𝜑 → (1r‘𝑅) = (1r‘(Scalar‘𝑃))) |
21 | 20 | oveq1d 7434 | . . . 4 ⊢ (𝜑 → ((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) = ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋))) |
22 | 3 | ply1lmod 22211 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
23 | 12, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ LMod) |
24 | 6, 8 | mgpbas 20109 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑁) |
25 | 3 | ply1ring 22207 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
26 | 6 | ringmgp 20208 | . . . . . . 7 ⊢ (𝑃 ∈ Ring → 𝑁 ∈ Mnd) |
27 | 12, 25, 26 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ Mnd) |
28 | 4, 3, 8 | vr1cl 22177 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
29 | 12, 28 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
30 | 24, 7, 27, 16, 29 | mulgnn0cld 19075 | . . . . 5 ⊢ (𝜑 → (𝐷 ↑ 𝑋) ∈ 𝐵) |
31 | eqid 2725 | . . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
32 | eqid 2725 | . . . . . 6 ⊢ (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃)) | |
33 | 8, 31, 5, 32 | lmodvs1 20802 | . . . . 5 ⊢ ((𝑃 ∈ LMod ∧ (𝐷 ↑ 𝑋) ∈ 𝐵) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) = (𝐷 ↑ 𝑋)) |
34 | 23, 30, 33 | syl2anc 582 | . . . 4 ⊢ (𝜑 → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) = (𝐷 ↑ 𝑋)) |
35 | 21, 34 | eqtrd 2765 | . . 3 ⊢ (𝜑 → ((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) = (𝐷 ↑ 𝑋)) |
36 | 35 | fvoveq1d 7441 | . 2 ⊢ (𝜑 → (coe1‘(((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) · 𝐴)) = (coe1‘((𝐷 ↑ 𝑋) · 𝐴))) |
37 | 12 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝑅 ∈ Ring) |
38 | eqid 2725 | . . . . . . . . 9 ⊢ (coe1‘𝐴) = (coe1‘𝐴) | |
39 | 38, 8, 3, 2 | coe1f 22171 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (coe1‘𝐴):ℕ0⟶(Base‘𝑅)) |
40 | 11, 39 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (coe1‘𝐴):ℕ0⟶(Base‘𝑅)) |
41 | 40 | ad2antrr 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (coe1‘𝐴):ℕ0⟶(Base‘𝑅)) |
42 | 16 | ad2antrr 724 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝐷 ∈ ℕ0) |
43 | simplr 767 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝑥 ∈ ℕ0) | |
44 | simpr 483 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝐷 ≤ 𝑥) | |
45 | nn0sub2 12661 | . . . . . . 7 ⊢ ((𝐷 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥) → (𝑥 − 𝐷) ∈ ℕ0) | |
46 | 42, 43, 44, 45 | syl3anc 1368 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (𝑥 − 𝐷) ∈ ℕ0) |
47 | 41, 46 | ffvelcdmd 7094 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → ((coe1‘𝐴)‘(𝑥 − 𝐷)) ∈ (Base‘𝑅)) |
48 | 2, 10, 13 | ringlidm 20234 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐴)‘(𝑥 − 𝐷)) ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)((coe1‘𝐴)‘(𝑥 − 𝐷))) = ((coe1‘𝐴)‘(𝑥 − 𝐷))) |
49 | 37, 47, 48 | syl2anc 582 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → ((1r‘𝑅)(.r‘𝑅)((coe1‘𝐴)‘(𝑥 − 𝐷))) = ((coe1‘𝐴)‘(𝑥 − 𝐷))) |
50 | 49 | ifeq1da 4561 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → if(𝐷 ≤ 𝑥, ((1r‘𝑅)(.r‘𝑅)((coe1‘𝐴)‘(𝑥 − 𝐷))), 0 ) = if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 )) |
51 | 50 | mpteq2dva 5249 | . 2 ⊢ (𝜑 → (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((1r‘𝑅)(.r‘𝑅)((coe1‘𝐴)‘(𝑥 − 𝐷))), 0 )) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))) |
52 | 17, 36, 51 | 3eqtr3d 2773 | 1 ⊢ (𝜑 → (coe1‘((𝐷 ↑ 𝑋) · 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ifcif 4530 class class class wbr 5149 ↦ cmpt 5232 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ≤ cle 11286 − cmin 11481 ℕ0cn0 12510 Basecbs 17199 .rcmulr 17253 Scalarcsca 17255 ·𝑠 cvsca 17256 0gc0g 17440 Mndcmnd 18713 .gcmg 19047 mulGrpcmgp 20103 1rcur 20150 Ringcrg 20202 LModclmod 20772 var1cv1 22135 Poly1cpl1 22136 coe1cco1 22137 |
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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-ofr 7686 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9393 df-sup 9472 df-oi 9540 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 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 14008 df-hash 14334 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-sca 17268 df-vsca 17269 df-ip 17270 df-tset 17271 df-ple 17272 df-ds 17274 df-hom 17276 df-cco 17277 df-0g 17442 df-gsum 17443 df-prds 17448 df-pws 17450 df-mre 17585 df-mrc 17586 df-acs 17588 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-mhm 18759 df-submnd 18760 df-grp 18917 df-minusg 18918 df-sbg 18919 df-mulg 19048 df-subg 19103 df-ghm 19193 df-cntz 19297 df-cmn 19766 df-abl 19767 df-mgp 20104 df-rng 20122 df-ur 20151 df-ring 20204 df-subrng 20512 df-subrg 20537 df-lmod 20774 df-lss 20845 df-psr 21876 df-mvr 21877 df-mpl 21878 df-opsr 21880 df-psr1 22139 df-vr1 22140 df-ply1 22141 df-coe1 22142 |
This theorem is referenced by: coe1pwmulfv 22241 |
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