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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 2736 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | coe1pwmul.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | coe1pwmul.x | . . 3 ⊢ 𝑋 = (var1‘𝑅) | |
5 | eqid 2736 | . . 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 2736 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | coe1pwmul.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
12 | coe1pwmul.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
13 | eqid 2736 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
14 | 2, 13 | ringidcl 19540 | . . . 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 21152 | . 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 21128 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
19 | 12, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
20 | 19 | fveq2d 6699 | . . . . 5 ⊢ (𝜑 → (1r‘𝑅) = (1r‘(Scalar‘𝑃))) |
21 | 20 | oveq1d 7206 | . . . 4 ⊢ (𝜑 → ((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) = ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋))) |
22 | 3 | ply1lmod 21127 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
23 | 12, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ LMod) |
24 | 3 | ply1ring 21123 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
25 | 6 | ringmgp 19522 | . . . . . . 7 ⊢ (𝑃 ∈ Ring → 𝑁 ∈ Mnd) |
26 | 12, 24, 25 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ Mnd) |
27 | 4, 3, 8 | vr1cl 21092 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
28 | 12, 27 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
29 | 6, 8 | mgpbas 19464 | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑁) |
30 | 29, 7 | mulgnn0cl 18462 | . . . . . 6 ⊢ ((𝑁 ∈ Mnd ∧ 𝐷 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝐷 ↑ 𝑋) ∈ 𝐵) |
31 | 26, 16, 28, 30 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝐷 ↑ 𝑋) ∈ 𝐵) |
32 | eqid 2736 | . . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
33 | eqid 2736 | . . . . . 6 ⊢ (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃)) | |
34 | 8, 32, 5, 33 | lmodvs1 19881 | . . . . 5 ⊢ ((𝑃 ∈ LMod ∧ (𝐷 ↑ 𝑋) ∈ 𝐵) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) = (𝐷 ↑ 𝑋)) |
35 | 23, 31, 34 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) = (𝐷 ↑ 𝑋)) |
36 | 21, 35 | eqtrd 2771 | . . 3 ⊢ (𝜑 → ((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) = (𝐷 ↑ 𝑋)) |
37 | 36 | fvoveq1d 7213 | . 2 ⊢ (𝜑 → (coe1‘(((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) · 𝐴)) = (coe1‘((𝐷 ↑ 𝑋) · 𝐴))) |
38 | 12 | ad2antrr 726 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝑅 ∈ Ring) |
39 | eqid 2736 | . . . . . . . . 9 ⊢ (coe1‘𝐴) = (coe1‘𝐴) | |
40 | 39, 8, 3, 2 | coe1f 21086 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (coe1‘𝐴):ℕ0⟶(Base‘𝑅)) |
41 | 11, 40 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (coe1‘𝐴):ℕ0⟶(Base‘𝑅)) |
42 | 41 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (coe1‘𝐴):ℕ0⟶(Base‘𝑅)) |
43 | 16 | ad2antrr 726 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝐷 ∈ ℕ0) |
44 | simplr 769 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝑥 ∈ ℕ0) | |
45 | simpr 488 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝐷 ≤ 𝑥) | |
46 | nn0sub2 12203 | . . . . . . 7 ⊢ ((𝐷 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥) → (𝑥 − 𝐷) ∈ ℕ0) | |
47 | 43, 44, 45, 46 | syl3anc 1373 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (𝑥 − 𝐷) ∈ ℕ0) |
48 | 42, 47 | ffvelrnd 6883 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → ((coe1‘𝐴)‘(𝑥 − 𝐷)) ∈ (Base‘𝑅)) |
49 | 2, 10, 13 | ringlidm 19543 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐴)‘(𝑥 − 𝐷)) ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)((coe1‘𝐴)‘(𝑥 − 𝐷))) = ((coe1‘𝐴)‘(𝑥 − 𝐷))) |
50 | 38, 48, 49 | syl2anc 587 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → ((1r‘𝑅)(.r‘𝑅)((coe1‘𝐴)‘(𝑥 − 𝐷))) = ((coe1‘𝐴)‘(𝑥 − 𝐷))) |
51 | 50 | ifeq1da 4456 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → if(𝐷 ≤ 𝑥, ((1r‘𝑅)(.r‘𝑅)((coe1‘𝐴)‘(𝑥 − 𝐷))), 0 ) = if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 )) |
52 | 51 | mpteq2dva 5135 | . 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 ))) |
53 | 17, 37, 52 | 3eqtr3d 2779 | 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 399 = wceq 1543 ∈ wcel 2112 ifcif 4425 class class class wbr 5039 ↦ cmpt 5120 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ≤ cle 10833 − cmin 11027 ℕ0cn0 12055 Basecbs 16666 .rcmulr 16750 Scalarcsca 16752 ·𝑠 cvsca 16753 0gc0g 16898 Mndcmnd 18127 .gcmg 18442 mulGrpcmgp 19458 1rcur 19470 Ringcrg 19516 LModclmod 19853 var1cv1 21051 Poly1cpl1 21052 coe1cco1 21053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofr 7448 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-tset 16768 df-ple 16769 df-0g 16900 df-gsum 16901 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-ghm 18574 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-subrg 19752 df-lmod 19855 df-lss 19923 df-psr 20822 df-mvr 20823 df-mpl 20824 df-opsr 20826 df-psr1 21055 df-vr1 21056 df-ply1 21057 df-coe1 21058 |
This theorem is referenced by: coe1pwmulfv 21155 |
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