Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2821 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | coe1pwmul.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | coe1pwmul.x | . . 3 ⊢ 𝑋 = (var1‘𝑅) | |
5 | eqid 2821 | . . 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 2821 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | coe1pwmul.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
12 | coe1pwmul.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
13 | eqid 2821 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
14 | 2, 13 | ringidcl 19312 | . . . 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 20439 | . 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 20415 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
19 | 12, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
20 | 19 | fveq2d 6668 | . . . . 5 ⊢ (𝜑 → (1r‘𝑅) = (1r‘(Scalar‘𝑃))) |
21 | 20 | oveq1d 7165 | . . . 4 ⊢ (𝜑 → ((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) = ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋))) |
22 | 3 | ply1lmod 20414 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
23 | 12, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ LMod) |
24 | 3 | ply1ring 20410 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
25 | 6 | ringmgp 19297 | . . . . . . 7 ⊢ (𝑃 ∈ Ring → 𝑁 ∈ Mnd) |
26 | 12, 24, 25 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ Mnd) |
27 | 4, 3, 8 | vr1cl 20379 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
28 | 12, 27 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
29 | 6, 8 | mgpbas 19239 | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑁) |
30 | 29, 7 | mulgnn0cl 18238 | . . . . . 6 ⊢ ((𝑁 ∈ Mnd ∧ 𝐷 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝐷 ↑ 𝑋) ∈ 𝐵) |
31 | 26, 16, 28, 30 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → (𝐷 ↑ 𝑋) ∈ 𝐵) |
32 | eqid 2821 | . . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
33 | eqid 2821 | . . . . . 6 ⊢ (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃)) | |
34 | 8, 32, 5, 33 | lmodvs1 19656 | . . . . 5 ⊢ ((𝑃 ∈ LMod ∧ (𝐷 ↑ 𝑋) ∈ 𝐵) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) = (𝐷 ↑ 𝑋)) |
35 | 23, 31, 34 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) = (𝐷 ↑ 𝑋)) |
36 | 21, 35 | eqtrd 2856 | . . 3 ⊢ (𝜑 → ((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) = (𝐷 ↑ 𝑋)) |
37 | 36 | fvoveq1d 7172 | . 2 ⊢ (𝜑 → (coe1‘(((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐷 ↑ 𝑋)) · 𝐴)) = (coe1‘((𝐷 ↑ 𝑋) · 𝐴))) |
38 | 12 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝑅 ∈ Ring) |
39 | eqid 2821 | . . . . . . . . 9 ⊢ (coe1‘𝐴) = (coe1‘𝐴) | |
40 | 39, 8, 3, 2 | coe1f 20373 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (coe1‘𝐴):ℕ0⟶(Base‘𝑅)) |
41 | 11, 40 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (coe1‘𝐴):ℕ0⟶(Base‘𝑅)) |
42 | 41 | ad2antrr 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (coe1‘𝐴):ℕ0⟶(Base‘𝑅)) |
43 | 16 | ad2antrr 724 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝐷 ∈ ℕ0) |
44 | simplr 767 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝑥 ∈ ℕ0) | |
45 | simpr 487 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝐷 ≤ 𝑥) | |
46 | nn0sub2 12037 | . . . . . . 7 ⊢ ((𝐷 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥) → (𝑥 − 𝐷) ∈ ℕ0) | |
47 | 43, 44, 45, 46 | syl3anc 1367 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (𝑥 − 𝐷) ∈ ℕ0) |
48 | 42, 47 | ffvelrnd 6846 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → ((coe1‘𝐴)‘(𝑥 − 𝐷)) ∈ (Base‘𝑅)) |
49 | 2, 10, 13 | ringlidm 19315 | . . . . 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 586 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → ((1r‘𝑅)(.r‘𝑅)((coe1‘𝐴)‘(𝑥 − 𝐷))) = ((coe1‘𝐴)‘(𝑥 − 𝐷))) |
51 | 50 | ifeq1da 4496 | . . 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 5153 | . 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 2864 | 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 398 = wceq 1533 ∈ wcel 2110 ifcif 4466 class class class wbr 5058 ↦ cmpt 5138 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ≤ cle 10670 − cmin 10864 ℕ0cn0 11891 Basecbs 16477 .rcmulr 16560 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 Mndcmnd 17905 .gcmg 18218 mulGrpcmgp 19233 1rcur 19245 Ringcrg 19291 LModclmod 19628 var1cv1 20338 Poly1cpl1 20339 coe1cco1 20340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-tset 16578 df-ple 16579 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-subrg 19527 df-lmod 19630 df-lss 19698 df-psr 20130 df-mvr 20131 df-mpl 20132 df-opsr 20134 df-psr1 20342 df-vr1 20343 df-ply1 20344 df-coe1 20345 |
This theorem is referenced by: coe1pwmulfv 20442 |
Copyright terms: Public domain | W3C validator |