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Mirrors > Home > MPE Home > Th. List > coe1sclmul2 | Structured version Visualization version GIF version |
Description: Coefficient vector of a polynomial multiplied on the right by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
Ref | Expression |
---|---|
coe1sclmul.p | ⊢ 𝑃 = (Poly1‘𝑅) |
coe1sclmul.b | ⊢ 𝐵 = (Base‘𝑃) |
coe1sclmul.k | ⊢ 𝐾 = (Base‘𝑅) |
coe1sclmul.a | ⊢ 𝐴 = (algSc‘𝑃) |
coe1sclmul.t | ⊢ ∙ = (.r‘𝑃) |
coe1sclmul.u | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
coe1sclmul2 | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘(𝑌 ∙ (𝐴‘𝑋))) = ((coe1‘𝑌) ∘f · (ℕ0 × {𝑋}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
2 | coe1sclmul.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
3 | coe1sclmul.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | eqid 2821 | . . 3 ⊢ (var1‘𝑅) = (var1‘𝑅) | |
5 | eqid 2821 | . . 3 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
6 | eqid 2821 | . . 3 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
7 | eqid 2821 | . . 3 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃)) | |
8 | coe1sclmul.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
9 | coe1sclmul.t | . . 3 ⊢ ∙ = (.r‘𝑃) | |
10 | coe1sclmul.u | . . 3 ⊢ · = (.r‘𝑅) | |
11 | simp3 1134 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
12 | simp1 1132 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑅 ∈ Ring) | |
13 | simp2 1133 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐾) | |
14 | 0nn0 11906 | . . . 4 ⊢ 0 ∈ ℕ0 | |
15 | 14 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 0 ∈ ℕ0) |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15 | coe1tmmul2 20438 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘(𝑌 ∙ (𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))) = (𝑥 ∈ ℕ0 ↦ if(0 ≤ 𝑥, (((coe1‘𝑌)‘(𝑥 − 0)) · 𝑋), (0g‘𝑅)))) |
17 | coe1sclmul.a | . . . . . 6 ⊢ 𝐴 = (algSc‘𝑃) | |
18 | 2, 3, 4, 5, 6, 7, 17 | ply1scltm 20443 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) |
19 | 18 | 3adant3 1128 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) |
20 | 19 | oveq2d 7166 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∙ (𝐴‘𝑋)) = (𝑌 ∙ (𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))) |
21 | 20 | fveq2d 6668 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘(𝑌 ∙ (𝐴‘𝑋))) = (coe1‘(𝑌 ∙ (𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))))) |
22 | nn0ex 11897 | . . . . 5 ⊢ ℕ0 ∈ V | |
23 | 22 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ℕ0 ∈ V) |
24 | fvexd 6679 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → ((coe1‘𝑌)‘𝑥) ∈ V) | |
25 | simpl2 1188 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑋 ∈ 𝐾) | |
26 | eqid 2821 | . . . . . . 7 ⊢ (coe1‘𝑌) = (coe1‘𝑌) | |
27 | 26, 8, 3, 2 | coe1f 20373 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (coe1‘𝑌):ℕ0⟶𝐾) |
28 | 27 | feqmptd 6727 | . . . . 5 ⊢ (𝑌 ∈ 𝐵 → (coe1‘𝑌) = (𝑥 ∈ ℕ0 ↦ ((coe1‘𝑌)‘𝑥))) |
29 | 28 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘𝑌) = (𝑥 ∈ ℕ0 ↦ ((coe1‘𝑌)‘𝑥))) |
30 | fconstmpt 5608 | . . . . 5 ⊢ (ℕ0 × {𝑋}) = (𝑥 ∈ ℕ0 ↦ 𝑋) | |
31 | 30 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (ℕ0 × {𝑋}) = (𝑥 ∈ ℕ0 ↦ 𝑋)) |
32 | 23, 24, 25, 29, 31 | offval2 7420 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ((coe1‘𝑌) ∘f · (ℕ0 × {𝑋})) = (𝑥 ∈ ℕ0 ↦ (((coe1‘𝑌)‘𝑥) · 𝑋))) |
33 | nn0ge0 11916 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → 0 ≤ 𝑥) | |
34 | 33 | iftrued 4474 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → if(0 ≤ 𝑥, (((coe1‘𝑌)‘(𝑥 − 0)) · 𝑋), (0g‘𝑅)) = (((coe1‘𝑌)‘(𝑥 − 0)) · 𝑋)) |
35 | nn0cn 11901 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ) | |
36 | 35 | subid1d 10980 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ0 → (𝑥 − 0) = 𝑥) |
37 | 36 | fveq2d 6668 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → ((coe1‘𝑌)‘(𝑥 − 0)) = ((coe1‘𝑌)‘𝑥)) |
38 | 37 | oveq1d 7165 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → (((coe1‘𝑌)‘(𝑥 − 0)) · 𝑋) = (((coe1‘𝑌)‘𝑥) · 𝑋)) |
39 | 34, 38 | eqtrd 2856 | . . . 4 ⊢ (𝑥 ∈ ℕ0 → if(0 ≤ 𝑥, (((coe1‘𝑌)‘(𝑥 − 0)) · 𝑋), (0g‘𝑅)) = (((coe1‘𝑌)‘𝑥) · 𝑋)) |
40 | 39 | mpteq2ia 5149 | . . 3 ⊢ (𝑥 ∈ ℕ0 ↦ if(0 ≤ 𝑥, (((coe1‘𝑌)‘(𝑥 − 0)) · 𝑋), (0g‘𝑅))) = (𝑥 ∈ ℕ0 ↦ (((coe1‘𝑌)‘𝑥) · 𝑋)) |
41 | 32, 40 | syl6eqr 2874 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ((coe1‘𝑌) ∘f · (ℕ0 × {𝑋})) = (𝑥 ∈ ℕ0 ↦ if(0 ≤ 𝑥, (((coe1‘𝑌)‘(𝑥 − 0)) · 𝑋), (0g‘𝑅)))) |
42 | 16, 21, 41 | 3eqtr4d 2866 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘(𝑌 ∙ (𝐴‘𝑋))) = ((coe1‘𝑌) ∘f · (ℕ0 × {𝑋}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ifcif 4466 {csn 4560 class class class wbr 5058 ↦ cmpt 5138 × cxp 5547 ‘cfv 6349 (class class class)co 7150 ∘f cof 7401 0cc0 10531 ≤ cle 10670 − cmin 10864 ℕ0cn0 11891 Basecbs 16477 .rcmulr 16560 ·𝑠 cvsca 16563 0gc0g 16707 .gcmg 18218 mulGrpcmgp 19233 Ringcrg 19291 algSccascl 20078 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-ascl 20081 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: (None) |
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