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Mirrors > Home > MPE Home > Th. List > coe1sclmul | Structured version Visualization version GIF version |
Description: Coefficient vector of a polynomial multiplied on the left 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 |
---|---|
coe1sclmul | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘((𝐴‘𝑋) ∙ 𝑌)) = ((ℕ0 × {𝑋}) ∘𝑓 · (coe1‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2758 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
2 | coe1sclmul.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
3 | coe1sclmul.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | eqid 2758 | . . 3 ⊢ (var1‘𝑅) = (var1‘𝑅) | |
5 | eqid 2758 | . . 3 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
6 | eqid 2758 | . . 3 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
7 | eqid 2758 | . . 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 1133 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
12 | simp1 1131 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑅 ∈ Ring) | |
13 | simp2 1132 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐾) | |
14 | 0nn0 11497 | . . . 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 | coe1tmmul 19847 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘((𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) ∙ 𝑌)) = (𝑥 ∈ ℕ0 ↦ if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅)))) |
17 | coe1sclmul.a | . . . . . 6 ⊢ 𝐴 = (algSc‘𝑃) | |
18 | 2, 3, 4, 5, 6, 7, 17 | ply1scltm 19851 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) |
19 | 18 | 3adant3 1127 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) |
20 | 19 | oveq1d 6826 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ((𝐴‘𝑋) ∙ 𝑌) = ((𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) ∙ 𝑌)) |
21 | 20 | fveq2d 6354 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘((𝐴‘𝑋) ∙ 𝑌)) = (coe1‘((𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) ∙ 𝑌))) |
22 | nn0ex 11488 | . . . . 5 ⊢ ℕ0 ∈ V | |
23 | 22 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ℕ0 ∈ V) |
24 | simpl2 1230 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑋 ∈ 𝐾) | |
25 | fvexd 6362 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → ((coe1‘𝑌)‘𝑥) ∈ V) | |
26 | fconstmpt 5318 | . . . . 5 ⊢ (ℕ0 × {𝑋}) = (𝑥 ∈ ℕ0 ↦ 𝑋) | |
27 | 26 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (ℕ0 × {𝑋}) = (𝑥 ∈ ℕ0 ↦ 𝑋)) |
28 | eqid 2758 | . . . . . . 7 ⊢ (coe1‘𝑌) = (coe1‘𝑌) | |
29 | 28, 8, 3, 2 | coe1f 19781 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (coe1‘𝑌):ℕ0⟶𝐾) |
30 | 29 | 3ad2ant3 1130 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘𝑌):ℕ0⟶𝐾) |
31 | 30 | feqmptd 6409 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘𝑌) = (𝑥 ∈ ℕ0 ↦ ((coe1‘𝑌)‘𝑥))) |
32 | 23, 24, 25, 27, 31 | offval2 7077 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ((ℕ0 × {𝑋}) ∘𝑓 · (coe1‘𝑌)) = (𝑥 ∈ ℕ0 ↦ (𝑋 · ((coe1‘𝑌)‘𝑥)))) |
33 | nn0ge0 11508 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → 0 ≤ 𝑥) | |
34 | 33 | iftrued 4236 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅)) = (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0)))) |
35 | nn0cn 11492 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ) | |
36 | 35 | subid1d 10571 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ0 → (𝑥 − 0) = 𝑥) |
37 | 36 | fveq2d 6354 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → ((coe1‘𝑌)‘(𝑥 − 0)) = ((coe1‘𝑌)‘𝑥)) |
38 | 37 | oveq2d 6827 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))) = (𝑋 · ((coe1‘𝑌)‘𝑥))) |
39 | 34, 38 | eqtrd 2792 | . . . 4 ⊢ (𝑥 ∈ ℕ0 → if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅)) = (𝑋 · ((coe1‘𝑌)‘𝑥))) |
40 | 39 | mpteq2ia 4890 | . . 3 ⊢ (𝑥 ∈ ℕ0 ↦ if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅))) = (𝑥 ∈ ℕ0 ↦ (𝑋 · ((coe1‘𝑌)‘𝑥))) |
41 | 32, 40 | syl6eqr 2810 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ((ℕ0 × {𝑋}) ∘𝑓 · (coe1‘𝑌)) = (𝑥 ∈ ℕ0 ↦ if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅)))) |
42 | 16, 21, 41 | 3eqtr4d 2802 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘((𝐴‘𝑋) ∙ 𝑌)) = ((ℕ0 × {𝑋}) ∘𝑓 · (coe1‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1630 ∈ wcel 2137 Vcvv 3338 ifcif 4228 {csn 4319 class class class wbr 4802 ↦ cmpt 4879 × cxp 5262 ⟶wf 6043 ‘cfv 6047 (class class class)co 6811 ∘𝑓 cof 7058 0cc0 10126 ≤ cle 10265 − cmin 10456 ℕ0cn0 11482 Basecbs 16057 .rcmulr 16142 ·𝑠 cvsca 16145 0gc0g 16300 .gcmg 17739 mulGrpcmgp 18687 Ringcrg 18745 algSccascl 19511 var1cv1 19746 Poly1cpl1 19747 coe1cco1 19748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-rep 4921 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 ax-inf2 8709 ax-cnex 10182 ax-resscn 10183 ax-1cn 10184 ax-icn 10185 ax-addcl 10186 ax-addrcl 10187 ax-mulcl 10188 ax-mulrcl 10189 ax-mulcom 10190 ax-addass 10191 ax-mulass 10192 ax-distr 10193 ax-i2m1 10194 ax-1ne0 10195 ax-1rid 10196 ax-rnegex 10197 ax-rrecex 10198 ax-cnre 10199 ax-pre-lttri 10200 ax-pre-lttrn 10201 ax-pre-ltadd 10202 ax-pre-mulgt0 10203 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-nel 3034 df-ral 3053 df-rex 3054 df-reu 3055 df-rmo 3056 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-pss 3729 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-tp 4324 df-op 4326 df-uni 4587 df-int 4626 df-iun 4672 df-iin 4673 df-br 4803 df-opab 4863 df-mpt 4880 df-tr 4903 df-id 5172 df-eprel 5177 df-po 5185 df-so 5186 df-fr 5223 df-se 5224 df-we 5225 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-pred 5839 df-ord 5885 df-on 5886 df-lim 5887 df-suc 5888 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-isom 6056 df-riota 6772 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-of 7060 df-ofr 7061 df-om 7229 df-1st 7331 df-2nd 7332 df-supp 7462 df-wrecs 7574 df-recs 7635 df-rdg 7673 df-1o 7727 df-2o 7728 df-oadd 7731 df-er 7909 df-map 8023 df-pm 8024 df-ixp 8073 df-en 8120 df-dom 8121 df-sdom 8122 df-fin 8123 df-fsupp 8439 df-oi 8578 df-card 8953 df-pnf 10266 df-mnf 10267 df-xr 10268 df-ltxr 10269 df-le 10270 df-sub 10458 df-neg 10459 df-nn 11211 df-2 11269 df-3 11270 df-4 11271 df-5 11272 df-6 11273 df-7 11274 df-8 11275 df-9 11276 df-n0 11483 df-z 11568 df-dec 11684 df-uz 11878 df-fz 12518 df-fzo 12658 df-seq 12994 df-hash 13310 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-ress 16065 df-plusg 16154 df-mulr 16155 df-sca 16157 df-vsca 16158 df-tset 16160 df-ple 16161 df-0g 16302 df-gsum 16303 df-mre 16446 df-mrc 16447 df-acs 16449 df-mgm 17441 df-sgrp 17483 df-mnd 17494 df-mhm 17534 df-submnd 17535 df-grp 17624 df-minusg 17625 df-sbg 17626 df-mulg 17740 df-subg 17790 df-ghm 17857 df-cntz 17948 df-cmn 18393 df-abl 18394 df-mgp 18688 df-ur 18700 df-ring 18747 df-subrg 18978 df-lmod 19065 df-lss 19133 df-ascl 19514 df-psr 19556 df-mvr 19557 df-mpl 19558 df-opsr 19560 df-psr1 19750 df-vr1 19751 df-ply1 19752 df-coe1 19753 |
This theorem is referenced by: coe1sclmulfv 19853 deg1mul3 24072 uc1pmon1p 24108 |
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