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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 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 11913 | . . . 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 20445 | . 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 | . . . . 5 ⊢ 𝐴 = (algSc‘𝑃) | |
18 | 2, 3, 4, 5, 6, 7, 17 | ply1scltm 20449 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) |
19 | 18 | 3adant3 1128 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) |
20 | 19 | fvoveq1d 7178 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘((𝐴‘𝑋) ∙ 𝑌)) = (coe1‘((𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) ∙ 𝑌))) |
21 | nn0ex 11904 | . . . . 5 ⊢ ℕ0 ∈ V | |
22 | 21 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ℕ0 ∈ V) |
23 | simpl2 1188 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑋 ∈ 𝐾) | |
24 | fvexd 6685 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → ((coe1‘𝑌)‘𝑥) ∈ V) | |
25 | fconstmpt 5614 | . . . . 5 ⊢ (ℕ0 × {𝑋}) = (𝑥 ∈ ℕ0 ↦ 𝑋) | |
26 | 25 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (ℕ0 × {𝑋}) = (𝑥 ∈ ℕ0 ↦ 𝑋)) |
27 | eqid 2821 | . . . . . . 7 ⊢ (coe1‘𝑌) = (coe1‘𝑌) | |
28 | 27, 8, 3, 2 | coe1f 20379 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (coe1‘𝑌):ℕ0⟶𝐾) |
29 | 28 | 3ad2ant3 1131 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘𝑌):ℕ0⟶𝐾) |
30 | 29 | feqmptd 6733 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘𝑌) = (𝑥 ∈ ℕ0 ↦ ((coe1‘𝑌)‘𝑥))) |
31 | 22, 23, 24, 26, 30 | offval2 7426 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ((ℕ0 × {𝑋}) ∘f · (coe1‘𝑌)) = (𝑥 ∈ ℕ0 ↦ (𝑋 · ((coe1‘𝑌)‘𝑥)))) |
32 | nn0ge0 11923 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → 0 ≤ 𝑥) | |
33 | 32 | iftrued 4475 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅)) = (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0)))) |
34 | nn0cn 11908 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ) | |
35 | 34 | subid1d 10986 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ0 → (𝑥 − 0) = 𝑥) |
36 | 35 | fveq2d 6674 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → ((coe1‘𝑌)‘(𝑥 − 0)) = ((coe1‘𝑌)‘𝑥)) |
37 | 36 | oveq2d 7172 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))) = (𝑋 · ((coe1‘𝑌)‘𝑥))) |
38 | 33, 37 | eqtrd 2856 | . . . 4 ⊢ (𝑥 ∈ ℕ0 → if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅)) = (𝑋 · ((coe1‘𝑌)‘𝑥))) |
39 | 38 | mpteq2ia 5157 | . . 3 ⊢ (𝑥 ∈ ℕ0 ↦ if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅))) = (𝑥 ∈ ℕ0 ↦ (𝑋 · ((coe1‘𝑌)‘𝑥))) |
40 | 31, 39 | syl6eqr 2874 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ((ℕ0 × {𝑋}) ∘f · (coe1‘𝑌)) = (𝑥 ∈ ℕ0 ↦ if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅)))) |
41 | 16, 20, 40 | 3eqtr4d 2866 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ifcif 4467 {csn 4567 class class class wbr 5066 ↦ cmpt 5146 × cxp 5553 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 0cc0 10537 ≤ cle 10676 − cmin 10870 ℕ0cn0 11898 Basecbs 16483 .rcmulr 16566 ·𝑠 cvsca 16569 0gc0g 16713 .gcmg 18224 mulGrpcmgp 19239 Ringcrg 19297 algSccascl 20084 var1cv1 20344 Poly1cpl1 20345 coe1cco1 20346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-tset 16584 df-ple 16585 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-subrg 19533 df-lmod 19636 df-lss 19704 df-ascl 20087 df-psr 20136 df-mvr 20137 df-mpl 20138 df-opsr 20140 df-psr1 20348 df-vr1 20349 df-ply1 20350 df-coe1 20351 |
This theorem is referenced by: coe1sclmulfv 20451 deg1mul3 24709 uc1pmon1p 24745 |
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