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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 2823 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
2 | coe1sclmul.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
3 | coe1sclmul.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | eqid 2823 | . . 3 ⊢ (var1‘𝑅) = (var1‘𝑅) | |
5 | eqid 2823 | . . 3 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
6 | eqid 2823 | . . 3 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
7 | eqid 2823 | . . 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 11915 | . . . 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 20447 | . 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 20451 | . . . 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 7180 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘((𝐴‘𝑋) ∙ 𝑌)) = (coe1‘((𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) ∙ 𝑌))) |
21 | nn0ex 11906 | . . . . 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 6687 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → ((coe1‘𝑌)‘𝑥) ∈ V) | |
25 | fconstmpt 5616 | . . . . 5 ⊢ (ℕ0 × {𝑋}) = (𝑥 ∈ ℕ0 ↦ 𝑋) | |
26 | 25 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (ℕ0 × {𝑋}) = (𝑥 ∈ ℕ0 ↦ 𝑋)) |
27 | eqid 2823 | . . . . . . 7 ⊢ (coe1‘𝑌) = (coe1‘𝑌) | |
28 | 27, 8, 3, 2 | coe1f 20381 | . . . . . 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 6735 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘𝑌) = (𝑥 ∈ ℕ0 ↦ ((coe1‘𝑌)‘𝑥))) |
31 | 22, 23, 24, 26, 30 | offval2 7428 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ((ℕ0 × {𝑋}) ∘f · (coe1‘𝑌)) = (𝑥 ∈ ℕ0 ↦ (𝑋 · ((coe1‘𝑌)‘𝑥)))) |
32 | nn0ge0 11925 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → 0 ≤ 𝑥) | |
33 | 32 | iftrued 4477 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅)) = (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0)))) |
34 | nn0cn 11910 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ) | |
35 | 34 | subid1d 10988 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ0 → (𝑥 − 0) = 𝑥) |
36 | 35 | fveq2d 6676 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → ((coe1‘𝑌)‘(𝑥 − 0)) = ((coe1‘𝑌)‘𝑥)) |
37 | 36 | oveq2d 7174 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))) = (𝑋 · ((coe1‘𝑌)‘𝑥))) |
38 | 33, 37 | eqtrd 2858 | . . . 4 ⊢ (𝑥 ∈ ℕ0 → if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅)) = (𝑋 · ((coe1‘𝑌)‘𝑥))) |
39 | 38 | mpteq2ia 5159 | . . 3 ⊢ (𝑥 ∈ ℕ0 ↦ if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅))) = (𝑥 ∈ ℕ0 ↦ (𝑋 · ((coe1‘𝑌)‘𝑥))) |
40 | 31, 39 | syl6eqr 2876 | . 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 2868 | 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 3496 ifcif 4469 {csn 4569 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 0cc0 10539 ≤ cle 10678 − cmin 10872 ℕ0cn0 11900 Basecbs 16485 .rcmulr 16568 ·𝑠 cvsca 16571 0gc0g 16715 .gcmg 18226 mulGrpcmgp 19241 Ringcrg 19299 algSccascl 20086 var1cv1 20346 Poly1cpl1 20347 coe1cco1 20348 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-tset 16586 df-ple 16587 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-subrg 19535 df-lmod 19638 df-lss 19706 df-ascl 20089 df-psr 20138 df-mvr 20139 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-vr1 20351 df-ply1 20352 df-coe1 20353 |
This theorem is referenced by: coe1sclmulfv 20453 deg1mul3 24711 uc1pmon1p 24747 |
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