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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 2610 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
2 | coe1sclmul.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
3 | coe1sclmul.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | eqid 2610 | . . 3 ⊢ (var1‘𝑅) = (var1‘𝑅) | |
5 | eqid 2610 | . . 3 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
6 | eqid 2610 | . . 3 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
7 | eqid 2610 | . . 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 1056 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
12 | simp1 1054 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑅 ∈ Ring) | |
13 | simp2 1055 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐾) | |
14 | 0nn0 11184 | . . . 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 19468 | . 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 19472 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) |
19 | 18 | 3adant3 1074 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) |
20 | 19 | oveq1d 6564 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ((𝐴‘𝑋) ∙ 𝑌) = ((𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) ∙ 𝑌)) |
21 | 20 | fveq2d 6107 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘((𝐴‘𝑋) ∙ 𝑌)) = (coe1‘((𝑋( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) ∙ 𝑌))) |
22 | nn0ex 11175 | . . . . 5 ⊢ ℕ0 ∈ V | |
23 | 22 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ℕ0 ∈ V) |
24 | simpl2 1058 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑋 ∈ 𝐾) | |
25 | fvex 6113 | . . . . 5 ⊢ ((coe1‘𝑌)‘𝑥) ∈ V | |
26 | 25 | a1i 11 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → ((coe1‘𝑌)‘𝑥) ∈ V) |
27 | fconstmpt 5085 | . . . . 5 ⊢ (ℕ0 × {𝑋}) = (𝑥 ∈ ℕ0 ↦ 𝑋) | |
28 | 27 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (ℕ0 × {𝑋}) = (𝑥 ∈ ℕ0 ↦ 𝑋)) |
29 | eqid 2610 | . . . . . . 7 ⊢ (coe1‘𝑌) = (coe1‘𝑌) | |
30 | 29, 8, 3, 2 | coe1f 19402 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (coe1‘𝑌):ℕ0⟶𝐾) |
31 | 30 | 3ad2ant3 1077 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘𝑌):ℕ0⟶𝐾) |
32 | 31 | feqmptd 6159 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘𝑌) = (𝑥 ∈ ℕ0 ↦ ((coe1‘𝑌)‘𝑥))) |
33 | 23, 24, 26, 28, 32 | offval2 6812 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ((ℕ0 × {𝑋}) ∘𝑓 · (coe1‘𝑌)) = (𝑥 ∈ ℕ0 ↦ (𝑋 · ((coe1‘𝑌)‘𝑥)))) |
34 | nn0ge0 11195 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → 0 ≤ 𝑥) | |
35 | 34 | iftrued 4044 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅)) = (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0)))) |
36 | nn0cn 11179 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ) | |
37 | 36 | subid1d 10260 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ0 → (𝑥 − 0) = 𝑥) |
38 | 37 | fveq2d 6107 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → ((coe1‘𝑌)‘(𝑥 − 0)) = ((coe1‘𝑌)‘𝑥)) |
39 | 38 | oveq2d 6565 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))) = (𝑋 · ((coe1‘𝑌)‘𝑥))) |
40 | 35, 39 | eqtrd 2644 | . . . 4 ⊢ (𝑥 ∈ ℕ0 → if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅)) = (𝑋 · ((coe1‘𝑌)‘𝑥))) |
41 | 40 | mpteq2ia 4668 | . . 3 ⊢ (𝑥 ∈ ℕ0 ↦ if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅))) = (𝑥 ∈ ℕ0 ↦ (𝑋 · ((coe1‘𝑌)‘𝑥))) |
42 | 33, 41 | syl6eqr 2662 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ((ℕ0 × {𝑋}) ∘𝑓 · (coe1‘𝑌)) = (𝑥 ∈ ℕ0 ↦ if(0 ≤ 𝑥, (𝑋 · ((coe1‘𝑌)‘(𝑥 − 0))), (0g‘𝑅)))) |
43 | 16, 21, 42 | 3eqtr4d 2654 | 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 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 0cc0 9815 ≤ cle 9954 − cmin 10145 ℕ0cn0 11169 Basecbs 15695 .rcmulr 15769 ·𝑠 cvsca 15772 0gc0g 15923 .gcmg 17363 mulGrpcmgp 18312 Ringcrg 18370 algSccascl 19132 var1cv1 19367 Poly1cpl1 19368 coe1cco1 19369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-ple 15788 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-subrg 18601 df-lmod 18688 df-lss 18754 df-ascl 19135 df-psr 19177 df-mvr 19178 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-vr1 19372 df-ply1 19373 df-coe1 19374 |
This theorem is referenced by: coe1sclmulfv 19474 deg1mul3 23679 uc1pmon1p 23715 |
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