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Mirrors > Home > MPE Home > Th. List > coe1pwmulfv | Structured version Visualization version GIF version |
Description: Function value of a right-multiplication by a variable power in the shifted domain. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
Ref | Expression |
---|---|
coe1pwmul.z | ⊢ 0 = (0g‘𝑅) |
coe1pwmul.p | ⊢ 𝑃 = (Poly1‘𝑅) |
coe1pwmul.x | ⊢ 𝑋 = (var1‘𝑅) |
coe1pwmul.n | ⊢ 𝑁 = (mulGrp‘𝑃) |
coe1pwmul.e | ⊢ ↑ = (.g‘𝑁) |
coe1pwmul.b | ⊢ 𝐵 = (Base‘𝑃) |
coe1pwmul.t | ⊢ · = (.r‘𝑃) |
coe1pwmul.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
coe1pwmul.a | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
coe1pwmul.d | ⊢ (𝜑 → 𝐷 ∈ ℕ0) |
coe1pwmulfv.y | ⊢ (𝜑 → 𝑌 ∈ ℕ0) |
Ref | Expression |
---|---|
coe1pwmulfv | ⊢ (𝜑 → ((coe1‘((𝐷 ↑ 𝑋) · 𝐴))‘(𝐷 + 𝑌)) = ((coe1‘𝐴)‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coe1pwmul.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
2 | coe1pwmul.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | coe1pwmul.x | . . . 4 ⊢ 𝑋 = (var1‘𝑅) | |
4 | coe1pwmul.n | . . . 4 ⊢ 𝑁 = (mulGrp‘𝑃) | |
5 | coe1pwmul.e | . . . 4 ⊢ ↑ = (.g‘𝑁) | |
6 | coe1pwmul.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
7 | coe1pwmul.t | . . . 4 ⊢ · = (.r‘𝑃) | |
8 | coe1pwmul.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
9 | coe1pwmul.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
10 | coe1pwmul.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℕ0) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | coe1pwmul 21200 | . . 3 ⊢ (𝜑 → (coe1‘((𝐷 ↑ 𝑋) · 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))) |
12 | 11 | fveq1d 6719 | . 2 ⊢ (𝜑 → ((coe1‘((𝐷 ↑ 𝑋) · 𝐴))‘(𝐷 + 𝑌)) = ((𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))‘(𝐷 + 𝑌))) |
13 | coe1pwmulfv.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℕ0) | |
14 | 10, 13 | nn0addcld 12154 | . . . 4 ⊢ (𝜑 → (𝐷 + 𝑌) ∈ ℕ0) |
15 | breq2 5057 | . . . . . 6 ⊢ (𝑥 = (𝐷 + 𝑌) → (𝐷 ≤ 𝑥 ↔ 𝐷 ≤ (𝐷 + 𝑌))) | |
16 | fvoveq1 7236 | . . . . . 6 ⊢ (𝑥 = (𝐷 + 𝑌) → ((coe1‘𝐴)‘(𝑥 − 𝐷)) = ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷))) | |
17 | 15, 16 | ifbieq1d 4463 | . . . . 5 ⊢ (𝑥 = (𝐷 + 𝑌) → if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ) = if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 )) |
18 | eqid 2737 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 )) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 )) | |
19 | fvex 6730 | . . . . . 6 ⊢ ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) ∈ V | |
20 | 1 | fvexi 6731 | . . . . . 6 ⊢ 0 ∈ V |
21 | 19, 20 | ifex 4489 | . . . . 5 ⊢ if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 ) ∈ V |
22 | 17, 18, 21 | fvmpt 6818 | . . . 4 ⊢ ((𝐷 + 𝑌) ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))‘(𝐷 + 𝑌)) = if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 )) |
23 | 14, 22 | syl 17 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))‘(𝐷 + 𝑌)) = if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 )) |
24 | 10 | nn0red 12151 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
25 | nn0addge1 12136 | . . . . 5 ⊢ ((𝐷 ∈ ℝ ∧ 𝑌 ∈ ℕ0) → 𝐷 ≤ (𝐷 + 𝑌)) | |
26 | 24, 13, 25 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐷 ≤ (𝐷 + 𝑌)) |
27 | 26 | iftrued 4447 | . . 3 ⊢ (𝜑 → if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 ) = ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷))) |
28 | 10 | nn0cnd 12152 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
29 | 13 | nn0cnd 12152 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℂ) |
30 | 28, 29 | pncan2d 11191 | . . . 4 ⊢ (𝜑 → ((𝐷 + 𝑌) − 𝐷) = 𝑌) |
31 | 30 | fveq2d 6721 | . . 3 ⊢ (𝜑 → ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) = ((coe1‘𝐴)‘𝑌)) |
32 | 23, 27, 31 | 3eqtrd 2781 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))‘(𝐷 + 𝑌)) = ((coe1‘𝐴)‘𝑌)) |
33 | 12, 32 | eqtrd 2777 | 1 ⊢ (𝜑 → ((coe1‘((𝐷 ↑ 𝑋) · 𝐴))‘(𝐷 + 𝑌)) = ((coe1‘𝐴)‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ifcif 4439 class class class wbr 5053 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 + caddc 10732 ≤ cle 10868 − cmin 11062 ℕ0cn0 12090 Basecbs 16760 .rcmulr 16803 0gc0g 16944 .gcmg 18488 mulGrpcmgp 19504 Ringcrg 19562 var1cv1 21097 Poly1cpl1 21098 coe1cco1 21099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-ofr 7470 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-fzo 13239 df-seq 13575 df-hash 13897 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-tset 16821 df-ple 16822 df-0g 16946 df-gsum 16947 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-mulg 18489 df-subg 18540 df-ghm 18620 df-cntz 18711 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-subrg 19798 df-lmod 19901 df-lss 19969 df-psr 20868 df-mvr 20869 df-mpl 20870 df-opsr 20872 df-psr1 21101 df-vr1 21102 df-ply1 21103 df-coe1 21104 |
This theorem is referenced by: hbtlem4 40654 |
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