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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 20446 | . . 3 ⊢ (𝜑 → (coe1‘((𝐷 ↑ 𝑋) · 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))) |
12 | 11 | fveq1d 6671 | . 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 11958 | . . . 4 ⊢ (𝜑 → (𝐷 + 𝑌) ∈ ℕ0) |
15 | breq2 5069 | . . . . . 6 ⊢ (𝑥 = (𝐷 + 𝑌) → (𝐷 ≤ 𝑥 ↔ 𝐷 ≤ (𝐷 + 𝑌))) | |
16 | fvoveq1 7178 | . . . . . 6 ⊢ (𝑥 = (𝐷 + 𝑌) → ((coe1‘𝐴)‘(𝑥 − 𝐷)) = ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷))) | |
17 | 15, 16 | ifbieq1d 4489 | . . . . 5 ⊢ (𝑥 = (𝐷 + 𝑌) → if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ) = if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 )) |
18 | eqid 2821 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 )) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 )) | |
19 | fvex 6682 | . . . . . 6 ⊢ ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) ∈ V | |
20 | 1 | fvexi 6683 | . . . . . 6 ⊢ 0 ∈ V |
21 | 19, 20 | ifex 4514 | . . . . 5 ⊢ if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 ) ∈ V |
22 | 17, 18, 21 | fvmpt 6767 | . . . 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 11955 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
25 | nn0addge1 11942 | . . . . 5 ⊢ ((𝐷 ∈ ℝ ∧ 𝑌 ∈ ℕ0) → 𝐷 ≤ (𝐷 + 𝑌)) | |
26 | 24, 13, 25 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐷 ≤ (𝐷 + 𝑌)) |
27 | 26 | iftrued 4474 | . . 3 ⊢ (𝜑 → if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 ) = ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷))) |
28 | 10 | nn0cnd 11956 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
29 | 13 | nn0cnd 11956 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℂ) |
30 | 28, 29 | pncan2d 10998 | . . . 4 ⊢ (𝜑 → ((𝐷 + 𝑌) − 𝐷) = 𝑌) |
31 | 30 | fveq2d 6673 | . . 3 ⊢ (𝜑 → ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) = ((coe1‘𝐴)‘𝑌)) |
32 | 23, 27, 31 | 3eqtrd 2860 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))‘(𝐷 + 𝑌)) = ((coe1‘𝐴)‘𝑌)) |
33 | 12, 32 | eqtrd 2856 | 1 ⊢ (𝜑 → ((coe1‘((𝐷 ↑ 𝑋) · 𝐴))‘(𝐷 + 𝑌)) = ((coe1‘𝐴)‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ifcif 4466 class class class wbr 5065 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 + caddc 10539 ≤ cle 10675 − cmin 10869 ℕ0cn0 11896 Basecbs 16482 .rcmulr 16565 0gc0g 16712 .gcmg 18223 mulGrpcmgp 19238 Ringcrg 19296 var1cv1 20343 Poly1cpl1 20344 coe1cco1 20345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-ofr 7409 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-tset 16583 df-ple 16584 df-0g 16714 df-gsum 16715 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-ghm 18355 df-cntz 18446 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-subrg 19532 df-lmod 19635 df-lss 19703 df-psr 20135 df-mvr 20136 df-mpl 20137 df-opsr 20139 df-psr1 20347 df-vr1 20348 df-ply1 20349 df-coe1 20350 |
This theorem is referenced by: hbtlem4 39724 |
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