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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 22270 | . . 3 ⊢ (𝜑 → (coe1‘((𝐷 ↑ 𝑋) · 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))) |
12 | 11 | fveq1d 6903 | . 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 12588 | . . . 4 ⊢ (𝜑 → (𝐷 + 𝑌) ∈ ℕ0) |
15 | breq2 5157 | . . . . . 6 ⊢ (𝑥 = (𝐷 + 𝑌) → (𝐷 ≤ 𝑥 ↔ 𝐷 ≤ (𝐷 + 𝑌))) | |
16 | fvoveq1 7447 | . . . . . 6 ⊢ (𝑥 = (𝐷 + 𝑌) → ((coe1‘𝐴)‘(𝑥 − 𝐷)) = ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷))) | |
17 | 15, 16 | ifbieq1d 4557 | . . . . 5 ⊢ (𝑥 = (𝐷 + 𝑌) → if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ) = if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 )) |
18 | eqid 2726 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 )) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 )) | |
19 | fvex 6914 | . . . . . 6 ⊢ ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) ∈ V | |
20 | 1 | fvexi 6915 | . . . . . 6 ⊢ 0 ∈ V |
21 | 19, 20 | ifex 4583 | . . . . 5 ⊢ if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 ) ∈ V |
22 | 17, 18, 21 | fvmpt 7009 | . . . 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 12585 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
25 | nn0addge1 12570 | . . . . 5 ⊢ ((𝐷 ∈ ℝ ∧ 𝑌 ∈ ℕ0) → 𝐷 ≤ (𝐷 + 𝑌)) | |
26 | 24, 13, 25 | syl2anc 582 | . . . 4 ⊢ (𝜑 → 𝐷 ≤ (𝐷 + 𝑌)) |
27 | 26 | iftrued 4541 | . . 3 ⊢ (𝜑 → if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 ) = ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷))) |
28 | 10 | nn0cnd 12586 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
29 | 13 | nn0cnd 12586 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℂ) |
30 | 28, 29 | pncan2d 11623 | . . . 4 ⊢ (𝜑 → ((𝐷 + 𝑌) − 𝐷) = 𝑌) |
31 | 30 | fveq2d 6905 | . . 3 ⊢ (𝜑 → ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) = ((coe1‘𝐴)‘𝑌)) |
32 | 23, 27, 31 | 3eqtrd 2770 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))‘(𝐷 + 𝑌)) = ((coe1‘𝐴)‘𝑌)) |
33 | 12, 32 | eqtrd 2766 | 1 ⊢ (𝜑 → ((coe1‘((𝐷 ↑ 𝑋) · 𝐴))‘(𝐷 + 𝑌)) = ((coe1‘𝐴)‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ifcif 4533 class class class wbr 5153 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 ℝcr 11157 + caddc 11161 ≤ cle 11299 − cmin 11494 ℕ0cn0 12524 Basecbs 17213 .rcmulr 17267 0gc0g 17454 .gcmg 19061 mulGrpcmgp 20117 Ringcrg 20216 var1cv1 22165 Poly1cpl1 22166 coe1cco1 22167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-ofr 7691 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-sup 9485 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-fz 13539 df-fzo 13682 df-seq 14022 df-hash 14348 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-hom 17290 df-cco 17291 df-0g 17456 df-gsum 17457 df-prds 17462 df-pws 17464 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-mulg 19062 df-subg 19117 df-ghm 19207 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-subrng 20528 df-subrg 20553 df-lmod 20838 df-lss 20909 df-psr 21906 df-mvr 21907 df-mpl 21908 df-opsr 21910 df-psr1 22169 df-vr1 22170 df-ply1 22171 df-coe1 22172 |
This theorem is referenced by: hbtlem4 42787 |
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