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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 20162 | . . 3 ⊢ (𝜑 → (coe1‘((𝐷 ↑ 𝑋) · 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))) |
12 | 11 | fveq1d 6498 | . 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 11769 | . . . 4 ⊢ (𝜑 → (𝐷 + 𝑌) ∈ ℕ0) |
15 | breq2 4929 | . . . . . 6 ⊢ (𝑥 = (𝐷 + 𝑌) → (𝐷 ≤ 𝑥 ↔ 𝐷 ≤ (𝐷 + 𝑌))) | |
16 | fvoveq1 6997 | . . . . . 6 ⊢ (𝑥 = (𝐷 + 𝑌) → ((coe1‘𝐴)‘(𝑥 − 𝐷)) = ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷))) | |
17 | 15, 16 | ifbieq1d 4367 | . . . . 5 ⊢ (𝑥 = (𝐷 + 𝑌) → if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ) = if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 )) |
18 | eqid 2772 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 )) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 )) | |
19 | fvex 6509 | . . . . . 6 ⊢ ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) ∈ V | |
20 | 1 | fvexi 6510 | . . . . . 6 ⊢ 0 ∈ V |
21 | 19, 20 | ifex 4392 | . . . . 5 ⊢ if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 ) ∈ V |
22 | 17, 18, 21 | fvmpt 6593 | . . . 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 11766 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
25 | nn0addge1 11753 | . . . . 5 ⊢ ((𝐷 ∈ ℝ ∧ 𝑌 ∈ ℕ0) → 𝐷 ≤ (𝐷 + 𝑌)) | |
26 | 24, 13, 25 | syl2anc 576 | . . . 4 ⊢ (𝜑 → 𝐷 ≤ (𝐷 + 𝑌)) |
27 | 26 | iftrued 4352 | . . 3 ⊢ (𝜑 → if(𝐷 ≤ (𝐷 + 𝑌), ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)), 0 ) = ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷))) |
28 | 10 | nn0cnd 11767 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
29 | 13 | nn0cnd 11767 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℂ) |
30 | 28, 29 | pncan2d 10798 | . . . 4 ⊢ (𝜑 → ((𝐷 + 𝑌) − 𝐷) = 𝑌) |
31 | 30 | fveq2d 6500 | . . 3 ⊢ (𝜑 → ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) = ((coe1‘𝐴)‘𝑌)) |
32 | 23, 27, 31 | 3eqtrd 2812 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 ))‘(𝐷 + 𝑌)) = ((coe1‘𝐴)‘𝑌)) |
33 | 12, 32 | eqtrd 2808 | 1 ⊢ (𝜑 → ((coe1‘((𝐷 ↑ 𝑋) · 𝐴))‘(𝐷 + 𝑌)) = ((coe1‘𝐴)‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ifcif 4344 class class class wbr 4925 ↦ cmpt 5004 ‘cfv 6185 (class class class)co 6974 ℝcr 10332 + caddc 10336 ≤ cle 10473 − cmin 10668 ℕ0cn0 11705 Basecbs 16337 .rcmulr 16420 0gc0g 16567 .gcmg 18023 mulGrpcmgp 18974 Ringcrg 19032 var1cv1 20059 Poly1cpl1 20060 coe1cco1 20061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-ofr 7226 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-fz 12707 df-fzo 12848 df-seq 13183 df-hash 13504 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-sca 16435 df-vsca 16436 df-tset 16438 df-ple 16439 df-0g 16569 df-gsum 16570 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-mulg 18024 df-subg 18072 df-ghm 18139 df-cntz 18230 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-subrg 19268 df-lmod 19370 df-lss 19438 df-psr 19862 df-mvr 19863 df-mpl 19864 df-opsr 19866 df-psr1 20063 df-vr1 20064 df-ply1 20065 df-coe1 20066 |
This theorem is referenced by: hbtlem4 39151 |
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