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Mirrors > Home > MPE Home > Th. List > evl1vsd | Structured version Visualization version GIF version |
Description: Polynomial evaluation builder for scalar multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
evl1addd.q | β’ π = (eval1βπ ) |
evl1addd.p | β’ π = (Poly1βπ ) |
evl1addd.b | β’ π΅ = (Baseβπ ) |
evl1addd.u | β’ π = (Baseβπ) |
evl1addd.1 | β’ (π β π β CRing) |
evl1addd.2 | β’ (π β π β π΅) |
evl1addd.3 | β’ (π β (π β π β§ ((πβπ)βπ) = π)) |
evl1vsd.4 | β’ (π β π β π΅) |
evl1vsd.s | β’ β = ( Β·π βπ) |
evl1vsd.t | β’ Β· = (.rβπ ) |
Ref | Expression |
---|---|
evl1vsd | β’ (π β ((π β π) β π β§ ((πβ(π β π))βπ) = (π Β· π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1addd.q | . . 3 β’ π = (eval1βπ ) | |
2 | evl1addd.p | . . 3 β’ π = (Poly1βπ ) | |
3 | evl1addd.b | . . 3 β’ π΅ = (Baseβπ ) | |
4 | evl1addd.u | . . 3 β’ π = (Baseβπ) | |
5 | evl1addd.1 | . . 3 β’ (π β π β CRing) | |
6 | evl1addd.2 | . . 3 β’ (π β π β π΅) | |
7 | eqid 2726 | . . . 4 β’ (algScβπ) = (algScβπ) | |
8 | evl1vsd.4 | . . . 4 β’ (π β π β π΅) | |
9 | 1, 2, 3, 7, 4, 5, 8, 6 | evl1scad 22204 | . . 3 β’ (π β (((algScβπ)βπ) β π β§ ((πβ((algScβπ)βπ))βπ) = π)) |
10 | evl1addd.3 | . . 3 β’ (π β (π β π β§ ((πβπ)βπ) = π)) | |
11 | eqid 2726 | . . 3 β’ (.rβπ) = (.rβπ) | |
12 | evl1vsd.t | . . 3 β’ Β· = (.rβπ ) | |
13 | 1, 2, 3, 4, 5, 6, 9, 10, 11, 12 | evl1muld 22212 | . 2 β’ (π β ((((algScβπ)βπ)(.rβπ)π) β π β§ ((πβ(((algScβπ)βπ)(.rβπ)π))βπ) = (π Β· π))) |
14 | 2 | ply1assa 22068 | . . . . . 6 β’ (π β CRing β π β AssAlg) |
15 | 5, 14 | syl 17 | . . . . 5 β’ (π β π β AssAlg) |
16 | 2 | ply1sca 22121 | . . . . . . . . 9 β’ (π β CRing β π = (Scalarβπ)) |
17 | 5, 16 | syl 17 | . . . . . . . 8 β’ (π β π = (Scalarβπ)) |
18 | 17 | fveq2d 6888 | . . . . . . 7 β’ (π β (Baseβπ ) = (Baseβ(Scalarβπ))) |
19 | 3, 18 | eqtrid 2778 | . . . . . 6 β’ (π β π΅ = (Baseβ(Scalarβπ))) |
20 | 8, 19 | eleqtrd 2829 | . . . . 5 β’ (π β π β (Baseβ(Scalarβπ))) |
21 | 10 | simpld 494 | . . . . 5 β’ (π β π β π) |
22 | eqid 2726 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
23 | eqid 2726 | . . . . . 6 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
24 | evl1vsd.s | . . . . . 6 β’ β = ( Β·π βπ) | |
25 | 7, 22, 23, 4, 11, 24 | asclmul1 21775 | . . . . 5 β’ ((π β AssAlg β§ π β (Baseβ(Scalarβπ)) β§ π β π) β (((algScβπ)βπ)(.rβπ)π) = (π β π)) |
26 | 15, 20, 21, 25 | syl3anc 1368 | . . . 4 β’ (π β (((algScβπ)βπ)(.rβπ)π) = (π β π)) |
27 | 26 | eleq1d 2812 | . . 3 β’ (π β ((((algScβπ)βπ)(.rβπ)π) β π β (π β π) β π)) |
28 | 26 | fveq2d 6888 | . . . . 5 β’ (π β (πβ(((algScβπ)βπ)(.rβπ)π)) = (πβ(π β π))) |
29 | 28 | fveq1d 6886 | . . . 4 β’ (π β ((πβ(((algScβπ)βπ)(.rβπ)π))βπ) = ((πβ(π β π))βπ)) |
30 | 29 | eqeq1d 2728 | . . 3 β’ (π β (((πβ(((algScβπ)βπ)(.rβπ)π))βπ) = (π Β· π) β ((πβ(π β π))βπ) = (π Β· π))) |
31 | 27, 30 | anbi12d 630 | . 2 β’ (π β (((((algScβπ)βπ)(.rβπ)π) β π β§ ((πβ(((algScβπ)βπ)(.rβπ)π))βπ) = (π Β· π)) β ((π β π) β π β§ ((πβ(π β π))βπ) = (π Β· π)))) |
32 | 13, 31 | mpbid 231 | 1 β’ (π β ((π β π) β π β§ ((πβ(π β π))βπ) = (π Β· π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 Basecbs 17150 .rcmulr 17204 Scalarcsca 17206 Β·π cvsca 17207 CRingccrg 20136 AssAlgcasa 21740 algSccascl 21742 Poly1cpl1 22046 eval1ce1 22183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14293 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-hom 17227 df-cco 17228 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-submnd 18711 df-grp 18863 df-minusg 18864 df-sbg 18865 df-mulg 18993 df-subg 19047 df-ghm 19136 df-cntz 19230 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-srg 20089 df-ring 20137 df-cring 20138 df-rhm 20371 df-subrng 20443 df-subrg 20468 df-lmod 20705 df-lss 20776 df-lsp 20816 df-assa 21743 df-asp 21744 df-ascl 21745 df-psr 21798 df-mvr 21799 df-mpl 21800 df-opsr 21802 df-evls 21972 df-evl 21973 df-psr1 22049 df-ply1 22051 df-evl1 22185 |
This theorem is referenced by: evl1scvarpwval 22233 fta1blem 26055 plypf1 26096 evls1vsca 33159 |
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