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| Mirrors > Home > MPE Home > Th. List > coe1sclmulfv | Structured version Visualization version GIF version | ||
| Description: A single coefficient of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
| Ref | Expression |
|---|---|
| coe1sclmul.p | β’ π = (Poly1βπ ) |
| coe1sclmul.b | β’ π΅ = (Baseβπ) |
| coe1sclmul.k | β’ πΎ = (Baseβπ ) |
| coe1sclmul.a | β’ π΄ = (algScβπ) |
| coe1sclmul.t | β’ β = (.rβπ) |
| coe1sclmul.u | β’ Β· = (.rβπ ) |
| Ref | Expression |
|---|---|
| coe1sclmulfv | β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β ((coe1β((π΄βπ) β π))β 0 ) = (π Β· ((coe1βπ)β 0 ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | coe1sclmul.p | . . . . . 6 β’ π = (Poly1βπ ) | |
| 2 | coe1sclmul.b | . . . . . 6 β’ π΅ = (Baseβπ) | |
| 3 | coe1sclmul.k | . . . . . 6 β’ πΎ = (Baseβπ ) | |
| 4 | coe1sclmul.a | . . . . . 6 β’ π΄ = (algScβπ) | |
| 5 | coe1sclmul.t | . . . . . 6 β’ β = (.rβπ) | |
| 6 | coe1sclmul.u | . . . . . 6 β’ Β· = (.rβπ ) | |
| 7 | 1, 2, 3, 4, 5, 6 | coe1sclmul 22205 | . . . . 5 β’ ((π β Ring β§ π β πΎ β§ π β π΅) β (coe1β((π΄βπ) β π)) = ((β0 Γ {π}) βf Β· (coe1βπ))) |
| 8 | 7 | 3expb 1117 | . . . 4 β’ ((π β Ring β§ (π β πΎ β§ π β π΅)) β (coe1β((π΄βπ) β π)) = ((β0 Γ {π}) βf Β· (coe1βπ))) |
| 9 | 8 | 3adant3 1129 | . . 3 β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β (coe1β((π΄βπ) β π)) = ((β0 Γ {π}) βf Β· (coe1βπ))) |
| 10 | 9 | fveq1d 6892 | . 2 β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β ((coe1β((π΄βπ) β π))β 0 ) = (((β0 Γ {π}) βf Β· (coe1βπ))β 0 )) |
| 11 | simp3 1135 | . . 3 β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β 0 β β0) | |
| 12 | nn0ex 12503 | . . . . 5 β’ β0 β V | |
| 13 | 12 | a1i 11 | . . . 4 β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β β0 β V) |
| 14 | simp2l 1196 | . . . 4 β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β π β πΎ) | |
| 15 | simp2r 1197 | . . . . 5 β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β π β π΅) | |
| 16 | eqid 2725 | . . . . . 6 β’ (coe1βπ) = (coe1βπ) | |
| 17 | eqid 2725 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
| 18 | 16, 2, 1, 17 | coe1f 22134 | . . . . 5 β’ (π β π΅ β (coe1βπ):β0βΆ(Baseβπ )) |
| 19 | ffn 6717 | . . . . 5 β’ ((coe1βπ):β0βΆ(Baseβπ ) β (coe1βπ) Fn β0) | |
| 20 | 15, 18, 19 | 3syl 18 | . . . 4 β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β (coe1βπ) Fn β0) |
| 21 | eqidd 2726 | . . . 4 β’ (((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β§ 0 β β0) β ((coe1βπ)β 0 ) = ((coe1βπ)β 0 )) | |
| 22 | 13, 14, 20, 21 | ofc1 7706 | . . 3 β’ (((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β§ 0 β β0) β (((β0 Γ {π}) βf Β· (coe1βπ))β 0 ) = (π Β· ((coe1βπ)β 0 ))) |
| 23 | 11, 22 | mpdan 685 | . 2 β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β (((β0 Γ {π}) βf Β· (coe1βπ))β 0 ) = (π Β· ((coe1βπ)β 0 ))) |
| 24 | 10, 23 | eqtrd 2765 | 1 β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β ((coe1β((π΄βπ) β π))β 0 ) = (π Β· ((coe1βπ)β 0 ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3463 {csn 4625 Γ cxp 5671 Fn wfn 6538 βΆwf 6539 βcfv 6543 (class class class)co 7413 βf cof 7677 β0cn0 12497 Basecbs 17174 .rcmulr 17228 Ringcrg 20172 algSccascl 21785 Poly1cpl1 22099 coe1cco1 22100 |
| 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 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-fzo 13655 df-seq 13994 df-hash 14317 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-0g 17417 df-gsum 17418 df-prds 17423 df-pws 17425 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-submnd 18735 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-ghm 19167 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-subrng 20482 df-subrg 20507 df-lmod 20744 df-lss 20815 df-ascl 21788 df-psr 21841 df-mvr 21842 df-mpl 21843 df-opsr 21845 df-psr1 22102 df-vr1 22103 df-ply1 22104 df-coe1 22105 |
| This theorem is referenced by: deg1mul3le 26065 hbtlem2 42609 coe1sclmulval 47561 |
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