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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 22175 | . . . . 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 1118 | . . . 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 1130 | . . 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 6893 | . 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 1136 | . . 3 β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β 0 β β0) | |
| 12 | nn0ex 12494 | . . . . 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 1197 | . . . 4 β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β π β πΎ) | |
| 15 | simp2r 1198 | . . . . 5 β’ ((π β Ring β§ (π β πΎ β§ π β π΅) β§ 0 β β0) β π β π΅) | |
| 16 | eqid 2727 | . . . . . 6 β’ (coe1βπ) = (coe1βπ) | |
| 17 | eqid 2727 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
| 18 | 16, 2, 1, 17 | coe1f 22104 | . . . . 5 β’ (π β π΅ β (coe1βπ):β0βΆ(Baseβπ )) |
| 19 | ffn 6716 | . . . . 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 2728 | . . . 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 7703 | . . 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 686 | . 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 2767 | 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 395 β§ w3a 1085 = wceq 1534 β wcel 2099 Vcvv 3469 {csn 4624 Γ cxp 5670 Fn wfn 6537 βΆwf 6538 βcfv 6542 (class class class)co 7414 βf cof 7675 β0cn0 12488 Basecbs 17165 .rcmulr 17219 Ringcrg 20157 algSccascl 21766 Poly1cpl1 22070 coe1cco1 22071 |
| 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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-ofr 7678 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-fzo 13646 df-seq 13985 df-hash 14308 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17408 df-gsum 17409 df-prds 17414 df-pws 17416 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-submnd 18726 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19008 df-subg 19062 df-ghm 19152 df-cntz 19252 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-subrng 20465 df-subrg 20490 df-lmod 20727 df-lss 20798 df-ascl 21769 df-psr 21822 df-mvr 21823 df-mpl 21824 df-opsr 21826 df-psr1 22073 df-vr1 22074 df-ply1 22075 df-coe1 22076 |
| This theorem is referenced by: deg1mul3le 26026 hbtlem2 42460 coe1sclmulval 47366 |
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