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Mirrors > Home > MPE Home > Th. List > cpmidpmatlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for cpmidpmat 22245. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) |
Ref | Expression |
---|---|
cpmidgsum.a | β’ π΄ = (π Mat π ) |
cpmidgsum.b | β’ π΅ = (Baseβπ΄) |
cpmidgsum.p | β’ π = (Poly1βπ ) |
cpmidgsum.y | β’ π = (π Mat π) |
cpmidgsum.x | β’ π = (var1βπ ) |
cpmidgsum.e | β’ β = (.gβ(mulGrpβπ)) |
cpmidgsum.m | β’ Β· = ( Β·π βπ) |
cpmidgsum.1 | β’ 1 = (1rβπ) |
cpmidgsum.u | β’ π = (algScβπ) |
cpmidgsum.c | β’ πΆ = (π CharPlyMat π ) |
cpmidgsum.k | β’ πΎ = (πΆβπ) |
cpmidgsum.h | β’ π» = (πΎ Β· 1 ) |
cpmidgsumm2pm.o | β’ π = (1rβπ΄) |
cpmidgsumm2pm.m | β’ β = ( Β·π βπ΄) |
cpmidgsumm2pm.t | β’ π = (π matToPolyMat π ) |
cpmidpmat.g | β’ πΊ = (π β β0 β¦ (((coe1βπΎ)βπ) β π)) |
Ref | Expression |
---|---|
cpmidpmatlem2 | β’ ((π β Fin β§ π β CRing β§ π β π΅) β πΊ β (π΅ βm β0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1192 | . . . 4 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β π β Fin) | |
2 | crngring 19984 | . . . . . 6 β’ (π β CRing β π β Ring) | |
3 | 2 | 3ad2ant2 1135 | . . . . 5 β’ ((π β Fin β§ π β CRing β§ π β π΅) β π β Ring) |
4 | 3 | adantr 482 | . . . 4 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β π β Ring) |
5 | cpmidgsum.k | . . . . . 6 β’ πΎ = (πΆβπ) | |
6 | cpmidgsum.c | . . . . . . 7 β’ πΆ = (π CharPlyMat π ) | |
7 | cpmidgsum.a | . . . . . . 7 β’ π΄ = (π Mat π ) | |
8 | cpmidgsum.b | . . . . . . 7 β’ π΅ = (Baseβπ΄) | |
9 | cpmidgsum.p | . . . . . . 7 β’ π = (Poly1βπ ) | |
10 | eqid 2733 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
11 | 6, 7, 8, 9, 10 | chpmatply1 22204 | . . . . . 6 β’ ((π β Fin β§ π β CRing β§ π β π΅) β (πΆβπ) β (Baseβπ)) |
12 | 5, 11 | eqeltrid 2838 | . . . . 5 β’ ((π β Fin β§ π β CRing β§ π β π΅) β πΎ β (Baseβπ)) |
13 | eqid 2733 | . . . . . 6 β’ (coe1βπΎ) = (coe1βπΎ) | |
14 | eqid 2733 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
15 | 13, 10, 9, 14 | coe1fvalcl 21606 | . . . . 5 β’ ((πΎ β (Baseβπ) β§ π β β0) β ((coe1βπΎ)βπ) β (Baseβπ )) |
16 | 12, 15 | sylan 581 | . . . 4 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β ((coe1βπΎ)βπ) β (Baseβπ )) |
17 | 2 | anim2i 618 | . . . . . . 7 β’ ((π β Fin β§ π β CRing) β (π β Fin β§ π β Ring)) |
18 | 7 | matring 21815 | . . . . . . 7 β’ ((π β Fin β§ π β Ring) β π΄ β Ring) |
19 | cpmidgsumm2pm.o | . . . . . . . 8 β’ π = (1rβπ΄) | |
20 | 8, 19 | ringidcl 19997 | . . . . . . 7 β’ (π΄ β Ring β π β π΅) |
21 | 17, 18, 20 | 3syl 18 | . . . . . 6 β’ ((π β Fin β§ π β CRing) β π β π΅) |
22 | 21 | 3adant3 1133 | . . . . 5 β’ ((π β Fin β§ π β CRing β§ π β π΅) β π β π΅) |
23 | 22 | adantr 482 | . . . 4 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β π β π΅) |
24 | cpmidgsumm2pm.m | . . . . 5 β’ β = ( Β·π βπ΄) | |
25 | 14, 7, 8, 24 | matvscl 21803 | . . . 4 β’ (((π β Fin β§ π β Ring) β§ (((coe1βπΎ)βπ) β (Baseβπ ) β§ π β π΅)) β (((coe1βπΎ)βπ) β π) β π΅) |
26 | 1, 4, 16, 23, 25 | syl22anc 838 | . . 3 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β (((coe1βπΎ)βπ) β π) β π΅) |
27 | cpmidpmat.g | . . 3 β’ πΊ = (π β β0 β¦ (((coe1βπΎ)βπ) β π)) | |
28 | 26, 27 | fmptd 7066 | . 2 β’ ((π β Fin β§ π β CRing β§ π β π΅) β πΊ:β0βΆπ΅) |
29 | 8 | fvexi 6860 | . . . 4 β’ π΅ β V |
30 | nn0ex 12427 | . . . 4 β’ β0 β V | |
31 | 29, 30 | pm3.2i 472 | . . 3 β’ (π΅ β V β§ β0 β V) |
32 | elmapg 8784 | . . 3 β’ ((π΅ β V β§ β0 β V) β (πΊ β (π΅ βm β0) β πΊ:β0βΆπ΅)) | |
33 | 31, 32 | mp1i 13 | . 2 β’ ((π β Fin β§ π β CRing β§ π β π΅) β (πΊ β (π΅ βm β0) β πΊ:β0βΆπ΅)) |
34 | 28, 33 | mpbird 257 | 1 β’ ((π β Fin β§ π β CRing β§ π β π΅) β πΊ β (π΅ βm β0)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3447 β¦ cmpt 5192 βΆwf 6496 βcfv 6500 (class class class)co 7361 βm cmap 8771 Fincfn 8889 β0cn0 12421 Basecbs 17091 Β·π cvsca 17145 .gcmg 18880 mulGrpcmgp 19904 1rcur 19921 Ringcrg 19972 CRingccrg 19973 algSccascl 21281 var1cv1 21570 Poly1cpl1 21571 coe1cco1 21572 Mat cmat 21777 matToPolyMat cmat2pmat 22076 CharPlyMat cchpmat 22198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-ot 4599 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-ofr 7622 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-pm 8774 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-sup 9386 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-xnn0 12494 df-z 12508 df-dec 12627 df-uz 12772 df-rp 12924 df-fz 13434 df-fzo 13577 df-seq 13916 df-exp 13977 df-hash 14240 df-word 14412 df-lsw 14460 df-concat 14468 df-s1 14493 df-substr 14538 df-pfx 14568 df-splice 14647 df-reverse 14656 df-s2 14746 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-0g 17331 df-gsum 17332 df-prds 17337 df-pws 17339 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-submnd 18610 df-efmnd 18687 df-grp 18759 df-minusg 18760 df-sbg 18761 df-mulg 18881 df-subg 18933 df-ghm 19014 df-gim 19057 df-cntz 19105 df-oppg 19132 df-symg 19157 df-pmtr 19232 df-psgn 19281 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-rnghom 20156 df-drng 20221 df-subrg 20262 df-lmod 20367 df-lss 20437 df-sra 20678 df-rgmod 20679 df-cnfld 20820 df-zring 20893 df-zrh 20927 df-dsmm 21161 df-frlm 21176 df-ascl 21284 df-psr 21334 df-mvr 21335 df-mpl 21336 df-opsr 21338 df-psr1 21574 df-vr1 21575 df-ply1 21576 df-coe1 21577 df-mamu 21756 df-mat 21778 df-mdet 21957 df-mat2pmat 22079 df-chpmat 22199 |
This theorem is referenced by: cpmidpmat 22245 |
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