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Mirrors > Home > MPE Home > Th. List > cpmidpmatlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for cpmidpmat 22245. (Contributed by AV, 13-Nov-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 |
---|---|
cpmidpmatlem1 | β’ (πΏ β β0 β (πΊβπΏ) = (((coe1βπΎ)βπΏ) β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6846 | . . 3 β’ (π = πΏ β ((coe1βπΎ)βπ) = ((coe1βπΎ)βπΏ)) | |
2 | 1 | oveq1d 7376 | . 2 β’ (π = πΏ β (((coe1βπΎ)βπ) β π) = (((coe1βπΎ)βπΏ) β π)) |
3 | cpmidpmat.g | . 2 β’ πΊ = (π β β0 β¦ (((coe1βπΎ)βπ) β π)) | |
4 | ovex 7394 | . 2 β’ (((coe1βπΎ)βπΏ) β π) β V | |
5 | 2, 3, 4 | fvmpt 6952 | 1 β’ (πΏ β β0 β (πΊβπΏ) = (((coe1βπΎ)βπΏ) β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β¦ cmpt 5192 βcfv 6500 (class class class)co 7361 β0cn0 12421 Basecbs 17091 Β·π cvsca 17145 .gcmg 18880 mulGrpcmgp 19904 1rcur 19921 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-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 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-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 |
This theorem is referenced by: cpmidpmat 22245 |
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