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Mirrors > Home > MPE Home > Th. List > matbas | Structured version Visualization version GIF version |
Description: The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Ref | Expression |
---|---|
matbas.a | β’ π΄ = (π Mat π ) |
matbas.g | β’ πΊ = (π freeLMod (π Γ π)) |
Ref | Expression |
---|---|
matbas | β’ ((π β Fin β§ π β π) β (BaseβπΊ) = (Baseβπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baseid 17012 | . . 3 β’ Base = Slot (Baseβndx) | |
2 | basendxnmulrndx 17102 | . . 3 β’ (Baseβndx) β (.rβndx) | |
3 | 1, 2 | setsnid 17007 | . 2 β’ (BaseβπΊ) = (Baseβ(πΊ sSet β¨(.rβndx), (π maMul β¨π, π, πβ©)β©)) |
4 | matbas.a | . . . 4 β’ π΄ = (π Mat π ) | |
5 | matbas.g | . . . 4 β’ πΊ = (π freeLMod (π Γ π)) | |
6 | eqid 2736 | . . . 4 β’ (π maMul β¨π, π, πβ©) = (π maMul β¨π, π, πβ©) | |
7 | 4, 5, 6 | matval 21664 | . . 3 β’ ((π β Fin β§ π β π) β π΄ = (πΊ sSet β¨(.rβndx), (π maMul β¨π, π, πβ©)β©)) |
8 | 7 | fveq2d 6829 | . 2 β’ ((π β Fin β§ π β π) β (Baseβπ΄) = (Baseβ(πΊ sSet β¨(.rβndx), (π maMul β¨π, π, πβ©)β©))) |
9 | 3, 8 | eqtr4id 2795 | 1 β’ ((π β Fin β§ π β π) β (BaseβπΊ) = (Baseβπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1540 β wcel 2105 β¨cop 4579 β¨cotp 4581 Γ cxp 5618 βcfv 6479 (class class class)co 7337 Fincfn 8804 sSet csts 16961 ndxcnx 16991 Basecbs 17009 .rcmulr 17060 freeLMod cfrlm 21059 maMul cmmul 21638 Mat cmat 21660 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-ot 4582 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-mulr 17073 df-mat 21661 |
This theorem is referenced by: mat0 21672 matinvg 21673 matbas2 21676 matplusg2 21682 matvsca2 21683 matlmod 21684 matsubg 21687 matsubgcell 21689 matgsum 21692 matdim 31996 matunitlindflem2 35887 matunitlindf 35888 |
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