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Mirrors > Home > MPE Home > Th. List > matbas0 | Structured version Visualization version GIF version |
Description: There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.) |
Ref | Expression |
---|---|
matbas0 | β’ (Β¬ (π β Fin β§ π β V) β (Baseβ(π Mat π )) = β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mat 21661 | . . . 4 β’ Mat = (π β Fin, π β V β¦ ((π freeLMod (π Γ π)) sSet β¨(.rβndx), (π maMul β¨π, π, πβ©)β©)) | |
2 | 1 | mpondm0 7572 | . . 3 β’ (Β¬ (π β Fin β§ π β V) β (π Mat π ) = β ) |
3 | 2 | fveq2d 6829 | . 2 β’ (Β¬ (π β Fin β§ π β V) β (Baseβ(π Mat π )) = (Baseββ )) |
4 | base0 17014 | . 2 β’ β = (Baseββ ) | |
5 | 3, 4 | eqtr4di 2794 | 1 β’ (Β¬ (π β Fin β§ π β V) β (Baseβ(π Mat π )) = β ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 = wceq 1540 β wcel 2105 Vcvv 3441 β c0 4269 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-1cn 11030 ax-addcl 11032 |
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-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-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-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-nn 12075 df-slot 16980 df-ndx 16992 df-base 17010 df-mat 21661 |
This theorem is referenced by: nfimdetndef 21844 mdetfval1 21845 |
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