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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 22128 | . . . 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 7649 | . . 3 β’ (Β¬ (π β Fin β§ π β V) β (π Mat π ) = β ) |
3 | 2 | fveq2d 6895 | . 2 β’ (Β¬ (π β Fin β§ π β V) β (Baseβ(π Mat π )) = (Baseββ )) |
4 | base0 17153 | . 2 β’ β = (Baseββ ) | |
5 | 3, 4 | eqtr4di 2790 | 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 1541 β wcel 2106 Vcvv 3474 β c0 4322 β¨cop 4634 β¨cotp 4636 Γ cxp 5674 βcfv 6543 (class class class)co 7411 Fincfn 8941 sSet csts 17100 ndxcnx 17130 Basecbs 17148 .rcmulr 17202 freeLMod cfrlm 21520 maMul cmmul 22105 Mat cmat 22127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-1cn 11170 ax-addcl 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-nn 12217 df-slot 17119 df-ndx 17131 df-base 17149 df-mat 22128 |
This theorem is referenced by: nfimdetndef 22311 mdetfval1 22312 |
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