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Mirrors > Home > MPE Home > Th. List > matbas0pc | Structured version Visualization version GIF version |
Description: There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.) |
Ref | Expression |
---|---|
matbas0pc | ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mat 21814 | . . . . 5 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉)) | |
2 | 1 | reldmmpo 7517 | . . . 4 ⊢ Rel dom Mat |
3 | 2 | ovprc 7422 | . . 3 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅) |
4 | 3 | fveq2d 6873 | . 2 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = (Base‘∅)) |
5 | base0 17121 | . 2 ⊢ ∅ = (Base‘∅) | |
6 | 4, 5 | eqtr4di 2789 | 1 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3466 ∅c0 4309 〈cop 4619 〈cotp 4621 × cxp 5658 ‘cfv 6523 (class class class)co 7384 Fincfn 8912 sSet csts 17068 ndxcnx 17098 Basecbs 17116 .rcmulr 17170 freeLMod cfrlm 21211 maMul cmmul 21791 Mat cmat 21813 |
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 2702 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 ax-cnex 11138 ax-1cn 11140 ax-addcl 11142 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3372 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-pss 3954 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-iun 4983 df-br 5133 df-opab 5195 df-mpt 5216 df-tr 5250 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5615 df-we 5617 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6280 df-ord 6347 df-on 6348 df-lim 6349 df-suc 6350 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-ov 7387 df-oprab 7388 df-mpo 7389 df-om 7830 df-2nd 7949 df-frecs 8239 df-wrecs 8270 df-recs 8344 df-rdg 8383 df-nn 12185 df-slot 17087 df-ndx 17099 df-base 17117 df-mat 21814 |
This theorem is referenced by: marrepfval 21968 marepvfval 21973 submafval 21987 minmar1fval 22054 |
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