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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 21019 | . . . . 5 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉)) | |
2 | 1 | reldmmpo 7287 | . . . 4 ⊢ Rel dom Mat |
3 | 2 | ovprc 7196 | . . 3 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅) |
4 | 3 | fveq2d 6676 | . 2 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = (Base‘∅)) |
5 | base0 16538 | . 2 ⊢ ∅ = (Base‘∅) | |
6 | 4, 5 | syl6eqr 2876 | 1 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 〈cop 4575 〈cotp 4577 × cxp 5555 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ndxcnx 16482 sSet csts 16483 Basecbs 16485 .rcmulr 16568 freeLMod cfrlm 20892 maMul cmmul 20996 Mat cmat 21018 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-slot 16489 df-base 16491 df-mat 21019 |
This theorem is referenced by: marrepfval 21171 marepvfval 21176 submafval 21190 minmar1fval 21257 |
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