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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 | ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mat 20214 | . . . 4 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉)) | |
2 | 1 | mpt2ndm0 6875 | . . 3 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅) |
3 | 2 | fveq2d 6195 | . 2 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = (Base‘∅)) |
4 | base0 15912 | . 2 ⊢ ∅ = (Base‘∅) | |
5 | 3, 4 | syl6eqr 2674 | 1 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 〈cop 4183 〈cotp 4185 × cxp 5112 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ndxcnx 15854 sSet csts 15855 Basecbs 15857 .rcmulr 15942 freeLMod cfrlm 20090 maMul cmmul 20189 Mat cmat 20213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-slot 15861 df-base 15863 df-mat 20214 |
This theorem is referenced by: nfimdetndef 20395 mdetfval1 20396 |
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