![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mat0dimbas0 | Structured version Visualization version GIF version |
Description: The empty set is the one and only matrix of dimension 0, called "the empty matrix". (Contributed by AV, 27-Feb-2019.) |
Ref | Expression |
---|---|
mat0dimbas0 | ⊢ (𝑅 ∈ 𝑉 → (Base‘(∅ Mat 𝑅)) = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xp 5735 | . . . . 5 ⊢ (∅ × ∅) = ∅ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (∅ × ∅) = ∅) |
3 | 2 | oveq2d 7378 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ((Base‘𝑅) ↑m (∅ × ∅)) = ((Base‘𝑅) ↑m ∅)) |
4 | fvex 6860 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
5 | map0e 8827 | . . . 4 ⊢ ((Base‘𝑅) ∈ V → ((Base‘𝑅) ↑m ∅) = 1o) | |
6 | 4, 5 | mp1i 13 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ((Base‘𝑅) ↑m ∅) = 1o) |
7 | 3, 6 | eqtrd 2777 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((Base‘𝑅) ↑m (∅ × ∅)) = 1o) |
8 | 0fin 9122 | . . 3 ⊢ ∅ ∈ Fin | |
9 | eqid 2737 | . . . 4 ⊢ (∅ Mat 𝑅) = (∅ Mat 𝑅) | |
10 | eqid 2737 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
11 | 9, 10 | matbas2 21786 | . . 3 ⊢ ((∅ ∈ Fin ∧ 𝑅 ∈ 𝑉) → ((Base‘𝑅) ↑m (∅ × ∅)) = (Base‘(∅ Mat 𝑅))) |
12 | 8, 11 | mpan 689 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((Base‘𝑅) ↑m (∅ × ∅)) = (Base‘(∅ Mat 𝑅))) |
13 | df1o2 8424 | . . 3 ⊢ 1o = {∅} | |
14 | 13 | a1i 11 | . 2 ⊢ (𝑅 ∈ 𝑉 → 1o = {∅}) |
15 | 7, 12, 14 | 3eqtr3d 2785 | 1 ⊢ (𝑅 ∈ 𝑉 → (Base‘(∅ Mat 𝑅)) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ∅c0 4287 {csn 4591 × cxp 5636 ‘cfv 6501 (class class class)co 7362 1oc1o 8410 ↑m cmap 8772 Fincfn 8890 Basecbs 17090 Mat cmat 21770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-sup 9385 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-hom 17164 df-cco 17165 df-0g 17330 df-prds 17336 df-pws 17338 df-sra 20649 df-rgmod 20650 df-dsmm 21154 df-frlm 21169 df-mat 21771 |
This theorem is referenced by: mat0dim0 21832 mat0dimid 21833 mat0dimscm 21834 mat0dimcrng 21835 mat0scmat 21903 mavmul0 21917 mdet0pr 21957 cramer0 22055 d0mat2pmat 22103 chpmat0d 22199 matunitlindf 36105 |
Copyright terms: Public domain | W3C validator |