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 5642 | . . . . 5 ⊢ (∅ × ∅) = ∅ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (∅ × ∅) = ∅) |
3 | 2 | oveq2d 7161 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ((Base‘𝑅) ↑m (∅ × ∅)) = ((Base‘𝑅) ↑m ∅)) |
4 | fvex 6676 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
5 | map0e 8435 | . . . 4 ⊢ ((Base‘𝑅) ∈ V → ((Base‘𝑅) ↑m ∅) = 1o) | |
6 | 4, 5 | mp1i 13 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ((Base‘𝑅) ↑m ∅) = 1o) |
7 | 3, 6 | eqtrd 2853 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((Base‘𝑅) ↑m (∅ × ∅)) = 1o) |
8 | 0fin 8734 | . . 3 ⊢ ∅ ∈ Fin | |
9 | eqid 2818 | . . . 4 ⊢ (∅ Mat 𝑅) = (∅ Mat 𝑅) | |
10 | eqid 2818 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
11 | 9, 10 | matbas2 20958 | . . 3 ⊢ ((∅ ∈ Fin ∧ 𝑅 ∈ 𝑉) → ((Base‘𝑅) ↑m (∅ × ∅)) = (Base‘(∅ Mat 𝑅))) |
12 | 8, 11 | mpan 686 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((Base‘𝑅) ↑m (∅ × ∅)) = (Base‘(∅ Mat 𝑅))) |
13 | df1o2 8105 | . . 3 ⊢ 1o = {∅} | |
14 | 13 | a1i 11 | . 2 ⊢ (𝑅 ∈ 𝑉 → 1o = {∅}) |
15 | 7, 12, 14 | 3eqtr3d 2861 | 1 ⊢ (𝑅 ∈ 𝑉 → (Base‘(∅ Mat 𝑅)) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 {csn 4557 × cxp 5546 ‘cfv 6348 (class class class)co 7145 1oc1o 8084 ↑m cmap 8395 Fincfn 8497 Basecbs 16471 Mat cmat 20944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-0g 16703 df-prds 16709 df-pws 16711 df-sra 19873 df-rgmod 19874 df-dsmm 20804 df-frlm 20819 df-mat 20945 |
This theorem is referenced by: mat0dim0 21004 mat0dimid 21005 mat0dimscm 21006 mat0dimcrng 21007 mat0scmat 21075 mavmul0 21089 mdet0pr 21129 cramer0 21227 d0mat2pmat 21274 chpmat0d 21370 matunitlindf 34771 |
Copyright terms: Public domain | W3C validator |