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Mirrors > Home > MPE Home > Th. List > mat0scmat | Structured version Visualization version GIF version |
Description: The empty matrix over a ring is a scalar matrix (and therefore, by scmatdmat 20321, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.) |
Ref | Expression |
---|---|
mat0scmat | ⊢ (𝑅 ∈ Ring → ∅ ∈ (∅ ScMat 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4790 | . . . 4 ⊢ ∅ ∈ V | |
2 | 1 | snid 4208 | . . 3 ⊢ ∅ ∈ {∅} |
3 | mat0dimbas0 20272 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(∅ Mat 𝑅)) = {∅}) | |
4 | 2, 3 | syl5eleqr 2708 | . 2 ⊢ (𝑅 ∈ Ring → ∅ ∈ (Base‘(∅ Mat 𝑅))) |
5 | eqid 2622 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2622 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | 5, 6 | ringidcl 18568 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
8 | oveq1 6657 | . . . . . 6 ⊢ (𝑐 = (1r‘𝑅) → (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅) = ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅)) | |
9 | 8 | eqeq2d 2632 | . . . . 5 ⊢ (𝑐 = (1r‘𝑅) → (∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅) ↔ ∅ = ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅))) |
10 | 9 | adantl 482 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑐 = (1r‘𝑅)) → (∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅) ↔ ∅ = ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅))) |
11 | eqid 2622 | . . . . . . 7 ⊢ (∅ Mat 𝑅) = (∅ Mat 𝑅) | |
12 | 11 | mat0dimscm 20275 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅)) → ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅) = ∅) |
13 | 7, 12 | mpdan 702 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅) = ∅) |
14 | 13 | eqcomd 2628 | . . . 4 ⊢ (𝑅 ∈ Ring → ∅ = ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅)) |
15 | 7, 10, 14 | rspcedvd 3317 | . . 3 ⊢ (𝑅 ∈ Ring → ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅)) |
16 | 11 | mat0dimid 20274 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘(∅ Mat 𝑅)) = ∅) |
17 | 16 | oveq2d 6666 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅))) = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅)) |
18 | 17 | eqeq2d 2632 | . . . 4 ⊢ (𝑅 ∈ Ring → (∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅))) ↔ ∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅))) |
19 | 18 | rexbidv 3052 | . . 3 ⊢ (𝑅 ∈ Ring → (∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅))) ↔ ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅))) |
20 | 15, 19 | mpbird 247 | . 2 ⊢ (𝑅 ∈ Ring → ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅)))) |
21 | 0fin 8188 | . . 3 ⊢ ∅ ∈ Fin | |
22 | eqid 2622 | . . . 4 ⊢ (Base‘(∅ Mat 𝑅)) = (Base‘(∅ Mat 𝑅)) | |
23 | eqid 2622 | . . . 4 ⊢ (1r‘(∅ Mat 𝑅)) = (1r‘(∅ Mat 𝑅)) | |
24 | eqid 2622 | . . . 4 ⊢ ( ·𝑠 ‘(∅ Mat 𝑅)) = ( ·𝑠 ‘(∅ Mat 𝑅)) | |
25 | eqid 2622 | . . . 4 ⊢ (∅ ScMat 𝑅) = (∅ ScMat 𝑅) | |
26 | 5, 11, 22, 23, 24, 25 | scmatel 20311 | . . 3 ⊢ ((∅ ∈ Fin ∧ 𝑅 ∈ Ring) → (∅ ∈ (∅ ScMat 𝑅) ↔ (∅ ∈ (Base‘(∅ Mat 𝑅)) ∧ ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅)))))) |
27 | 21, 26 | mpan 706 | . 2 ⊢ (𝑅 ∈ Ring → (∅ ∈ (∅ ScMat 𝑅) ↔ (∅ ∈ (Base‘(∅ Mat 𝑅)) ∧ ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅)))))) |
28 | 4, 20, 27 | mpbir2and 957 | 1 ⊢ (𝑅 ∈ Ring → ∅ ∈ (∅ ScMat 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ∅c0 3915 {csn 4177 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 Basecbs 15857 ·𝑠 cvsca 15945 1rcur 18501 Ringcrg 18547 Mat cmat 20213 ScMat cscmat 20295 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-dsmm 20076 df-frlm 20091 df-mamu 20190 df-mat 20214 df-scmat 20297 |
This theorem is referenced by: (None) |
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