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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 21946, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.) |
Ref | Expression |
---|---|
mat0scmat | ⊢ (𝑅 ∈ Ring → ∅ ∈ (∅ ScMat 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5300 | . . . 4 ⊢ ∅ ∈ V | |
2 | 1 | snid 4658 | . . 3 ⊢ ∅ ∈ {∅} |
3 | mat0dimbas0 21897 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(∅ Mat 𝑅)) = {∅}) | |
4 | 2, 3 | eleqtrrid 2839 | . 2 ⊢ (𝑅 ∈ Ring → ∅ ∈ (Base‘(∅ Mat 𝑅))) |
5 | eqid 2731 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2731 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | 5, 6 | ringidcl 20040 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
8 | oveq1 7400 | . . . . . 6 ⊢ (𝑐 = (1r‘𝑅) → (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅) = ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅)) | |
9 | 8 | eqeq2d 2742 | . . . . 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 2731 | . . . . . . 7 ⊢ (∅ Mat 𝑅) = (∅ Mat 𝑅) | |
12 | 11 | mat0dimscm 21900 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅)) → ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅) = ∅) |
13 | 7, 12 | mpdan 685 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅) = ∅) |
14 | 13 | eqcomd 2737 | . . . 4 ⊢ (𝑅 ∈ Ring → ∅ = ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅)) |
15 | 7, 10, 14 | rspcedvd 3611 | . . 3 ⊢ (𝑅 ∈ Ring → ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅)) |
16 | 11 | mat0dimid 21899 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘(∅ Mat 𝑅)) = ∅) |
17 | 16 | oveq2d 7409 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅))) = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅)) |
18 | 17 | eqeq2d 2742 | . . . 4 ⊢ (𝑅 ∈ Ring → (∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅))) ↔ ∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅))) |
19 | 18 | rexbidv 3177 | . . 3 ⊢ (𝑅 ∈ Ring → (∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅))) ↔ ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅))) |
20 | 15, 19 | mpbird 256 | . 2 ⊢ (𝑅 ∈ Ring → ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅)))) |
21 | 0fin 9154 | . . 3 ⊢ ∅ ∈ Fin | |
22 | eqid 2731 | . . . 4 ⊢ (Base‘(∅ Mat 𝑅)) = (Base‘(∅ Mat 𝑅)) | |
23 | eqid 2731 | . . . 4 ⊢ (1r‘(∅ Mat 𝑅)) = (1r‘(∅ Mat 𝑅)) | |
24 | eqid 2731 | . . . 4 ⊢ ( ·𝑠 ‘(∅ Mat 𝑅)) = ( ·𝑠 ‘(∅ Mat 𝑅)) | |
25 | eqid 2731 | . . . 4 ⊢ (∅ ScMat 𝑅) = (∅ ScMat 𝑅) | |
26 | 5, 11, 22, 23, 24, 25 | scmatel 21936 | . . 3 ⊢ ((∅ ∈ Fin ∧ 𝑅 ∈ Ring) → (∅ ∈ (∅ ScMat 𝑅) ↔ (∅ ∈ (Base‘(∅ Mat 𝑅)) ∧ ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅)))))) |
27 | 21, 26 | mpan 688 | . 2 ⊢ (𝑅 ∈ Ring → (∅ ∈ (∅ ScMat 𝑅) ↔ (∅ ∈ (Base‘(∅ Mat 𝑅)) ∧ ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅)))))) |
28 | 4, 20, 27 | mpbir2and 711 | 1 ⊢ (𝑅 ∈ Ring → ∅ ∈ (∅ ScMat 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3069 ∅c0 4318 {csn 4622 ‘cfv 6532 (class class class)co 7393 Fincfn 8922 Basecbs 17126 ·𝑠 cvsca 17183 1rcur 19963 Ringcrg 20014 Mat cmat 21836 ScMat cscmat 21920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-ot 4631 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-sup 9419 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-fzo 13610 df-seq 13949 df-hash 14273 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17369 df-gsum 17370 df-prds 17375 df-pws 17377 df-mre 17512 df-mrc 17513 df-acs 17515 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-mhm 18647 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-mulg 18923 df-subg 18975 df-ghm 19056 df-cntz 19147 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-subrg 20310 df-lmod 20422 df-lss 20492 df-sra 20734 df-rgmod 20735 df-dsmm 21220 df-frlm 21235 df-mamu 21815 df-mat 21837 df-scmat 21922 |
This theorem is referenced by: (None) |
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