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Mirrors > Home > MPE Home > Th. List > mat0dim0 | Structured version Visualization version GIF version |
Description: The zero of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.) |
Ref | Expression |
---|---|
mat0dim.a | ⊢ 𝐴 = (∅ Mat 𝑅) |
Ref | Expression |
---|---|
mat0dim0 | ⊢ (𝑅 ∈ Ring → (0g‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0fin 9074 | . . . 4 ⊢ ∅ ∈ Fin | |
2 | mat0dim.a | . . . . 5 ⊢ 𝐴 = (∅ Mat 𝑅) | |
3 | 2 | matring 21744 | . . . 4 ⊢ ((∅ ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
4 | 1, 3 | mpan 689 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐴 ∈ Ring) |
5 | ringgrp 19923 | . . 3 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ Grp) | |
6 | eqid 2738 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
7 | eqid 2738 | . . . 4 ⊢ (0g‘𝐴) = (0g‘𝐴) | |
8 | 6, 7 | grpidcl 18738 | . . 3 ⊢ (𝐴 ∈ Grp → (0g‘𝐴) ∈ (Base‘𝐴)) |
9 | 4, 5, 8 | 3syl 18 | . 2 ⊢ (𝑅 ∈ Ring → (0g‘𝐴) ∈ (Base‘𝐴)) |
10 | 2 | fveq2i 6843 | . . . . 5 ⊢ (Base‘𝐴) = (Base‘(∅ Mat 𝑅)) |
11 | mat0dimbas0 21767 | . . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘(∅ Mat 𝑅)) = {∅}) | |
12 | 10, 11 | eqtrid 2790 | . . . 4 ⊢ (𝑅 ∈ Ring → (Base‘𝐴) = {∅}) |
13 | 12 | eleq2d 2824 | . . 3 ⊢ (𝑅 ∈ Ring → ((0g‘𝐴) ∈ (Base‘𝐴) ↔ (0g‘𝐴) ∈ {∅})) |
14 | elsni 4602 | . . 3 ⊢ ((0g‘𝐴) ∈ {∅} → (0g‘𝐴) = ∅) | |
15 | 13, 14 | syl6bi 253 | . 2 ⊢ (𝑅 ∈ Ring → ((0g‘𝐴) ∈ (Base‘𝐴) → (0g‘𝐴) = ∅)) |
16 | 9, 15 | mpd 15 | 1 ⊢ (𝑅 ∈ Ring → (0g‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∅c0 4281 {csn 4585 ‘cfv 6494 (class class class)co 7352 Fincfn 8842 Basecbs 17043 0gc0g 17281 Grpcgrp 18708 Ringcrg 19918 Mat cmat 21706 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7610 df-om 7796 df-1st 7914 df-2nd 7915 df-supp 8086 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-map 8726 df-ixp 8795 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-fsupp 9265 df-sup 9337 df-oi 9405 df-card 9834 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-z 12459 df-dec 12578 df-uz 12723 df-fz 13380 df-fzo 13523 df-seq 13862 df-hash 14185 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-sca 17109 df-vsca 17110 df-ip 17111 df-tset 17112 df-ple 17113 df-ds 17115 df-hom 17117 df-cco 17118 df-0g 17283 df-gsum 17284 df-prds 17289 df-pws 17291 df-mre 17426 df-mrc 17427 df-acs 17429 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-mhm 18561 df-submnd 18562 df-grp 18711 df-minusg 18712 df-sbg 18713 df-mulg 18832 df-subg 18884 df-ghm 18965 df-cntz 19056 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-ring 19920 df-subrg 20173 df-lmod 20277 df-lss 20346 df-sra 20586 df-rgmod 20587 df-dsmm 21091 df-frlm 21106 df-mamu 21685 df-mat 21707 |
This theorem is referenced by: chpmat0d 22135 |
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