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Mirrors > Home > MPE Home > Th. List > mat0dimid | Structured version Visualization version GIF version |
Description: The identity of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.) |
Ref | Expression |
---|---|
mat0dim.a | ⊢ 𝐴 = (∅ Mat 𝑅) |
Ref | Expression |
---|---|
mat0dimid | ⊢ (𝑅 ∈ Ring → (1r‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0fin 8735 | . . . 4 ⊢ ∅ ∈ Fin | |
2 | mat0dim.a | . . . . 5 ⊢ 𝐴 = (∅ Mat 𝑅) | |
3 | 2 | matring 20982 | . . . 4 ⊢ ((∅ ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
4 | 1, 3 | mpan 686 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐴 ∈ Ring) |
5 | eqid 2821 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
6 | eqid 2821 | . . . 4 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
7 | 5, 6 | ringidcl 19249 | . . 3 ⊢ (𝐴 ∈ Ring → (1r‘𝐴) ∈ (Base‘𝐴)) |
8 | 4, 7 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → (1r‘𝐴) ∈ (Base‘𝐴)) |
9 | 2 | fveq2i 6667 | . . . . 5 ⊢ (Base‘𝐴) = (Base‘(∅ Mat 𝑅)) |
10 | mat0dimbas0 21005 | . . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘(∅ Mat 𝑅)) = {∅}) | |
11 | 9, 10 | syl5eq 2868 | . . . 4 ⊢ (𝑅 ∈ Ring → (Base‘𝐴) = {∅}) |
12 | 11 | eleq2d 2898 | . . 3 ⊢ (𝑅 ∈ Ring → ((1r‘𝐴) ∈ (Base‘𝐴) ↔ (1r‘𝐴) ∈ {∅})) |
13 | elsni 4576 | . . 3 ⊢ ((1r‘𝐴) ∈ {∅} → (1r‘𝐴) = ∅) | |
14 | 12, 13 | syl6bi 254 | . 2 ⊢ (𝑅 ∈ Ring → ((1r‘𝐴) ∈ (Base‘𝐴) → (1r‘𝐴) = ∅)) |
15 | 8, 14 | mpd 15 | 1 ⊢ (𝑅 ∈ Ring → (1r‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∅c0 4290 {csn 4559 ‘cfv 6349 (class class class)co 7145 Fincfn 8498 Basecbs 16473 1rcur 19182 Ringcrg 19228 Mat cmat 20946 |
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 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
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 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-ot 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-supp 7822 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-map 8398 df-ixp 8451 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-fsupp 8823 df-sup 8895 df-oi 8963 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-fz 12883 df-fzo 13024 df-seq 13360 df-hash 13681 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-sca 16571 df-vsca 16572 df-ip 16573 df-tset 16574 df-ple 16575 df-ds 16577 df-hom 16579 df-cco 16580 df-0g 16705 df-gsum 16706 df-prds 16711 df-pws 16713 df-mre 16847 df-mrc 16848 df-acs 16850 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-mhm 17946 df-submnd 17947 df-grp 18046 df-minusg 18047 df-sbg 18048 df-mulg 18165 df-subg 18216 df-ghm 18296 df-cntz 18387 df-cmn 18839 df-abl 18840 df-mgp 19171 df-ur 19183 df-ring 19230 df-subrg 19464 df-lmod 19567 df-lss 19635 df-sra 19875 df-rgmod 19876 df-dsmm 20806 df-frlm 20821 df-mamu 20925 df-mat 20947 |
This theorem is referenced by: mat0scmat 21077 chpmat0d 21372 matunitlindf 34772 |
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