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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 20140, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.) |
Ref | Expression |
---|---|
mat0scmat | ⊢ (𝑅 ∈ Ring → ∅ ∈ (∅ ScMat 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4718 | . . . 4 ⊢ ∅ ∈ V | |
2 | 1 | snid 4155 | . . 3 ⊢ ∅ ∈ {∅} |
3 | mat0dimbas0 20091 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(∅ Mat 𝑅)) = {∅}) | |
4 | 2, 3 | syl5eleqr 2695 | . 2 ⊢ (𝑅 ∈ Ring → ∅ ∈ (Base‘(∅ Mat 𝑅))) |
5 | eqid 2610 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2610 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | 5, 6 | ringidcl 18391 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
8 | oveq1 6556 | . . . . . 6 ⊢ (𝑐 = (1r‘𝑅) → (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅) = ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅)) | |
9 | 8 | eqeq2d 2620 | . . . . 5 ⊢ (𝑐 = (1r‘𝑅) → (∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅) ↔ ∅ = ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅))) |
10 | 9 | adantl 481 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑐 = (1r‘𝑅)) → (∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅) ↔ ∅ = ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅))) |
11 | eqid 2610 | . . . . . . 7 ⊢ (∅ Mat 𝑅) = (∅ Mat 𝑅) | |
12 | 11 | mat0dimscm 20094 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅)) → ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅) = ∅) |
13 | 7, 12 | mpdan 699 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅) = ∅) |
14 | 13 | eqcomd 2616 | . . . 4 ⊢ (𝑅 ∈ Ring → ∅ = ((1r‘𝑅)( ·𝑠 ‘(∅ Mat 𝑅))∅)) |
15 | 7, 10, 14 | rspcedvd 3289 | . . 3 ⊢ (𝑅 ∈ Ring → ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅)) |
16 | 11 | mat0dimid 20093 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘(∅ Mat 𝑅)) = ∅) |
17 | 16 | oveq2d 6565 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅))) = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅)) |
18 | 17 | eqeq2d 2620 | . . . 4 ⊢ (𝑅 ∈ Ring → (∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅))) ↔ ∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅))) |
19 | 18 | rexbidv 3034 | . . 3 ⊢ (𝑅 ∈ Ring → (∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅))) ↔ ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))∅))) |
20 | 15, 19 | mpbird 246 | . 2 ⊢ (𝑅 ∈ Ring → ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅)))) |
21 | 0fin 8073 | . . 3 ⊢ ∅ ∈ Fin | |
22 | eqid 2610 | . . . 4 ⊢ (Base‘(∅ Mat 𝑅)) = (Base‘(∅ Mat 𝑅)) | |
23 | eqid 2610 | . . . 4 ⊢ (1r‘(∅ Mat 𝑅)) = (1r‘(∅ Mat 𝑅)) | |
24 | eqid 2610 | . . . 4 ⊢ ( ·𝑠 ‘(∅ Mat 𝑅)) = ( ·𝑠 ‘(∅ Mat 𝑅)) | |
25 | eqid 2610 | . . . 4 ⊢ (∅ ScMat 𝑅) = (∅ ScMat 𝑅) | |
26 | 5, 11, 22, 23, 24, 25 | scmatel 20130 | . . 3 ⊢ ((∅ ∈ Fin ∧ 𝑅 ∈ Ring) → (∅ ∈ (∅ ScMat 𝑅) ↔ (∅ ∈ (Base‘(∅ Mat 𝑅)) ∧ ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅)))))) |
27 | 21, 26 | mpan 702 | . 2 ⊢ (𝑅 ∈ Ring → (∅ ∈ (∅ ScMat 𝑅) ↔ (∅ ∈ (Base‘(∅ Mat 𝑅)) ∧ ∃𝑐 ∈ (Base‘𝑅)∅ = (𝑐( ·𝑠 ‘(∅ Mat 𝑅))(1r‘(∅ Mat 𝑅)))))) |
28 | 4, 20, 27 | mpbir2and 959 | 1 ⊢ (𝑅 ∈ Ring → ∅ ∈ (∅ ScMat 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∅c0 3874 {csn 4125 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 Basecbs 15695 ·𝑠 cvsca 15772 1rcur 18324 Ringcrg 18370 Mat cmat 20032 ScMat cscmat 20114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-subrg 18601 df-lmod 18688 df-lss 18754 df-sra 18993 df-rgmod 18994 df-dsmm 19895 df-frlm 19910 df-mamu 20009 df-mat 20033 df-scmat 20116 |
This theorem is referenced by: (None) |
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