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Mirrors > Home > MPE Home > Th. List > matunit | Structured version Visualization version GIF version |
Description: A matrix is a unit in the ring of matrices iff its determinant is a unit in the underlying ring. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
Ref | Expression |
---|---|
matunit.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
matunit.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
matunit.b | ⊢ 𝐵 = (Base‘𝐴) |
matunit.u | ⊢ 𝑈 = (Unit‘𝐴) |
matunit.v | ⊢ 𝑉 = (Unit‘𝑅) |
Ref | Expression |
---|---|
matunit | ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑈 ↔ (𝐷‘𝑀) ∈ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2821 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | eqid 2821 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
4 | matunit.v | . . . 4 ⊢ 𝑉 = (Unit‘𝑅) | |
5 | eqid 2821 | . . . 4 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
6 | crngring 19239 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
7 | 6 | ad2antrr 722 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → 𝑅 ∈ Ring) |
8 | matunit.d | . . . . . 6 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
9 | matunit.a | . . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
10 | matunit.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
11 | 8, 9, 10, 1 | mdetcl 21135 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) ∈ (Base‘𝑅)) |
12 | 11 | adantr 481 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → (𝐷‘𝑀) ∈ (Base‘𝑅)) |
13 | 8, 9, 10, 1 | mdetf 21134 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅)) |
14 | 13 | ad2antrr 722 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → 𝐷:𝐵⟶(Base‘𝑅)) |
15 | 9, 10 | matrcl 20951 | . . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
16 | 15 | simpld 495 | . . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
17 | 16 | ad2antlr 723 | . . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → 𝑁 ∈ Fin) |
18 | 9 | matring 20982 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
19 | 17, 7, 18 | syl2anc 584 | . . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → 𝐴 ∈ Ring) |
20 | matunit.u | . . . . . . 7 ⊢ 𝑈 = (Unit‘𝐴) | |
21 | eqid 2821 | . . . . . . 7 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
22 | 20, 21, 10 | ringinvcl 19357 | . . . . . 6 ⊢ ((𝐴 ∈ Ring ∧ 𝑀 ∈ 𝑈) → ((invr‘𝐴)‘𝑀) ∈ 𝐵) |
23 | 19, 22 | sylancom 588 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → ((invr‘𝐴)‘𝑀) ∈ 𝐵) |
24 | 14, 23 | ffvelrnd 6845 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → (𝐷‘((invr‘𝐴)‘𝑀)) ∈ (Base‘𝑅)) |
25 | eqid 2821 | . . . . . . . 8 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
26 | eqid 2821 | . . . . . . . 8 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
27 | 20, 21, 25, 26 | unitrinv 19359 | . . . . . . 7 ⊢ ((𝐴 ∈ Ring ∧ 𝑀 ∈ 𝑈) → (𝑀(.r‘𝐴)((invr‘𝐴)‘𝑀)) = (1r‘𝐴)) |
28 | 19, 27 | sylancom 588 | . . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → (𝑀(.r‘𝐴)((invr‘𝐴)‘𝑀)) = (1r‘𝐴)) |
29 | 28 | fveq2d 6668 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → (𝐷‘(𝑀(.r‘𝐴)((invr‘𝐴)‘𝑀))) = (𝐷‘(1r‘𝐴))) |
30 | simpll 763 | . . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → 𝑅 ∈ CRing) | |
31 | simplr 765 | . . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → 𝑀 ∈ 𝐵) | |
32 | 9, 10, 8, 2, 25 | mdetmul 21162 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ ((invr‘𝐴)‘𝑀) ∈ 𝐵) → (𝐷‘(𝑀(.r‘𝐴)((invr‘𝐴)‘𝑀))) = ((𝐷‘𝑀)(.r‘𝑅)(𝐷‘((invr‘𝐴)‘𝑀)))) |
33 | 30, 31, 23, 32 | syl3anc 1363 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → (𝐷‘(𝑀(.r‘𝐴)((invr‘𝐴)‘𝑀))) = ((𝐷‘𝑀)(.r‘𝑅)(𝐷‘((invr‘𝐴)‘𝑀)))) |
34 | 8, 9, 26, 3 | mdet1 21140 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐷‘(1r‘𝐴)) = (1r‘𝑅)) |
35 | 30, 17, 34 | syl2anc 584 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → (𝐷‘(1r‘𝐴)) = (1r‘𝑅)) |
36 | 29, 33, 35 | 3eqtr3d 2864 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → ((𝐷‘𝑀)(.r‘𝑅)(𝐷‘((invr‘𝐴)‘𝑀))) = (1r‘𝑅)) |
37 | 20, 21, 25, 26 | unitlinv 19358 | . . . . . . 7 ⊢ ((𝐴 ∈ Ring ∧ 𝑀 ∈ 𝑈) → (((invr‘𝐴)‘𝑀)(.r‘𝐴)𝑀) = (1r‘𝐴)) |
38 | 19, 37 | sylancom 588 | . . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → (((invr‘𝐴)‘𝑀)(.r‘𝐴)𝑀) = (1r‘𝐴)) |
39 | 38 | fveq2d 6668 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → (𝐷‘(((invr‘𝐴)‘𝑀)(.r‘𝐴)𝑀)) = (𝐷‘(1r‘𝐴))) |
40 | 9, 10, 8, 2, 25 | mdetmul 21162 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ ((invr‘𝐴)‘𝑀) ∈ 𝐵 ∧ 𝑀 ∈ 𝐵) → (𝐷‘(((invr‘𝐴)‘𝑀)(.r‘𝐴)𝑀)) = ((𝐷‘((invr‘𝐴)‘𝑀))(.r‘𝑅)(𝐷‘𝑀))) |
41 | 30, 23, 31, 40 | syl3anc 1363 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → (𝐷‘(((invr‘𝐴)‘𝑀)(.r‘𝐴)𝑀)) = ((𝐷‘((invr‘𝐴)‘𝑀))(.r‘𝑅)(𝐷‘𝑀))) |
42 | 39, 41, 35 | 3eqtr3d 2864 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → ((𝐷‘((invr‘𝐴)‘𝑀))(.r‘𝑅)(𝐷‘𝑀)) = (1r‘𝑅)) |
43 | 1, 2, 3, 4, 5, 7, 12, 24, 36, 42 | invrvald 21215 | . . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → ((𝐷‘𝑀) ∈ 𝑉 ∧ ((invr‘𝑅)‘(𝐷‘𝑀)) = (𝐷‘((invr‘𝐴)‘𝑀)))) |
44 | 43 | simpld 495 | . 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ 𝑈) → (𝐷‘𝑀) ∈ 𝑉) |
45 | eqid 2821 | . . . . 5 ⊢ (𝑁 maAdju 𝑅) = (𝑁 maAdju 𝑅) | |
46 | eqid 2821 | . . . . 5 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴) | |
47 | 9, 45, 8, 10, 20, 4, 5, 21, 46 | matinv 21216 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ (𝐷‘𝑀) ∈ 𝑉) → (𝑀 ∈ 𝑈 ∧ ((invr‘𝐴)‘𝑀) = (((invr‘𝑅)‘(𝐷‘𝑀))( ·𝑠 ‘𝐴)((𝑁 maAdju 𝑅)‘𝑀)))) |
48 | 47 | simpld 495 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ (𝐷‘𝑀) ∈ 𝑉) → 𝑀 ∈ 𝑈) |
49 | 48 | 3expa 1110 | . 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐷‘𝑀) ∈ 𝑉) → 𝑀 ∈ 𝑈) |
50 | 44, 49 | impbida 797 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑈 ↔ (𝐷‘𝑀) ∈ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3495 ⟶wf 6345 ‘cfv 6349 (class class class)co 7145 Fincfn 8498 Basecbs 16473 .rcmulr 16556 ·𝑠 cvsca 16559 1rcur 19182 Ringcrg 19228 CRingccrg 19229 Unitcui 19320 invrcinvr 19352 Mat cmat 20946 maDet cmdat 21123 maAdju cmadu 21171 |
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 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-xor 1496 df-tru 1531 df-fal 1541 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-tpos 7883 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-pm 8399 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-div 11287 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-xnn0 11957 df-z 11971 df-dec 12088 df-uz 12233 df-rp 12380 df-fz 12883 df-fzo 13024 df-seq 13360 df-exp 13420 df-hash 13681 df-word 13852 df-lsw 13905 df-concat 13913 df-s1 13940 df-substr 13993 df-pfx 14023 df-splice 14102 df-reverse 14111 df-s2 14200 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-starv 16570 df-sca 16571 df-vsca 16572 df-ip 16573 df-tset 16574 df-ple 16575 df-ds 16577 df-unif 16578 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-gim 18339 df-cntz 18387 df-oppg 18414 df-symg 18436 df-pmtr 18501 df-psgn 18550 df-evpm 18551 df-cmn 18839 df-abl 18840 df-mgp 19171 df-ur 19183 df-srg 19187 df-ring 19230 df-cring 19231 df-oppr 19304 df-dvdsr 19322 df-unit 19323 df-invr 19353 df-dvr 19364 df-rnghom 19398 df-drng 19435 df-subrg 19464 df-lmod 19567 df-lss 19635 df-sra 19875 df-rgmod 19876 df-assa 20015 df-cnfld 20476 df-zring 20548 df-zrh 20581 df-dsmm 20806 df-frlm 20821 df-mamu 20925 df-mat 20947 df-mdet 21124 df-madu 21173 |
This theorem is referenced by: slesolinv 21219 slesolinvbi 21220 slesolex 21221 matunitlindf 34772 |
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