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Mirrors > Home > MPE Home > Th. List > mdet1 | Structured version Visualization version GIF version |
Description: The determinant of the identity matrix is 1, i.e. the determinant function is normalized, see also definition in [Lang] p. 513. (Contributed by SO, 10-Jul-2018.) (Proof shortened by AV, 25-Nov-2019.) |
Ref | Expression |
---|---|
mdet1.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
mdet1.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mdet1.n | ⊢ 𝐼 = (1r‘𝐴) |
mdet1.o | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
mdet1 | ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐷‘𝐼) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑅 ∈ CRing ∧ 𝑁 ∈ Fin)) | |
2 | crngring 19574 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
3 | 2 | anim1ci 619 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
4 | mdet1.a | . . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
5 | 4 | matring 21340 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
6 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
7 | mdet1.n | . . . . . 6 ⊢ 𝐼 = (1r‘𝐴) | |
8 | 6, 7 | ringidcl 19586 | . . . . 5 ⊢ (𝐴 ∈ Ring → 𝐼 ∈ (Base‘𝐴)) |
9 | 3, 5, 8 | 3syl 18 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝐼 ∈ (Base‘𝐴)) |
10 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
11 | mdet1.o | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
12 | 10, 11 | ringidcl 19586 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
13 | 2, 12 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ CRing → 1 ∈ (Base‘𝑅)) |
14 | 13 | adantr 484 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 1 ∈ (Base‘𝑅)) |
15 | 1, 9, 14 | jca32 519 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ (Base‘𝐴) ∧ 1 ∈ (Base‘𝑅)))) |
16 | eqid 2737 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | simplr 769 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin) | |
18 | 2 | adantr 484 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring) |
19 | 18 | adantr 484 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring) |
20 | simprl 771 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁) | |
21 | simprr 773 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁) | |
22 | 4, 11, 16, 17, 19, 20, 21, 7 | mat1ov 21345 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, 1 , (0g‘𝑅))) |
23 | 22 | ralrimivva 3112 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐼𝑗) = if(𝑖 = 𝑗, 1 , (0g‘𝑅))) |
24 | mdet1.d | . . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
25 | eqid 2737 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
26 | eqid 2737 | . . . 4 ⊢ (.g‘(mulGrp‘𝑅)) = (.g‘(mulGrp‘𝑅)) | |
27 | 24, 4, 6, 25, 16, 10, 26 | mdetdiagid 21497 | . . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ (Base‘𝐴) ∧ 1 ∈ (Base‘𝑅))) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐼𝑗) = if(𝑖 = 𝑗, 1 , (0g‘𝑅)) → (𝐷‘𝐼) = ((♯‘𝑁)(.g‘(mulGrp‘𝑅)) 1 ))) |
28 | 15, 23, 27 | sylc 65 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐷‘𝐼) = ((♯‘𝑁)(.g‘(mulGrp‘𝑅)) 1 )) |
29 | ringsrg 19607 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing) | |
30 | 2, 29 | syl 17 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ SRing) |
31 | hashcl 13923 | . . 3 ⊢ (𝑁 ∈ Fin → (♯‘𝑁) ∈ ℕ0) | |
32 | 25, 26, 11 | srg1expzeq1 19554 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ (♯‘𝑁) ∈ ℕ0) → ((♯‘𝑁)(.g‘(mulGrp‘𝑅)) 1 ) = 1 ) |
33 | 30, 31, 32 | syl2an 599 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((♯‘𝑁)(.g‘(mulGrp‘𝑅)) 1 ) = 1 ) |
34 | 28, 33 | eqtrd 2777 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐷‘𝐼) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ifcif 4439 ‘cfv 6380 (class class class)co 7213 Fincfn 8626 ℕ0cn0 12090 ♯chash 13896 Basecbs 16760 0gc0g 16944 .gcmg 18488 mulGrpcmgp 19504 1rcur 19516 SRingcsrg 19520 Ringcrg 19562 CRingccrg 19563 Mat cmat 21304 maDet cmdat 21481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-addf 10808 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-xor 1508 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-ot 4550 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-sup 9058 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-xnn0 12163 df-z 12177 df-dec 12294 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-word 14070 df-lsw 14118 df-concat 14126 df-s1 14153 df-substr 14206 df-pfx 14236 df-splice 14315 df-reverse 14324 df-s2 14413 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-0g 16946 df-gsum 16947 df-prds 16952 df-pws 16954 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-efmnd 18296 df-grp 18368 df-minusg 18369 df-sbg 18370 df-mulg 18489 df-subg 18540 df-ghm 18620 df-gim 18663 df-cntz 18711 df-oppg 18738 df-symg 18760 df-pmtr 18834 df-psgn 18883 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-srg 19521 df-ring 19564 df-cring 19565 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-rnghom 19735 df-drng 19769 df-subrg 19798 df-lmod 19901 df-lss 19969 df-sra 20209 df-rgmod 20210 df-cnfld 20364 df-zring 20436 df-zrh 20470 df-dsmm 20694 df-frlm 20709 df-mamu 21283 df-mat 21305 df-mdet 21482 |
This theorem is referenced by: mdetuni0 21518 matunit 21575 cramerimplem1 21580 matunitlindflem2 35511 |
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