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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 19906 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
3 | 2 | anim1ci 617 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
4 | mdet1.a | . . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
5 | 4 | matring 21720 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
6 | eqid 2738 | . . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
7 | mdet1.n | . . . . . 6 ⊢ 𝐼 = (1r‘𝐴) | |
8 | 6, 7 | ringidcl 19919 | . . . . 5 ⊢ (𝐴 ∈ Ring → 𝐼 ∈ (Base‘𝐴)) |
9 | 3, 5, 8 | 3syl 18 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝐼 ∈ (Base‘𝐴)) |
10 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
11 | mdet1.o | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
12 | 10, 11 | ringidcl 19919 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
13 | 2, 12 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ CRing → 1 ∈ (Base‘𝑅)) |
14 | 13 | adantr 482 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 1 ∈ (Base‘𝑅)) |
15 | 1, 9, 14 | jca32 517 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ (Base‘𝐴) ∧ 1 ∈ (Base‘𝑅)))) |
16 | eqid 2738 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | simplr 768 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin) | |
18 | 2 | adantr 482 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring) |
19 | 18 | adantr 482 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring) |
20 | simprl 770 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁) | |
21 | simprr 772 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁) | |
22 | 4, 11, 16, 17, 19, 20, 21, 7 | mat1ov 21725 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, 1 , (0g‘𝑅))) |
23 | 22 | ralrimivva 3196 | . . 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 2738 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
26 | eqid 2738 | . . . 4 ⊢ (.g‘(mulGrp‘𝑅)) = (.g‘(mulGrp‘𝑅)) | |
27 | 24, 4, 6, 25, 16, 10, 26 | mdetdiagid 21877 | . . 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 19942 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing) | |
30 | 2, 29 | syl 17 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ SRing) |
31 | hashcl 14185 | . . 3 ⊢ (𝑁 ∈ Fin → (♯‘𝑁) ∈ ℕ0) | |
32 | 25, 26, 11 | srg1expzeq1 19886 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ (♯‘𝑁) ∈ ℕ0) → ((♯‘𝑁)(.g‘(mulGrp‘𝑅)) 1 ) = 1 ) |
33 | 30, 31, 32 | syl2an 597 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((♯‘𝑁)(.g‘(mulGrp‘𝑅)) 1 ) = 1 ) |
34 | 28, 33 | eqtrd 2778 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐷‘𝐼) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3063 ifcif 4485 ‘cfv 6492 (class class class)co 7350 Fincfn 8817 ℕ0cn0 12347 ♯chash 14159 Basecbs 17019 0gc0g 17257 .gcmg 18807 mulGrpcmgp 19831 1rcur 19848 SRingcsrg 19852 Ringcrg 19894 CRingccrg 19895 Mat cmat 21682 maDet cmdat 21861 |
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 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-addf 11064 ax-mulf 11065 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 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 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-of 7608 df-om 7794 df-1st 7912 df-2nd 7913 df-supp 8061 df-tpos 8125 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8582 df-map 8701 df-ixp 8770 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fsupp 9240 df-sup 9312 df-oi 9380 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-xnn0 12420 df-z 12434 df-dec 12553 df-uz 12698 df-rp 12846 df-fz 13355 df-fzo 13498 df-seq 13837 df-exp 13898 df-hash 14160 df-word 14332 df-lsw 14380 df-concat 14388 df-s1 14413 df-substr 14462 df-pfx 14492 df-splice 14571 df-reverse 14580 df-s2 14670 df-struct 16955 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-mulr 17083 df-starv 17084 df-sca 17085 df-vsca 17086 df-ip 17087 df-tset 17088 df-ple 17089 df-ds 17091 df-unif 17092 df-hom 17093 df-cco 17094 df-0g 17259 df-gsum 17260 df-prds 17265 df-pws 17267 df-mre 17402 df-mrc 17403 df-acs 17405 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-mhm 18537 df-submnd 18538 df-efmnd 18615 df-grp 18687 df-minusg 18688 df-sbg 18689 df-mulg 18808 df-subg 18860 df-ghm 18941 df-gim 18984 df-cntz 19032 df-oppg 19059 df-symg 19084 df-pmtr 19159 df-psgn 19208 df-cmn 19499 df-abl 19500 df-mgp 19832 df-ur 19849 df-srg 19853 df-ring 19896 df-cring 19897 df-oppr 19978 df-dvdsr 19999 df-unit 20000 df-invr 20030 df-dvr 20041 df-rnghom 20075 df-drng 20116 df-subrg 20149 df-lmod 20253 df-lss 20322 df-sra 20562 df-rgmod 20563 df-cnfld 20726 df-zring 20799 df-zrh 20833 df-dsmm 21067 df-frlm 21082 df-mamu 21661 df-mat 21683 df-mdet 21862 |
This theorem is referenced by: mdetuni0 21898 matunit 21955 cramerimplem1 21960 matunitlindflem2 36006 |
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