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Theorem mdet1 20898

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.)

**Hypotheses**
Ref	Expression
mdet1.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdet1.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mdet1.n	⊢ 𝐼 = (1_r‘𝐴)
mdet1.o	⊢ 1 = (1_r‘𝑅)

**Assertion**
Ref	Expression
mdet1	⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐷‘𝐼) = 1 )

Proof of Theorem mdet1 Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑅 ∈ CRing ∧ 𝑁 ∈ Fin))
2		crngring 19002	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	anim1ci 615	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
4		mdet1.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
5	4	matring 20740	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
6		eqid 2797	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴)
7		mdet1.n	. . . . . 6 ⊢ 𝐼 = (1_r‘𝐴)
8	6, 7	ringidcl 19012	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝐼 ∈ (Base‘𝐴))
9	3, 5, 8	3syl 18	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝐼 ∈ (Base‘𝐴))
10		eqid 2797	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		mdet1.o	. . . . . . 7 ⊢ 1 = (1_r‘𝑅)
12	10, 11	ringidcl 19012	. . . . . 6 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
13	2, 12	syl 17	. . . . 5 ⊢ (𝑅 ∈ CRing → 1 ∈ (Base‘𝑅))
14	13	adantr 481	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 1 ∈ (Base‘𝑅))
15	1, 9, 14	jca32 516	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ (Base‘𝐴) ∧ 1 ∈ (Base‘𝑅))))
16		eqid 2797	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
17		simplr 765	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
18	2	adantr 481	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
19	18	adantr 481	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
20		simprl 767	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
21		simprr 769	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
22	4, 11, 16, 17, 19, 20, 21, 7	mat1ov 20745	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, 1 , (0_g‘𝑅)))
23	22	ralrimivva 3160	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐼𝑗) = if(𝑖 = 𝑗, 1 , (0_g‘𝑅)))
24		mdet1.d	. . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅)
25		eqid 2797	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
26		eqid 2797	. . . 4 ⊢ (._g‘(mulGrp‘𝑅)) = (._g‘(mulGrp‘𝑅))
27	24, 4, 6, 25, 16, 10, 26	mdetdiagid 20897	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ (Base‘𝐴) ∧ 1 ∈ (Base‘𝑅))) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐼𝑗) = if(𝑖 = 𝑗, 1 , (0_g‘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 19033	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
30	2, 29	syl 17	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ SRing)
31		hashcl 13571	. . 3 ⊢ (𝑁 ∈ Fin → (♯‘𝑁) ∈ ℕ₀)
32	25, 26, 11	srg1expzeq1 18983	. . 3 ⊢ ((𝑅 ∈ SRing ∧ (♯‘𝑁) ∈ ℕ₀) → ((♯‘𝑁)(._g‘(mulGrp‘𝑅)) 1 ) = 1 )
33	30, 31, 32	syl2an 595	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((♯‘𝑁)(._g‘(mulGrp‘𝑅)) 1 ) = 1 )
34	28, 33	eqtrd 2833	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐷‘𝐼) = 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∀wral 3107 ifcif 4387 ‘cfv 6232 (class class class)co 7023 Fincfn 8364 ℕ₀cn0 11751 ♯chash 13544 Basecbs 16316 0_gc0g 16546 ._gcmg 17985 mulGrpcmgp 18933 1_rcur 18945 SRingcsrg 18949 Ringcrg 18991 CRingccrg 18992 Mat cmat 20704 maDet cmdat 20881

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-addf 10469 ax-mulf 10470

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-xor 1497 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-ot 4487 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-tpos 7750 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-er 8146 df-map 8265 df-ixp 8318 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fsupp 8687 df-sup 8759 df-oi 8827 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-xnn0 11822 df-z 11836 df-dec 11953 df-uz 12098 df-rp 12244 df-fz 12747 df-fzo 12888 df-seq 13224 df-exp 13284 df-hash 13545 df-word 13712 df-lsw 13765 df-concat 13773 df-s1 13798 df-substr 13843 df-pfx 13873 df-splice 13952 df-reverse 13961 df-s2 14050 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-starv 16413 df-sca 16414 df-vsca 16415 df-ip 16416 df-tset 16417 df-ple 16418 df-ds 16420 df-unif 16421 df-hom 16422 df-cco 16423 df-0g 16548 df-gsum 16549 df-prds 16554 df-pws 16556 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-mhm 17778 df-submnd 17779 df-grp 17868 df-minusg 17869 df-sbg 17870 df-mulg 17986 df-subg 18034 df-ghm 18101 df-gim 18144 df-cntz 18192 df-oppg 18219 df-symg 18241 df-pmtr 18305 df-psgn 18354 df-cmn 18639 df-abl 18640 df-mgp 18934 df-ur 18946 df-srg 18950 df-ring 18993 df-cring 18994 df-oppr 19067 df-dvdsr 19085 df-unit 19086 df-invr 19116 df-dvr 19127 df-rnghom 19161 df-drng 19198 df-subrg 19227 df-lmod 19330 df-lss 19398 df-sra 19638 df-rgmod 19639 df-cnfld 20232 df-zring 20304 df-zrh 20337 df-dsmm 20562 df-frlm 20577 df-mamu 20681 df-mat 20705 df-mdet 20882

This theorem is referenced by: mdetuni0 20918 matunit 20975 cramerimplem1 20980 matunitlindflem2 34441

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