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Theorem mdet1 20407

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 18558	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	anim1i 592	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
4	3	ancomd 467	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
5		mdet1.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6	5	matring 20249	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
7		eqid 2622	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴)
8		mdet1.n	. . . . . 6 ⊢ 𝐼 = (1_r‘𝐴)
9	7, 8	ringidcl 18568	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝐼 ∈ (Base‘𝐴))
10	4, 6, 9	3syl 18	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝐼 ∈ (Base‘𝐴))
11		eqid 2622	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
12		mdet1.o	. . . . . . 7 ⊢ 1 = (1_r‘𝑅)
13	11, 12	ringidcl 18568	. . . . . 6 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
14	2, 13	syl 17	. . . . 5 ⊢ (𝑅 ∈ CRing → 1 ∈ (Base‘𝑅))
15	14	adantr 481	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 1 ∈ (Base‘𝑅))
16	1, 10, 15	jca32 558	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ (Base‘𝐴) ∧ 1 ∈ (Base‘𝑅))))
17		eqid 2622	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
18		simplr 792	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
19	2	adantr 481	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
20	19	adantr 481	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
21		simprl 794	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
22		simprr 796	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
23	5, 12, 17, 18, 20, 21, 22, 8	mat1ov 20254	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, 1 , (0_g‘𝑅)))
24	23	ralrimivva 2971	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐼𝑗) = if(𝑖 = 𝑗, 1 , (0_g‘𝑅)))
25		mdet1.d	. . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅)
26		eqid 2622	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
27		eqid 2622	. . . 4 ⊢ (._g‘(mulGrp‘𝑅)) = (._g‘(mulGrp‘𝑅))
28	25, 5, 7, 26, 17, 11, 27	mdetdiagid 20406	. . 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 )))
29	16, 24, 28	sylc 65	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐷‘𝐼) = ((#‘𝑁)(._g‘(mulGrp‘𝑅)) 1 ))
30		ringsrg 18589	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
31	2, 30	syl 17	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ SRing)
32		hashcl 13147	. . 3 ⊢ (𝑁 ∈ Fin → (#‘𝑁) ∈ ℕ₀)
33	26, 27, 12	srg1expzeq1 18539	. . 3 ⊢ ((𝑅 ∈ SRing ∧ (#‘𝑁) ∈ ℕ₀) → ((#‘𝑁)(._g‘(mulGrp‘𝑅)) 1 ) = 1 )
34	31, 32, 33	syl2an 494	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → ((#‘𝑁)(._g‘(mulGrp‘𝑅)) 1 ) = 1 )
35	29, 34	eqtrd 2656	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐷‘𝐼) = 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ifcif 4086 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℕ₀cn0 11292 #chash 13117 Basecbs 15857 0_gc0g 16100 ._gcmg 17540 mulGrpcmgp 18489 1_rcur 18501 SRingcsrg 18505 Ringcrg 18547 CRingccrg 18548 Mat cmat 20213 maDet cmdat 20390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-xor 1465 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-reverse 13305 df-s2 13593 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-gim 17701 df-cntz 17750 df-oppg 17776 df-symg 17798 df-pmtr 17862 df-psgn 17911 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-srg 18506 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-dsmm 20076 df-frlm 20091 df-mamu 20190 df-mat 20214 df-mdet 20391

This theorem is referenced by: mdetuni0 20427 matunit 20484 cramerimplem1 20489 matunitlindflem2 33406

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