Theorem mdetdiagid 20406

Description: The determinant of a diagonal matrix with identical entries is the power of the entry in the diagonal. (Contributed by AV, 17-Aug-2019.)

Hypotheses

Ref	Expression
mdetdiag.d	|- D = ( N maDet R )
mdetdiag.a	|- A = ( N Mat R )
mdetdiag.b	|- B = ( Base ` A )
mdetdiag.g	|- G = (mulGrp ` R )
mdetdiag.0	|- .0. = ( 0g ` R )
mdetdiagid.c	|- C = ( Base ` R )
mdetdiagid.t	|- .x. = (.g ` G )

Assertion

Ref	Expression
mdetdiagid	|- ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) -> ( A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) -> ( D ` M ) = ( ( # ` N ) .x. X ) ) )

Distinct variable groups:   i, M, j   i, N, j   .0. , i, j   B, i, j   C, i, j   R, i, j   i, X, j

Allowed substitution hints:   A( i, j)   D( i, j)   .x. ( i, j)   G( i, j)

Proof of Theorem mdetdiagid

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . . 7 |- ( ( R e. CRing /\ N e. Fin ) -> R e. CRing )
2	1	adantr 481	. . . . . 6 |- ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) -> R e. CRing )
3		simpr 477	. . . . . . 7 |- ( ( R e. CRing /\ N e. Fin ) -> N e. Fin )
4	3	adantr 481	. . . . . 6 |- ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) -> N e. Fin )
5		simpl 473	. . . . . . 7 |- ( ( M e. B /\ X e. C ) -> M e. B )
6	5	adantl 482	. . . . . 6 |- ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) -> M e. B )
7	2, 4, 6	3jca 1242	. . . . 5 |- ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) -> ( R e. CRing /\ N e. Fin /\ M e. B ) )
8	7	adantr 481	. . . 4 |- ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) ) -> ( R e. CRing /\ N e. Fin /\ M e. B ) )
9		id 22	. . . . . . . . . 10 |- ( ( i M j ) = if ( i = j , X , .0. ) -> ( i M j ) = if ( i = j , X , .0. ) )
10		ifnefalse 4098	. . . . . . . . . . 11 |- ( i =/= j -> if ( i = j , X , .0. ) = .0. )
11	10	adantl 482	. . . . . . . . . 10 |- ( ( ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ i e. N ) /\ j e. N ) /\ i =/= j ) -> if ( i = j , X , .0. ) = .0. )
12	9, 11	sylan9eqr 2678	. . . . . . . . 9 |- ( ( ( ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ i e. N ) /\ j e. N ) /\ i =/= j ) /\ ( i M j ) = if ( i = j , X , .0. ) ) -> ( i M j ) = .0. )
13	12	exp31 630	. . . . . . . 8 |- ( ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ i e. N ) /\ j e. N ) -> ( i =/= j -> ( ( i M j ) = if ( i = j , X , .0. ) -> ( i M j ) = .0. ) ) )
14	13	com23 86	. . . . . . 7 |- ( ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ i e. N ) /\ j e. N ) -> ( ( i M j ) = if ( i = j , X , .0. ) -> ( i =/= j -> ( i M j ) = .0. ) ) )
15	14	ralimdva 2962	. . . . . 6 |- ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ i e. N ) -> ( A. j e. N ( i M j ) = if ( i = j , X , .0. ) -> A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) )
16	15	ralimdva 2962	. . . . 5 |- ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) -> ( A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) -> A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) )
17	16	imp 445	. . . 4 |- ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) ) -> A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) )
18		mdetdiag.d	. . . . 5 |- D = ( N maDet R )
19		mdetdiag.a	. . . . 5 |- A = ( N Mat R )
20		mdetdiag.b	. . . . 5 |- B = ( Base ` A )
21		mdetdiag.g	. . . . 5 |- G = (mulGrp ` R )
22		mdetdiag.0	. . . . 5 |- .0. = ( 0g ` R )
23	18, 19, 20, 21, 22	mdetdiag 20405	. . . 4 |- ( ( R e. CRing /\ N e. Fin /\ M e. B ) -> ( A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) -> ( D ` M ) = ( G gsumg ( k e. N |-> ( k M k ) ) ) ) )
24	8, 17, 23	sylc 65	. . 3 |- ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) ) -> ( D ` M ) = ( G gsumg ( k e. N |-> ( k M k ) ) ) )
25		oveq1 6657	. . . . . . . . . . . 12 |- ( i = k -> ( i M j ) = ( k M j ) )
26		equequ1 1952	. . . . . . . . . . . . 13 |- ( i = k -> ( i = j <-> k = j ) )
27	26	ifbid 4108	. . . . . . . . . . . 12 |- ( i = k -> if ( i = j , X , .0. ) = if ( k = j , X , .0. ) )
28	25, 27	eqeq12d 2637	. . . . . . . . . . 11 |- ( i = k -> ( ( i M j ) = if ( i = j , X , .0. ) <-> ( k M j ) = if ( k = j , X , .0. ) ) )
29		oveq2 6658	. . . . . . . . . . . 12 |- ( j = k -> ( k M j ) = ( k M k ) )
30		equequ2 1953	. . . . . . . . . . . . 13 |- ( j = k -> ( k = j <-> k = k ) )
31	30	ifbid 4108	. . . . . . . . . . . 12 |- ( j = k -> if ( k = j , X , .0. ) = if ( k = k , X , .0. ) )
32	29, 31	eqeq12d 2637	. . . . . . . . . . 11 |- ( j = k -> ( ( k M j ) = if ( k = j , X , .0. ) <-> ( k M k ) = if ( k = k , X , .0. ) ) )
33	28, 32	rspc2v 3322	. . . . . . . . . 10 |- ( ( k e. N /\ k e. N ) -> ( A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) -> ( k M k ) = if ( k = k , X , .0. ) ) )
34	33	anidms 677	. . . . . . . . 9 |- ( k e. N -> ( A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) -> ( k M k ) = if ( k = k , X , .0. ) ) )
35	34	adantl 482	. . . . . . . 8 |- ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ k e. N ) -> ( A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) -> ( k M k ) = if ( k = k , X , .0. ) ) )
36	35	imp 445	. . . . . . 7 |- ( ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ k e. N ) /\ A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) ) -> ( k M k ) = if ( k = k , X , .0. ) )
37		equid 1939	. . . . . . . 8 |- k = k
38	37	iftruei 4093	. . . . . . 7 |- if ( k = k , X , .0. ) = X
39	36, 38	syl6eq 2672	. . . . . 6 |- ( ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ k e. N ) /\ A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) ) -> ( k M k ) = X )
40	39	an32s 846	. . . . 5 |- ( ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) ) /\ k e. N ) -> ( k M k ) = X )
41	40	mpteq2dva 4744	. . . 4 |- ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) ) -> ( k e. N |-> ( k M k ) ) = ( k e. N |-> X ) )
42	41	oveq2d 6666	. . 3 |- ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) ) -> ( G gsumg ( k e. N |-> ( k M k ) ) ) = ( G gsumg ( k e. N |-> X ) ) )
43	21	crngmgp 18555	. . . . . . . 8 |- ( R e. CRing -> G e. CMnd )
44		cmnmnd 18208	. . . . . . . 8 |- ( G e. CMnd -> G e. Mnd )
45	43, 44	syl 17	. . . . . . 7 |- ( R e. CRing -> G e. Mnd )
46	45	adantr 481	. . . . . 6 |- ( ( R e. CRing /\ N e. Fin ) -> G e. Mnd )
47	46	adantr 481	. . . . 5 |- ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) -> G e. Mnd )
48		simpr 477	. . . . . 6 |- ( ( M e. B /\ X e. C ) -> X e. C )
49	48	adantl 482	. . . . 5 |- ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) -> X e. C )
50		mdetdiagid.c	. . . . . . 7 |- C = ( Base ` R )
51	21, 50	mgpbas 18495	. . . . . 6 |- C = ( Base ` G )
52		mdetdiagid.t	. . . . . 6 |- .x. = (.g ` G )
53	51, 52	gsumconst 18334	. . . . 5 |- ( ( G e. Mnd /\ N e. Fin /\ X e. C ) -> ( G gsumg ( k e. N |-> X ) ) = ( ( # ` N ) .x. X ) )
54	47, 4, 49, 53	syl3anc 1326	. . . 4 |- ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) -> ( G gsumg ( k e. N |-> X ) ) = ( ( # ` N ) .x. X ) )
55	54	adantr 481	. . 3 |- ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) ) -> ( G gsumg ( k e. N |-> X ) ) = ( ( # ` N ) .x. X ) )
56	24, 42, 55	3eqtrd 2660	. 2 |- ( ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) /\ A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) ) -> ( D ` M ) = ( ( # ` N ) .x. X ) )
57	56	ex 450	1 |- ( ( ( R e. CRing /\ N e. Fin ) /\ ( M e. B /\ X e. C ) ) -> ( A. i e. N A. j e. N ( i M j ) = if ( i = j , X , .0. ) -> ( D ` M ) = ( ( # ` N ) .x. X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   #chash 13117   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  ._gcmg 17540  CMndccmn 18193  mulGrpcmgp 18489   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-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-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-sra 19172  df-rgmod 19173  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-dsmm 20076  df-frlm 20091  df-mat 20214  df-mdet 20391

This theorem is referenced by:  mdet1  20407