Theorem decpmatid 19731

Description: The matrix consisting of the coefficients in the polynomial entries of the identity matrix is an identity or a zero matrix. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 2-Dec-2019.)

Hypotheses

Ref	Expression
decpmatid.p	|- P = (Poly1 ` R )
decpmatid.c	|- C = ( N Mat P )
decpmatid.i	|- I = ( 1r ` C )
decpmatid.a	|- A = ( N Mat R )
decpmatid.0	|- .0. = ( 0g ` A )
decpmatid.1	|- .1. = ( 1r ` A )

Assertion

Ref	Expression
decpmatid	|- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( I decompPMat K ) = if ( K = 0 , .1. , .0. ) )

Proof of Theorem decpmatid

Dummy variables i j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		decpmatid.p	. . . . . 6 |- P = (Poly1 ` R )
2		decpmatid.c	. . . . . 6 |- C = ( N Mat P )
3	1, 2	pmatring 19654	. . . . 5 |- ( ( N e. Fin /\ R e. Ring ) -> C e. Ring )
4	3	3adant3 1025	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> C e. Ring )
5		eqid 2420	. . . . 5 |- ( Base ` C ) = ( Base ` C )
6		decpmatid.i	. . . . 5 |- I = ( 1r ` C )
7	5, 6	ringidcl 17742	. . . 4 |- ( C e. Ring -> I e. ( Base ` C ) )
8	4, 7	syl 17	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> I e. ( Base ` C ) )
9		simp3 1007	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> K e. NN0 )
10	2, 5	decpmatval 19726	. . 3 |- ( ( I e. ( Base ` C ) /\ K e. NN0 ) -> ( I decompPMat K ) = ( i e. N , j e. N |-> ( (coe1 ` ( i I j ) ) ` K ) ) )
11	8, 9, 10	syl2anc 665	. 2 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( I decompPMat K ) = ( i e. N , j e. N |-> ( (coe1 ` ( i I j ) ) ` K ) ) )
12		eqid 2420	. . . . . . 7 |- ( 0g ` P ) = ( 0g ` P )
13		eqid 2420	. . . . . . 7 |- ( 1r ` P ) = ( 1r ` P )
14		simp11 1035	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> N e. Fin )
15		simp12 1036	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> R e. Ring )
16		simp2 1006	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> i e. N )
17		simp3 1007	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> j e. N )
18	1, 2, 12, 13, 14, 15, 16, 17, 6	pmat1ovd 19658	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> ( i I j ) = if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) )
19	18	fveq2d 5876	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> (coe1 ` ( i I j ) ) = (coe1 ` if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) ) )
20	19	fveq1d 5874	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> ( (coe1 ` ( i I j ) ) ` K ) = ( (coe1 ` if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) ) ` K ) )
21		fvif 5883	. . . . . . 7 |- (coe1 ` if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) ) = if ( i = j , (coe1 ` ( 1r ` P ) ) , (coe1 ` ( 0g ` P ) ) )
22	21	fveq1i 5873	. . . . . 6 |- ( (coe1 ` if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) ) ` K ) = ( if ( i = j , (coe1 ` ( 1r ` P ) ) , (coe1 ` ( 0g ` P ) ) ) ` K )
23		iffv 5884	. . . . . 6 |- ( if ( i = j , (coe1 ` ( 1r ` P ) ) , (coe1 ` ( 0g ` P ) ) ) ` K ) = if ( i = j , ( (coe1 ` ( 1r ` P ) ) ` K ) , ( (coe1 ` ( 0g ` P ) ) ` K ) )
24	22, 23	eqtri 2449	. . . . 5 |- ( (coe1 ` if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) ) ` K ) = if ( i = j , ( (coe1 ` ( 1r ` P ) ) ` K ) , ( (coe1 ` ( 0g ` P ) ) ` K ) )
25		eqid 2420	. . . . . . . . . . . . 13 |- (var1 ` R ) = (var1 ` R )
26		eqid 2420	. . . . . . . . . . . . 13 |- (mulGrp ` P ) = (mulGrp ` P )
27		eqid 2420	. . . . . . . . . . . . 13 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
28	1, 25, 26, 27	ply1idvr1 18827	. . . . . . . . . . . 12 |- ( R e. Ring -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) = ( 1r ` P ) )
29	28	3ad2ant2 1027	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) = ( 1r ` P ) )
30	29	eqcomd 2428	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 1r ` P ) = ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) )
31	30	fveq2d 5876	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> (coe1 ` ( 1r ` P ) ) = (coe1 ` ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
32	31	fveq1d 5874	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( (coe1 ` ( 1r ` P ) ) ` K ) = ( (coe1 ` ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ` K ) )
33	1	ply1lmod 18786	. . . . . . . . . . . . 13 |- ( R e. Ring -> P e. LMod )
34	33	3ad2ant2 1027	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> P e. LMod )
35		0nn0 10873	. . . . . . . . . . . . . 14 |- 0 e. NN0
36		eqid 2420	. . . . . . . . . . . . . . 15 |- ( Base ` P ) = ( Base ` P )
37	1, 25, 26, 27, 36	ply1moncl 18805	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ 0 e. NN0 ) -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) )
38	35, 37	mpan2 675	. . . . . . . . . . . . 13 |- ( R e. Ring -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) )
39	38	3ad2ant2 1027	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) )
40		eqid 2420	. . . . . . . . . . . . 13 |- (Scalar ` P ) = (Scalar ` P )
41		eqid 2420	. . . . . . . . . . . . 13 |- ( .s ` P ) = ( .s ` P )
42		eqid 2420	. . . . . . . . . . . . 13 |- ( 1r ` (Scalar ` P ) ) = ( 1r ` (Scalar ` P ) )
43	36, 40, 41, 42	lmodvs1 18060	. . . . . . . . . . . 12 |- ( ( P e. LMod /\ ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) ) -> ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) )
44	34, 39, 43	syl2anc 665	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) )
45	44	eqcomd 2428	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) = ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
46	45	fveq2d 5876	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> (coe1 ` ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = (coe1 ` ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) )
47	46	fveq1d 5874	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( (coe1 ` ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ` K ) = ( (coe1 ` ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ` K ) )
48		simp2 1006	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> R e. Ring )
49	1	ply1sca 18787	. . . . . . . . . . . . . 14 |- ( R e. Ring -> R = (Scalar ` P ) )
50	49	3ad2ant2 1027	. . . . . . . . . . . . 13 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> R = (Scalar ` P ) )
51	50	eqcomd 2428	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> (Scalar ` P ) = R )
52	51	fveq2d 5876	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 1r ` (Scalar ` P ) ) = ( 1r ` R ) )
53		eqid 2420	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
54		eqid 2420	. . . . . . . . . . . . 13 |- ( 1r ` R ) = ( 1r ` R )
55	53, 54	ringidcl 17742	. . . . . . . . . . . 12 |- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) )
56	55	3ad2ant2 1027	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 1r ` R ) e. ( Base ` R ) )
57	52, 56	eqeltrd 2508	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 1r ` (Scalar ` P ) ) e. ( Base ` R ) )
58	35	a1i 11	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> 0 e. NN0 )
59		eqid 2420	. . . . . . . . . . 11 |- ( 0g ` R ) = ( 0g ` R )
60	59, 53, 1, 25, 41, 26, 27	coe1tm 18807	. . . . . . . . . 10 |- ( ( R e. Ring /\ ( 1r ` (Scalar ` P ) ) e. ( Base ` R ) /\ 0 e. NN0 ) -> (coe1 ` ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = ( k e. NN0 |-> if ( k = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) ) )
61	48, 57, 58, 60	syl3anc 1264	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> (coe1 ` ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = ( k e. NN0 |-> if ( k = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) ) )
62		eqeq1 2424	. . . . . . . . . . 11 |- ( k = K -> ( k = 0 <-> K = 0 ) )
63	62	ifbid 3928	. . . . . . . . . 10 |- ( k = K -> if ( k = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) )
64	63	adantl 467	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ k = K ) -> if ( k = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) )
65		fvex 5882	. . . . . . . . . . 11 |- ( 1r ` (Scalar ` P ) ) e. _V
66		fvex 5882	. . . . . . . . . . 11 |- ( 0g ` R ) e. _V
67	65, 66	ifex 3974	. . . . . . . . . 10 |- if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) e. _V
68	67	a1i 11	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) e. _V )
69	61, 64, 9, 68	fvmptd 5961	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( (coe1 ` ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ` K ) = if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) )
70	32, 47, 69	3eqtrd 2465	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( (coe1 ` ( 1r ` P ) ) ` K ) = if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) )
71	1, 12, 59	coe1z 18797	. . . . . . . . . 10 |- ( R e. Ring -> (coe1 ` ( 0g ` P ) ) = ( NN0 X. { ( 0g ` R ) } ) )
72	71	3ad2ant2 1027	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> (coe1 ` ( 0g ` P ) ) = ( NN0 X. { ( 0g ` R ) } ) )
73	72	fveq1d 5874	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( (coe1 ` ( 0g ` P ) ) ` K ) = ( ( NN0 X. { ( 0g ` R ) } ) ` K ) )
74	66	a1i 11	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 0g ` R ) e. _V )
75		fvconst2g 6124	. . . . . . . . 9 |- ( ( ( 0g ` R ) e. _V /\ K e. NN0 ) -> ( ( NN0 X. { ( 0g ` R ) } ) ` K ) = ( 0g ` R ) )
76	74, 9, 75	syl2anc 665	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( ( NN0 X. { ( 0g ` R ) } ) ` K ) = ( 0g ` R ) )
77	73, 76	eqtrd 2461	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( (coe1 ` ( 0g ` P ) ) ` K ) = ( 0g ` R ) )
78	70, 77	ifeq12d 3926	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> if ( i = j , ( (coe1 ` ( 1r ` P ) ) ` K ) , ( (coe1 ` ( 0g ` P ) ) ` K ) ) = if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) )
79	78	3ad2ant1 1026	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> if ( i = j , ( (coe1 ` ( 1r ` P ) ) ` K ) , ( (coe1 ` ( 0g ` P ) ) ` K ) ) = if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) )
80	24, 79	syl5eq 2473	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> ( (coe1 ` if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) ) ` K ) = if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) )
81	20, 80	eqtrd 2461	. . 3 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> ( (coe1 ` ( i I j ) ) ` K ) = if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) )
82	81	mpt2eq3dva 6360	. 2 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( i e. N , j e. N |-> ( (coe1 ` ( i I j ) ) ` K ) ) = ( i e. N , j e. N |-> if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) ) )
83	50	adantl 467	. . . . . . . . 9 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> R = (Scalar ` P ) )
84	83	eqcomd 2428	. . . . . . . 8 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> (Scalar ` P ) = R )
85	84	fveq2d 5876	. . . . . . 7 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( 1r ` (Scalar ` P ) ) = ( 1r ` R ) )
86	85	ifeq1d 3924	. . . . . 6 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> if ( i = j , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = if ( i = j , ( 1r ` R ) , ( 0g ` R ) ) )
87	86	mpt2eq3dv 6362	. . . . 5 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( i e. N , j e. N |-> if ( i = j , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) ) = ( i e. N , j e. N |-> if ( i = j , ( 1r ` R ) , ( 0g ` R ) ) ) )
88		iftrue 3912	. . . . . . . 8 |- ( K = 0 -> if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = ( 1r ` (Scalar ` P ) ) )
89	88	ifeq1d 3924	. . . . . . 7 |- ( K = 0 -> if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) = if ( i = j , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) )
90	89	adantr 466	. . . . . 6 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) = if ( i = j , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) )
91	90	mpt2eq3dv 6362	. . . . 5 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( i e. N , j e. N |-> if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) ) = ( i e. N , j e. N |-> if ( i = j , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) ) )
92		decpmatid.1	. . . . . . . 8 |- .1. = ( 1r ` A )
93		decpmatid.a	. . . . . . . . 9 |- A = ( N Mat R )
94	93, 54, 59	mat1 19409	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring ) -> ( 1r ` A ) = ( i e. N , j e. N |-> if ( i = j , ( 1r ` R ) , ( 0g ` R ) ) ) )
95	92, 94	syl5eq 2473	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> .1. = ( i e. N , j e. N |-> if ( i = j , ( 1r ` R ) , ( 0g ` R ) ) ) )
96	95	3adant3 1025	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> .1. = ( i e. N , j e. N |-> if ( i = j , ( 1r ` R ) , ( 0g ` R ) ) ) )
97	96	adantl 467	. . . . 5 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> .1. = ( i e. N , j e. N |-> if ( i = j , ( 1r ` R ) , ( 0g ` R ) ) ) )
98	87, 91, 97	3eqtr4d 2471	. . . 4 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( i e. N , j e. N |-> if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) ) = .1. )
99		iftrue 3912	. . . . . 6 |- ( K = 0 -> if ( K = 0 , .1. , .0. ) = .1. )
100	99	eqcomd 2428	. . . . 5 |- ( K = 0 -> .1. = if ( K = 0 , .1. , .0. ) )
101	100	adantr 466	. . . 4 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> .1. = if ( K = 0 , .1. , .0. ) )
102	98, 101	eqtrd 2461	. . 3 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( i e. N , j e. N |-> if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) ) = if ( K = 0 , .1. , .0. ) )
103		ifid 3943	. . . . . . 7 |- if ( i = j , ( 0g ` R ) , ( 0g ` R ) ) = ( 0g ` R )
104	103	a1i 11	. . . . . 6 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> if ( i = j , ( 0g ` R ) , ( 0g ` R ) ) = ( 0g ` R ) )
105	104	mpt2eq3dv 6362	. . . . 5 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( i e. N , j e. N |-> if ( i = j , ( 0g ` R ) , ( 0g ` R ) ) ) = ( i e. N , j e. N |-> ( 0g ` R ) ) )
106		iffalse 3915	. . . . . . . 8 |- ( -. K = 0 -> if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = ( 0g ` R ) )
107	106	adantr 466	. . . . . . 7 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = ( 0g ` R ) )
108	107	ifeq1d 3924	. . . . . 6 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) = if ( i = j , ( 0g ` R ) , ( 0g ` R ) ) )
109	108	mpt2eq3dv 6362	. . . . 5 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( i e. N , j e. N |-> if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) ) = ( i e. N , j e. N |-> if ( i = j , ( 0g ` R ) , ( 0g ` R ) ) ) )
110		3simpa 1002	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( N e. Fin /\ R e. Ring ) )
111	110	adantl 467	. . . . . 6 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( N e. Fin /\ R e. Ring ) )
112		decpmatid.0	. . . . . . 7 |- .0. = ( 0g ` A )
113	93, 59	mat0op 19381	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> ( 0g ` A ) = ( i e. N , j e. N |-> ( 0g ` R ) ) )
114	112, 113	syl5eq 2473	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring ) -> .0. = ( i e. N , j e. N |-> ( 0g ` R ) ) )
115	111, 114	syl 17	. . . . 5 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> .0. = ( i e. N , j e. N |-> ( 0g ` R ) ) )
116	105, 109, 115	3eqtr4d 2471	. . . 4 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( i e. N , j e. N |-> if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) ) = .0. )
117		iffalse 3915	. . . . . 6 |- ( -. K = 0 -> if ( K = 0 , .1. , .0. ) = .0. )
118	117	eqcomd 2428	. . . . 5 |- ( -. K = 0 -> .0. = if ( K = 0 , .1. , .0. ) )
119	118	adantr 466	. . . 4 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> .0. = if ( K = 0 , .1. , .0. ) )
120	116, 119	eqtrd 2461	. . 3 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( i e. N , j e. N |-> if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) ) = if ( K = 0 , .1. , .0. ) )
121	102, 120	pm2.61ian 797	. 2 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( i e. N , j e. N |-> if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) ) = if ( K = 0 , .1. , .0. ) )
122	11, 82, 121	3eqtrd 2465	1 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( I decompPMat K ) = if ( K = 0 , .1. , .0. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1867   _Vcvv 3078   ifcif 3906   {csn 3993   |-> cmpt 4475   X. cxp 4843   ` cfv 5592  (class class class)co 6296   |-> cmpt2 6298   Fincfn 7568   0cc0 9528   NN0cn0 10858   Basecbs 15081  Scalarcsca 15153   .scvsca 15154   0gc0g 15298  ._gcmg 16624  mulGrpcmgp 17664   1rcur 17676   Ringcrg 17721   LModclmod 18032  var₁cv1 18710  Poly₁cpl1 18711  coe₁cco1 18712   Mat cmat 19369   decompPMat cdecpmat 19723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-ot 4002  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-ofr 6537  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-sup 7953  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-fz 11772  df-fzo 11903  df-seq 12200  df-hash 12502  df-struct 15083  df-ndx 15084  df-slot 15085  df-base 15086  df-sets 15087  df-ress 15088  df-plusg 15163  df-mulr 15164  df-sca 15166  df-vsca 15167  df-ip 15168  df-tset 15169  df-ple 15170  df-ds 15172  df-hom 15174  df-cco 15175  df-0g 15300  df-gsum 15301  df-prds 15306  df-pws 15308  df-mre 15444  df-mrc 15445  df-acs 15447  df-mgm 16440  df-sgrp 16479  df-mnd 16489  df-mhm 16534  df-submnd 16535  df-grp 16625  df-minusg 16626  df-sbg 16627  df-mulg 16628  df-subg 16766  df-ghm 16833  df-cntz 16923  df-cmn 17373  df-abl 17374  df-mgp 17665  df-ur 17677  df-ring 17723  df-subrg 17947  df-lmod 18034  df-lss 18097  df-sra 18336  df-rgmod 18337  df-ascl 18479  df-psr 18521  df-mvr 18522  df-mpl 18523  df-opsr 18525  df-psr1 18714  df-vr1 18715  df-ply1 18716  df-coe1 18717  df-dsmm 19232  df-frlm 19247  df-mamu 19346  df-mat 19370  df-decpmat 19724

This theorem is referenced by:  idpm2idmp  19762