Theorem decpmatid 19787

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 19710	. . . . 5 |- ( ( N e. Fin /\ R e. Ring ) -> C e. Ring )
4	3	3adant3 1027	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> C e. Ring )
5		eqid 2450	. . . . 5 |- ( Base ` C ) = ( Base ` C )
6		decpmatid.i	. . . . 5 |- I = ( 1r ` C )
7	5, 6	ringidcl 17794	. . . 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 1009	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> K e. NN0 )
10	2, 5	decpmatval 19782	. . 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 666	. 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 2450	. . . . . . 7 |- ( 0g ` P ) = ( 0g ` P )
13		eqid 2450	. . . . . . 7 |- ( 1r ` P ) = ( 1r ` P )
14		simp11 1037	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> N e. Fin )
15		simp12 1038	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> R e. Ring )
16		simp2 1008	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> i e. N )
17		simp3 1009	. . . . . . 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 19714	. . . . . 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 5867	. . . . 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 5865	. . . 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 5874	. . . . . . 7 |- (coe1 ` if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) ) = if ( i = j , (coe1 ` ( 1r ` P ) ) , (coe1 ` ( 0g ` P ) ) )
22	21	fveq1i 5864	. . . . . 6 |- ( (coe1 ` if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) ) ` K ) = ( if ( i = j , (coe1 ` ( 1r ` P ) ) , (coe1 ` ( 0g ` P ) ) ) ` K )
23		iffv 5875	. . . . . 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 2472	. . . . 5 |- ( (coe1 ` if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) ) ` K ) = if ( i = j , ( (coe1 ` ( 1r ` P ) ) ` K ) , ( (coe1 ` ( 0g ` P ) ) ` K ) )
25		eqid 2450	. . . . . . . . . . . . 13 |- (var1 ` R ) = (var1 ` R )
26		eqid 2450	. . . . . . . . . . . . 13 |- (mulGrp ` P ) = (mulGrp ` P )
27		eqid 2450	. . . . . . . . . . . . 13 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
28	1, 25, 26, 27	ply1idvr1 18879	. . . . . . . . . . . 12 |- ( R e. Ring -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) = ( 1r ` P ) )
29	28	3ad2ant2 1029	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) = ( 1r ` P ) )
30	29	eqcomd 2456	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 1r ` P ) = ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) )
31	30	fveq2d 5867	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> (coe1 ` ( 1r ` P ) ) = (coe1 ` ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
32	31	fveq1d 5865	. . . . . . . 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 18838	. . . . . . . . . . . . 13 |- ( R e. Ring -> P e. LMod )
34	33	3ad2ant2 1029	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> P e. LMod )
35		0nn0 10881	. . . . . . . . . . . . . 14 |- 0 e. NN0
36		eqid 2450	. . . . . . . . . . . . . . 15 |- ( Base ` P ) = ( Base ` P )
37	1, 25, 26, 27, 36	ply1moncl 18857	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ 0 e. NN0 ) -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) )
38	35, 37	mpan2 676	. . . . . . . . . . . . 13 |- ( R e. Ring -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) )
39	38	3ad2ant2 1029	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) )
40		eqid 2450	. . . . . . . . . . . . 13 |- (Scalar ` P ) = (Scalar ` P )
41		eqid 2450	. . . . . . . . . . . . 13 |- ( .s ` P ) = ( .s ` P )
42		eqid 2450	. . . . . . . . . . . . 13 |- ( 1r ` (Scalar ` P ) ) = ( 1r ` (Scalar ` P ) )
43	36, 40, 41, 42	lmodvs1 18112	. . . . . . . . . . . 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 666	. . . . . . . . . . 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 2456	. . . . . . . . . 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 5867	. . . . . . . . 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 5865	. . . . . . . 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 1008	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> R e. Ring )
49	1	ply1sca 18839	. . . . . . . . . . . . . 14 |- ( R e. Ring -> R = (Scalar ` P ) )
50	49	3ad2ant2 1029	. . . . . . . . . . . . 13 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> R = (Scalar ` P ) )
51	50	eqcomd 2456	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> (Scalar ` P ) = R )
52	51	fveq2d 5867	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 1r ` (Scalar ` P ) ) = ( 1r ` R ) )
53		eqid 2450	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
54		eqid 2450	. . . . . . . . . . . . 13 |- ( 1r ` R ) = ( 1r ` R )
55	53, 54	ringidcl 17794	. . . . . . . . . . . 12 |- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) )
56	55	3ad2ant2 1029	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 1r ` R ) e. ( Base ` R ) )
57	52, 56	eqeltrd 2528	. . . . . . . . . 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 2450	. . . . . . . . . . 11 |- ( 0g ` R ) = ( 0g ` R )
60	59, 53, 1, 25, 41, 26, 27	coe1tm 18859	. . . . . . . . . 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 1267	. . . . . . . . 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 2454	. . . . . . . . . . 11 |- ( k = K -> ( k = 0 <-> K = 0 ) )
63	62	ifbid 3902	. . . . . . . . . 10 |- ( k = K -> if ( k = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) )
64	63	adantl 468	. . . . . . . . 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 5873	. . . . . . . . . . 11 |- ( 1r ` (Scalar ` P ) ) e. _V
66		fvex 5873	. . . . . . . . . . 11 |- ( 0g ` R ) e. _V
67	65, 66	ifex 3948	. . . . . . . . . 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 5952	. . . . . . . 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 2488	. . . . . . 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 18849	. . . . . . . . . 10 |- ( R e. Ring -> (coe1 ` ( 0g ` P ) ) = ( NN0 X. { ( 0g ` R ) } ) )
72	71	3ad2ant2 1029	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> (coe1 ` ( 0g ` P ) ) = ( NN0 X. { ( 0g ` R ) } ) )
73	72	fveq1d 5865	. . . . . . . 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 6116	. . . . . . . . 9 |- ( ( ( 0g ` R ) e. _V /\ K e. NN0 ) -> ( ( NN0 X. { ( 0g ` R ) } ) ` K ) = ( 0g ` R ) )
76	74, 9, 75	syl2anc 666	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( ( NN0 X. { ( 0g ` R ) } ) ` K ) = ( 0g ` R ) )
77	73, 76	eqtrd 2484	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( (coe1 ` ( 0g ` P ) ) ` K ) = ( 0g ` R ) )
78	70, 77	ifeq12d 3900	. . . . . 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 1028	. . . . 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 2496	. . . 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 2484	. . 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 6352	. 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 468	. . . . . . . . 9 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> R = (Scalar ` P ) )
84	83	eqcomd 2456	. . . . . . . 8 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> (Scalar ` P ) = R )
85	84	fveq2d 5867	. . . . . . 7 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( 1r ` (Scalar ` P ) ) = ( 1r ` R ) )
86	85	ifeq1d 3898	. . . . . 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 6354	. . . . 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 3886	. . . . . . . 8 |- ( K = 0 -> if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = ( 1r ` (Scalar ` P ) ) )
89	88	ifeq1d 3898	. . . . . . 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 467	. . . . . 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 6354	. . . . 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 19465	. . . . . . . 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 2496	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> .1. = ( i e. N , j e. N |-> if ( i = j , ( 1r ` R ) , ( 0g ` R ) ) ) )
96	95	3adant3 1027	. . . . . 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 468	. . . . 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 2494	. . . 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 3886	. . . . . 6 |- ( K = 0 -> if ( K = 0 , .1. , .0. ) = .1. )
100	99	eqcomd 2456	. . . . 5 |- ( K = 0 -> .1. = if ( K = 0 , .1. , .0. ) )
101	100	adantr 467	. . . 4 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> .1. = if ( K = 0 , .1. , .0. ) )
102	98, 101	eqtrd 2484	. . 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 3917	. . . . . . 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 6354	. . . . 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 3889	. . . . . . . 8 |- ( -. K = 0 -> if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = ( 0g ` R ) )
107	106	adantr 467	. . . . . . 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 3898	. . . . . 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 6354	. . . . 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 1004	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( N e. Fin /\ R e. Ring ) )
111	110	adantl 468	. . . . . 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 19437	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> ( 0g ` A ) = ( i e. N , j e. N |-> ( 0g ` R ) ) )
114	112, 113	syl5eq 2496	. . . . . 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 2494	. . . 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 3889	. . . . . 6 |- ( -. K = 0 -> if ( K = 0 , .1. , .0. ) = .0. )
118	117	eqcomd 2456	. . . . 5 |- ( -. K = 0 -> .0. = if ( K = 0 , .1. , .0. ) )
119	118	adantr 467	. . . 4 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> .0. = if ( K = 0 , .1. , .0. ) )
120	116, 119	eqtrd 2484	. . 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 798	. 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 2488	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 371   /\ w3a 984   = wceq 1443   e. wcel 1886   _Vcvv 3044   ifcif 3880   {csn 3967   |-> cmpt 4460   X. cxp 4831   ` cfv 5581  (class class class)co 6288   |-> cmpt2 6290   Fincfn 7566   0cc0 9536   NN0cn0 10866   Basecbs 15114  Scalarcsca 15186   .scvsca 15187   0gc0g 15331  ._gcmg 16665  mulGrpcmgp 17716   1rcur 17728   Ringcrg 17773   LModclmod 18084  var₁cv1 18762  Poly₁cpl1 18763  coe₁cco1 18764   Mat cmat 19425   decompPMat cdecpmat 19779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-ot 3976  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-ofr 6529  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-sup 7953  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-fz 11782  df-fzo 11913  df-seq 12211  df-hash 12513  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-hom 15207  df-cco 15208  df-0g 15333  df-gsum 15334  df-prds 15339  df-pws 15341  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-submnd 16576  df-grp 16666  df-minusg 16667  df-sbg 16668  df-mulg 16669  df-subg 16807  df-ghm 16874  df-cntz 16964  df-cmn 17425  df-abl 17426  df-mgp 17717  df-ur 17729  df-ring 17775  df-subrg 17999  df-lmod 18086  df-lss 18149  df-sra 18388  df-rgmod 18389  df-ascl 18531  df-psr 18573  df-mvr 18574  df-mpl 18575  df-opsr 18577  df-psr1 18766  df-vr1 18767  df-ply1 18768  df-coe1 18769  df-dsmm 19288  df-frlm 19303  df-mamu 19402  df-mat 19426  df-decpmat 19780

This theorem is referenced by:  idpm2idmp  19818