Theorem decpmatid 19078

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	pmatrng 19001	. . . . 5 |- ( ( N e. Fin /\ R e. Ring ) -> C e. Ring )
4	3	3adant3 1016	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> C e. Ring )
5		eqid 2467	. . . . 5 |- ( Base ` C ) = ( Base ` C )
6		decpmatid.i	. . . . 5 |- I = ( 1r ` C )
7	5, 6	rngidcl 17032	. . . 4 |- ( C e. Ring -> I e. ( Base ` C ) )
8	4, 7	syl 16	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> I e. ( Base ` C ) )
9		simp3 998	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> K e. NN0 )
10	2, 5	decpmatval 19073	. . 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 661	. 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 2467	. . . . . . 7 |- ( 0g ` P ) = ( 0g ` P )
13		eqid 2467	. . . . . . 7 |- ( 1r ` P ) = ( 1r ` P )
14		simp11 1026	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> N e. Fin )
15		simp12 1027	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> R e. Ring )
16		simp2 997	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) /\ i e. N /\ j e. N ) -> i e. N )
17		simp3 998	. . . . . . 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 19005	. . . . . 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 5870	. . . . 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 5868	. . . 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 5877	. . . . . . 7 |- (coe1 ` if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) ) = if ( i = j , (coe1 ` ( 1r ` P ) ) , (coe1 ` ( 0g ` P ) ) )
22	21	fveq1i 5867	. . . . . 6 |- ( (coe1 ` if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) ) ` K ) = ( if ( i = j , (coe1 ` ( 1r ` P ) ) , (coe1 ` ( 0g ` P ) ) ) ` K )
23		iffv 5878	. . . . . 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 2496	. . . . 5 |- ( (coe1 ` if ( i = j , ( 1r ` P ) , ( 0g ` P ) ) ) ` K ) = if ( i = j , ( (coe1 ` ( 1r ` P ) ) ` K ) , ( (coe1 ` ( 0g ` P ) ) ` K ) )
25		eqid 2467	. . . . . . . . . . . . 13 |- (var1 ` R ) = (var1 ` R )
26		eqid 2467	. . . . . . . . . . . . 13 |- (mulGrp ` P ) = (mulGrp ` P )
27		eqid 2467	. . . . . . . . . . . . 13 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
28	1, 25, 26, 27	ply1idvr1 18145	. . . . . . . . . . . 12 |- ( R e. Ring -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) = ( 1r ` P ) )
29	28	3ad2ant2 1018	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) = ( 1r ` P ) )
30	29	eqcomd 2475	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 1r ` P ) = ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) )
31	30	fveq2d 5870	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> (coe1 ` ( 1r ` P ) ) = (coe1 ` ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
32	31	fveq1d 5868	. . . . . . . 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 18104	. . . . . . . . . . . . 13 |- ( R e. Ring -> P e. LMod )
34	33	3ad2ant2 1018	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> P e. LMod )
35		0nn0 10811	. . . . . . . . . . . . . 14 |- 0 e. NN0
36		eqid 2467	. . . . . . . . . . . . . . 15 |- ( Base ` P ) = ( Base ` P )
37	1, 25, 26, 27, 36	ply1moncl 18123	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ 0 e. NN0 ) -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) )
38	35, 37	mpan2 671	. . . . . . . . . . . . 13 |- ( R e. Ring -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) )
39	38	3ad2ant2 1018	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) )
40		eqid 2467	. . . . . . . . . . . . 13 |- (Scalar ` P ) = (Scalar ` P )
41		eqid 2467	. . . . . . . . . . . . 13 |- ( .s ` P ) = ( .s ` P )
42		eqid 2467	. . . . . . . . . . . . 13 |- ( 1r ` (Scalar ` P ) ) = ( 1r ` (Scalar ` P ) )
43	36, 40, 41, 42	lmodvs1 17352	. . . . . . . . . . . 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 661	. . . . . . . . . . 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 2475	. . . . . . . . . 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 5870	. . . . . . . . 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 5868	. . . . . . . 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 997	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> R e. Ring )
49	1	ply1sca 18105	. . . . . . . . . . . . . . 15 |- ( R e. Ring -> R = (Scalar ` P ) )
50	49	3ad2ant2 1018	. . . . . . . . . . . . . 14 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> R = (Scalar ` P ) )
51	50	eqcomd 2475	. . . . . . . . . . . . 13 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> (Scalar ` P ) = R )
52	51	fveq2d 5870	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 1r ` (Scalar ` P ) ) = ( 1r ` R ) )
53		eqid 2467	. . . . . . . . . . . . . 14 |- ( Base ` R ) = ( Base ` R )
54		eqid 2467	. . . . . . . . . . . . . 14 |- ( 1r ` R ) = ( 1r ` R )
55	53, 54	rngidcl 17032	. . . . . . . . . . . . 13 |- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) )
56	55	3ad2ant2 1018	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 1r ` R ) e. ( Base ` R ) )
57	52, 56	eqeltrd 2555	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 1r ` (Scalar ` P ) ) e. ( Base ` R ) )
58	35	a1i 11	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> 0 e. NN0 )
59		eqid 2467	. . . . . . . . . . . 12 |- ( 0g ` R ) = ( 0g ` R )
60	59, 53, 1, 25, 41, 26, 27	coe1tm 18125	. . . . . . . . . . 11 |- ( ( 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 1228	. . . . . . . . . 10 |- ( ( 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	61	fveq1d 5868	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( (coe1 ` ( ( 1r ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ` K ) = ( ( k e. NN0 |-> if ( k = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) ) ` K ) )
63		eqidd 2468	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( k e. NN0 |-> if ( k = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) ) = ( k e. NN0 |-> if ( k = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) ) )
64		eqeq1 2471	. . . . . . . . . . . 12 |- ( k = K -> ( k = 0 <-> K = 0 ) )
65	64	ifbid 3961	. . . . . . . . . . 11 |- ( k = K -> if ( k = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) )
66	65	adantl 466	. . . . . . . . . 10 |- ( ( ( 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 ) ) )
67		fvex 5876	. . . . . . . . . . . 12 |- ( 1r ` (Scalar ` P ) ) e. _V
68		fvex 5876	. . . . . . . . . . . 12 |- ( 0g ` R ) e. _V
69	67, 68	ifex 4008	. . . . . . . . . . 11 |- if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) e. _V
70	69	a1i 11	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) e. _V )
71	63, 66, 9, 70	fvmptd 5956	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( ( k e. NN0 |-> if ( k = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) ) ` K ) = if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) )
72	62, 71	eqtrd 2508	. . . . . . . 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 ) ) )
73	32, 47, 72	3eqtrd 2512	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( (coe1 ` ( 1r ` P ) ) ` K ) = if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) )
74	1, 12, 59	coe1z 18115	. . . . . . . . . 10 |- ( R e. Ring -> (coe1 ` ( 0g ` P ) ) = ( NN0 X. { ( 0g ` R ) } ) )
75	74	3ad2ant2 1018	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> (coe1 ` ( 0g ` P ) ) = ( NN0 X. { ( 0g ` R ) } ) )
76	75	fveq1d 5868	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( (coe1 ` ( 0g ` P ) ) ` K ) = ( ( NN0 X. { ( 0g ` R ) } ) ` K ) )
77	68	a1i 11	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( 0g ` R ) e. _V )
78		fvconst2g 6115	. . . . . . . . 9 |- ( ( ( 0g ` R ) e. _V /\ K e. NN0 ) -> ( ( NN0 X. { ( 0g ` R ) } ) ` K ) = ( 0g ` R ) )
79	77, 9, 78	syl2anc 661	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( ( NN0 X. { ( 0g ` R ) } ) ` K ) = ( 0g ` R ) )
80	76, 79	eqtrd 2508	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( (coe1 ` ( 0g ` P ) ) ` K ) = ( 0g ` R ) )
81	73, 80	ifeq12d 3959	. . . . . 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 ) ) )
82	81	3ad2ant1 1017	. . . . 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 ) ) )
83	24, 82	syl5eq 2520	. . . 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 ) ) )
84	20, 83	eqtrd 2508	. . 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 ) ) )
85	84	mpt2eq3dva 6346	. 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 ) ) ) )
86	50	adantl 466	. . . . . . . . 9 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> R = (Scalar ` P ) )
87	86	eqcomd 2475	. . . . . . . 8 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> (Scalar ` P ) = R )
88	87	fveq2d 5870	. . . . . . 7 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( 1r ` (Scalar ` P ) ) = ( 1r ` R ) )
89	88	ifeq1d 3957	. . . . . 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 ) ) )
90	89	mpt2eq3dv 6348	. . . . 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 ) ) ) )
91		iftrue 3945	. . . . . . . 8 |- ( K = 0 -> if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = ( 1r ` (Scalar ` P ) ) )
92	91	ifeq1d 3957	. . . . . . 7 |- ( K = 0 -> if ( i = j , if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) , ( 0g ` R ) ) = if ( i = j , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) )
93	92	adantr 465	. . . . . 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 ) ) )
94	93	mpt2eq3dv 6348	. . . . 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 ) ) ) )
95		decpmatid.1	. . . . . . . 8 |- .1. = ( 1r ` A )
96		decpmatid.a	. . . . . . . . 9 |- A = ( N Mat R )
97	96, 54, 59	mat1 18756	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring ) -> ( 1r ` A ) = ( i e. N , j e. N |-> if ( i = j , ( 1r ` R ) , ( 0g ` R ) ) ) )
98	95, 97	syl5eq 2520	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> .1. = ( i e. N , j e. N |-> if ( i = j , ( 1r ` R ) , ( 0g ` R ) ) ) )
99	98	3adant3 1016	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> .1. = ( i e. N , j e. N |-> if ( i = j , ( 1r ` R ) , ( 0g ` R ) ) ) )
100	99	adantl 466	. . . . 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 ) ) ) )
101	90, 94, 100	3eqtr4d 2518	. . . 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. )
102		iftrue 3945	. . . . . 6 |- ( K = 0 -> if ( K = 0 , .1. , .0. ) = .1. )
103	102	eqcomd 2475	. . . . 5 |- ( K = 0 -> .1. = if ( K = 0 , .1. , .0. ) )
104	103	adantr 465	. . . 4 |- ( ( K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> .1. = if ( K = 0 , .1. , .0. ) )
105	101, 104	eqtrd 2508	. . 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. ) )
106		ifid 3976	. . . . . . 7 |- if ( i = j , ( 0g ` R ) , ( 0g ` R ) ) = ( 0g ` R )
107	106	a1i 11	. . . . . 6 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> if ( i = j , ( 0g ` R ) , ( 0g ` R ) ) = ( 0g ` R ) )
108	107	mpt2eq3dv 6348	. . . . 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 ) ) )
109		iffalse 3948	. . . . . . . 8 |- ( -. K = 0 -> if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = ( 0g ` R ) )
110	109	adantr 465	. . . . . . 7 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> if ( K = 0 , ( 1r ` (Scalar ` P ) ) , ( 0g ` R ) ) = ( 0g ` R ) )
111	110	ifeq1d 3957	. . . . . 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 ) ) )
112	111	mpt2eq3dv 6348	. . . . 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 ) ) ) )
113		3simpa 993	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( N e. Fin /\ R e. Ring ) )
114	113	adantl 466	. . . . . 6 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> ( N e. Fin /\ R e. Ring ) )
115		decpmatid.0	. . . . . . 7 |- .0. = ( 0g ` A )
116	96, 59	mat0op 18728	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> ( 0g ` A ) = ( i e. N , j e. N |-> ( 0g ` R ) ) )
117	115, 116	syl5eq 2520	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring ) -> .0. = ( i e. N , j e. N |-> ( 0g ` R ) ) )
118	114, 117	syl 16	. . . . 5 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> .0. = ( i e. N , j e. N |-> ( 0g ` R ) ) )
119	108, 112, 118	3eqtr4d 2518	. . . 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. )
120		iffalse 3948	. . . . . 6 |- ( -. K = 0 -> if ( K = 0 , .1. , .0. ) = .0. )
121	120	eqcomd 2475	. . . . 5 |- ( -. K = 0 -> .0. = if ( K = 0 , .1. , .0. ) )
122	121	adantr 465	. . . 4 |- ( ( -. K = 0 /\ ( N e. Fin /\ R e. Ring /\ K e. NN0 ) ) -> .0. = if ( K = 0 , .1. , .0. ) )
123	119, 122	eqtrd 2508	. . 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. ) )
124	105, 123	pm2.61ian 788	. 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. ) )
125	11, 85, 124	3eqtrd 2512	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 369   /\ w3a 973   = wceq 1379   e. wcel 1767   _Vcvv 3113   ifcif 3939   {csn 4027   |-> cmpt 4505   X. cxp 4997   ` cfv 5588  (class class class)co 6285   |-> cmpt2 6287   Fincfn 7517   0cc0 9493   NN0cn0 10796   Basecbs 14493  Scalarcsca 14561   .scvsca 14562   0gc0g 14698  ._gcmg 15734  mulGrpcmgp 16955   1rcur 16967   Ringcrg 17012   LModclmod 17324  var₁cv1 18026  Poly₁cpl1 18027  coe₁cco1 18028   Mat cmat 18716   decompPMat cdecpmat 19070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-sup 7902  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-fz 11674  df-fzo 11794  df-seq 12077  df-hash 12375  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-hom 14582  df-cco 14583  df-0g 14700  df-gsum 14701  df-prds 14706  df-pws 14708  df-mre 14844  df-mrc 14845  df-acs 14847  df-mnd 15735  df-mhm 15789  df-submnd 15790  df-grp 15871  df-minusg 15872  df-sbg 15873  df-mulg 15874  df-subg 16012  df-ghm 16079  df-cntz 16169  df-cmn 16615  df-abl 16616  df-mgp 16956  df-ur 16968  df-rng 17014  df-subrg 17239  df-lmod 17326  df-lss 17391  df-sra 17630  df-rgmod 17631  df-ascl 17774  df-psr 17816  df-mvr 17817  df-mpl 17818  df-opsr 17820  df-psr1 18030  df-vr1 18031  df-ply1 18032  df-coe1 18033  df-dsmm 18570  df-frlm 18585  df-mamu 18693  df-mat 18717  df-decpmat 19071

This theorem is referenced by:  idpm2idmp  19109