Theorem pmatcollpwscmatlem1 20594

Description: Lemma 1 for pmatcollpwscmat 20596. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)

Hypotheses

Ref	Expression
pmatcollpwscmat.p	|- P = (Poly1 ` R )
pmatcollpwscmat.c	|- C = ( N Mat P )
pmatcollpwscmat.b	|- B = ( Base ` C )
pmatcollpwscmat.m1	|- .* = ( .s ` C )
pmatcollpwscmat.e1	|- .^ = (.g ` (mulGrp ` P ) )
pmatcollpwscmat.x	|- X = (var1 ` R )
pmatcollpwscmat.t	|- T = ( N matToPolyMat R )
pmatcollpwscmat.a	|- A = ( N Mat R )
pmatcollpwscmat.d	|- D = ( Base ` A )
pmatcollpwscmat.u	|- U = (algSc ` P )
pmatcollpwscmat.k	|- K = ( Base ` R )
pmatcollpwscmat.e2	|- E = ( Base ` P )
pmatcollpwscmat.s	|- S = (algSc ` P )
pmatcollpwscmat.1	|- .1. = ( 1r ` C )
pmatcollpwscmat.m2	|- M = ( Q .* .1. )

Assertion

Ref	Expression
pmatcollpwscmatlem1	|- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( ( (coe1 ` ( a M b ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = if ( a = b , ( U ` ( (coe1 ` Q ) ` L ) ) , ( 0g ` P ) ) )

Proof of Theorem pmatcollpwscmatlem1

Step	Hyp	Ref	Expression
1		pmatcollpwscmat.m2	. . . . . . . 8 |- M = ( Q .* .1. )
2	1	oveqi 6663	. . . . . . 7 |- ( a M b ) = ( a ( Q .* .1. ) b )
3		pmatcollpwscmat.p	. . . . . . . . . . . 12 |- P = (Poly1 ` R )
4	3	ply1ring 19618	. . . . . . . . . . 11 |- ( R e. Ring -> P e. Ring )
5	4	anim2i 593	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring ) -> ( N e. Fin /\ P e. Ring ) )
6		simpr 477	. . . . . . . . . 10 |- ( ( L e. NN0 /\ Q e. E ) -> Q e. E )
7	5, 6	anim12i 590	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( N e. Fin /\ P e. Ring ) /\ Q e. E ) )
8		df-3an 1039	. . . . . . . . 9 |- ( ( N e. Fin /\ P e. Ring /\ Q e. E ) <-> ( ( N e. Fin /\ P e. Ring ) /\ Q e. E ) )
9	7, 8	sylibr 224	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( N e. Fin /\ P e. Ring /\ Q e. E ) )
10		pmatcollpwscmat.c	. . . . . . . . 9 |- C = ( N Mat P )
11		pmatcollpwscmat.e2	. . . . . . . . 9 |- E = ( Base ` P )
12		eqid 2622	. . . . . . . . 9 |- ( 0g ` P ) = ( 0g ` P )
13		pmatcollpwscmat.1	. . . . . . . . 9 |- .1. = ( 1r ` C )
14		pmatcollpwscmat.m1	. . . . . . . . 9 |- .* = ( .s ` C )
15	10, 11, 12, 13, 14	scmatscmide 20313	. . . . . . . 8 |- ( ( ( N e. Fin /\ P e. Ring /\ Q e. E ) /\ ( a e. N /\ b e. N ) ) -> ( a ( Q .* .1. ) b ) = if ( a = b , Q , ( 0g ` P ) ) )
16	9, 15	sylan 488	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( a ( Q .* .1. ) b ) = if ( a = b , Q , ( 0g ` P ) ) )
17	2, 16	syl5eq 2668	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( a M b ) = if ( a = b , Q , ( 0g ` P ) ) )
18	17	fveq2d 6195	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> (coe1 ` ( a M b ) ) = (coe1 ` if ( a = b , Q , ( 0g ` P ) ) ) )
19	18	fveq1d 6193	. . . 4 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( (coe1 ` ( a M b ) ) ` L ) = ( (coe1 ` if ( a = b , Q , ( 0g ` P ) ) ) ` L ) )
20		fvif 6204	. . . . . 6 |- (coe1 ` if ( a = b , Q , ( 0g ` P ) ) ) = if ( a = b , (coe1 ` Q ) , (coe1 ` ( 0g ` P ) ) )
21	20	fveq1i 6192	. . . . 5 |- ( (coe1 ` if ( a = b , Q , ( 0g ` P ) ) ) ` L ) = ( if ( a = b , (coe1 ` Q ) , (coe1 ` ( 0g ` P ) ) ) ` L )
22		iffv 6205	. . . . 5 |- ( if ( a = b , (coe1 ` Q ) , (coe1 ` ( 0g ` P ) ) ) ` L ) = if ( a = b , ( (coe1 ` Q ) ` L ) , ( (coe1 ` ( 0g ` P ) ) ` L ) )
23	21, 22	eqtri 2644	. . . 4 |- ( (coe1 ` if ( a = b , Q , ( 0g ` P ) ) ) ` L ) = if ( a = b , ( (coe1 ` Q ) ` L ) , ( (coe1 ` ( 0g ` P ) ) ` L ) )
24	19, 23	syl6eq 2672	. . 3 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( (coe1 ` ( a M b ) ) ` L ) = if ( a = b , ( (coe1 ` Q ) ` L ) , ( (coe1 ` ( 0g ` P ) ) ` L ) ) )
25	24	oveq1d 6665	. 2 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( ( (coe1 ` ( a M b ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( if ( a = b , ( (coe1 ` Q ) ` L ) , ( (coe1 ` ( 0g ` P ) ) ` L ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
26		ovif 6737	. . 3 |- ( if ( a = b , ( (coe1 ` Q ) ` L ) , ( (coe1 ` ( 0g ` P ) ) ` L ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = if ( a = b , ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) , ( ( (coe1 ` ( 0g ` P ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
27		eqid 2622	. . . . . . . . . . 11 |- ( 0g ` R ) = ( 0g ` R )
28	3, 12, 27	coe1z 19633	. . . . . . . . . 10 |- ( R e. Ring -> (coe1 ` ( 0g ` P ) ) = ( NN0 X. { ( 0g ` R ) } ) )
29	28	ad2antlr 763	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> (coe1 ` ( 0g ` P ) ) = ( NN0 X. { ( 0g ` R ) } ) )
30	29	fveq1d 6193	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( (coe1 ` ( 0g ` P ) ) ` L ) = ( ( NN0 X. { ( 0g ` R ) } ) ` L ) )
31		fvexd 6203	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring ) -> ( 0g ` R ) e. _V )
32		simpl 473	. . . . . . . . . 10 |- ( ( L e. NN0 /\ Q e. E ) -> L e. NN0 )
33	31, 32	anim12i 590	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( 0g ` R ) e. _V /\ L e. NN0 ) )
34		fvconst2g 6467	. . . . . . . . 9 |- ( ( ( 0g ` R ) e. _V /\ L e. NN0 ) -> ( ( NN0 X. { ( 0g ` R ) } ) ` L ) = ( 0g ` R ) )
35	33, 34	syl 17	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( NN0 X. { ( 0g ` R ) } ) ` L ) = ( 0g ` R ) )
36	30, 35	eqtrd 2656	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( (coe1 ` ( 0g ` P ) ) ` L ) = ( 0g ` R ) )
37	36	oveq1d 6665	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( (coe1 ` ( 0g ` P ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( ( 0g ` R ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
38	3	ply1lmod 19622	. . . . . . . . 9 |- ( R e. Ring -> P e. LMod )
39	38	ad2antlr 763	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> P e. LMod )
40		eqid 2622	. . . . . . . . . . . 12 |- (mulGrp ` P ) = (mulGrp ` P )
41	40	ringmgp 18553	. . . . . . . . . . 11 |- ( P e. Ring -> (mulGrp ` P ) e. Mnd )
42	4, 41	syl 17	. . . . . . . . . 10 |- ( R e. Ring -> (mulGrp ` P ) e. Mnd )
43		0nn0 11307	. . . . . . . . . . 11 |- 0 e. NN0
44	43	a1i 11	. . . . . . . . . 10 |- ( R e. Ring -> 0 e. NN0 )
45		eqid 2622	. . . . . . . . . . 11 |- (var1 ` R ) = (var1 ` R )
46	45, 3, 11	vr1cl 19587	. . . . . . . . . 10 |- ( R e. Ring -> (var1 ` R ) e. E )
47	40, 11	mgpbas 18495	. . . . . . . . . . 11 |- E = ( Base ` (mulGrp ` P ) )
48		eqid 2622	. . . . . . . . . . 11 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
49	47, 48	mulgnn0cl 17558	. . . . . . . . . 10 |- ( ( (mulGrp ` P ) e. Mnd /\ 0 e. NN0 /\ (var1 ` R ) e. E ) -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. E )
50	42, 44, 46, 49	syl3anc 1326	. . . . . . . . 9 |- ( R e. Ring -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. E )
51	50	ad2antlr 763	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. E )
52		eqid 2622	. . . . . . . . 9 |- (Scalar ` P ) = (Scalar ` P )
53		eqid 2622	. . . . . . . . 9 |- ( .s ` P ) = ( .s ` P )
54		eqid 2622	. . . . . . . . 9 |- ( 0g ` (Scalar ` P ) ) = ( 0g ` (Scalar ` P ) )
55	11, 52, 53, 54, 12	lmod0vs 18896	. . . . . . . 8 |- ( ( P e. LMod /\ ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. E ) -> ( ( 0g ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( 0g ` P ) )
56	39, 51, 55	syl2anc 693	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( 0g ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( 0g ` P ) )
57	3	ply1sca 19623	. . . . . . . . . . . 12 |- ( R e. Ring -> R = (Scalar ` P ) )
58	57	adantl 482	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring ) -> R = (Scalar ` P ) )
59	58	fveq2d 6195	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring ) -> ( 0g ` R ) = ( 0g ` (Scalar ` P ) ) )
60	59	oveq1d 6665	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring ) -> ( ( 0g ` R ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( ( 0g ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
61	60	eqeq1d 2624	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring ) -> ( ( ( 0g ` R ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( 0g ` P ) <-> ( ( 0g ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( 0g ` P ) ) )
62	61	adantr 481	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( ( 0g ` R ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( 0g ` P ) <-> ( ( 0g ` (Scalar ` P ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( 0g ` P ) ) )
63	56, 62	mpbird 247	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( 0g ` R ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( 0g ` P ) )
64	37, 63	eqtrd 2656	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( (coe1 ` ( 0g ` P ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( 0g ` P ) )
65	64	ifeq2d 4105	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> if ( a = b , ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) , ( ( (coe1 ` ( 0g ` P ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = if ( a = b , ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) , ( 0g ` P ) ) )
66	65	adantr 481	. . 3 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> if ( a = b , ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) , ( ( (coe1 ` ( 0g ` P ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = if ( a = b , ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) , ( 0g ` P ) ) )
67	26, 66	syl5eq 2668	. 2 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( if ( a = b , ( (coe1 ` Q ) ` L ) , ( (coe1 ` ( 0g ` P ) ) ` L ) ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = if ( a = b , ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) , ( 0g ` P ) ) )
68		simpr 477	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( L e. NN0 /\ Q e. E ) )
69	68	ancomd 467	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( Q e. E /\ L e. NN0 ) )
70		eqid 2622	. . . . . . . . 9 |- (coe1 ` Q ) = (coe1 ` Q )
71		pmatcollpwscmat.k	. . . . . . . . 9 |- K = ( Base ` R )
72	70, 11, 3, 71	coe1fvalcl 19582	. . . . . . . 8 |- ( ( Q e. E /\ L e. NN0 ) -> ( (coe1 ` Q ) ` L ) e. K )
73	69, 72	syl 17	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( (coe1 ` Q ) ` L ) e. K )
74	57	eqcomd 2628	. . . . . . . . . . . 12 |- ( R e. Ring -> (Scalar ` P ) = R )
75	74	adantl 482	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring ) -> (Scalar ` P ) = R )
76	75	fveq2d 6195	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring ) -> ( Base ` (Scalar ` P ) ) = ( Base ` R ) )
77	76, 71	syl6eqr 2674	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring ) -> ( Base ` (Scalar ` P ) ) = K )
78	77	eleq2d 2687	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring ) -> ( ( (coe1 ` Q ) ` L ) e. ( Base ` (Scalar ` P ) ) <-> ( (coe1 ` Q ) ` L ) e. K ) )
79	78	adantr 481	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( (coe1 ` Q ) ` L ) e. ( Base ` (Scalar ` P ) ) <-> ( (coe1 ` Q ) ` L ) e. K ) )
80	73, 79	mpbird 247	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( (coe1 ` Q ) ` L ) e. ( Base ` (Scalar ` P ) ) )
81		pmatcollpwscmat.u	. . . . . . 7 |- U = (algSc ` P )
82		eqid 2622	. . . . . . 7 |- ( Base ` (Scalar ` P ) ) = ( Base ` (Scalar ` P ) )
83		eqid 2622	. . . . . . 7 |- ( 1r ` P ) = ( 1r ` P )
84	81, 52, 82, 53, 83	asclval 19335	. . . . . 6 |- ( ( (coe1 ` Q ) ` L ) e. ( Base ` (Scalar ` P ) ) -> ( U ` ( (coe1 ` Q ) ` L ) ) = ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 1r ` P ) ) )
85	80, 84	syl 17	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( U ` ( (coe1 ` Q ) ` L ) ) = ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 1r ` P ) ) )
86	3, 45, 40, 48	ply1idvr1 19663	. . . . . . . 8 |- ( R e. Ring -> ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) = ( 1r ` P ) )
87	86	eqcomd 2628	. . . . . . 7 |- ( R e. Ring -> ( 1r ` P ) = ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) )
88	87	ad2antlr 763	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( 1r ` P ) = ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) )
89	88	oveq2d 6666	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 1r ` P ) ) = ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
90	85, 89	eqtr2d 2657	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( U ` ( (coe1 ` Q ) ` L ) ) )
91	90	ifeq1d 4104	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> if ( a = b , ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) , ( 0g ` P ) ) = if ( a = b , ( U ` ( (coe1 ` Q ) ` L ) ) , ( 0g ` P ) ) )
92	91	adantr 481	. 2 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> if ( a = b , ( ( (coe1 ` Q ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) , ( 0g ` P ) ) = if ( a = b , ( U ` ( (coe1 ` Q ) ` L ) ) , ( 0g ` P ) ) )
93	25, 67, 92	3eqtrd 2660	1 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( ( (coe1 ` ( a M b ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = if ( a = b , ( U ` ( (coe1 ` Q ) ` L ) ) , ( 0g ` P ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   {csn 4177   X. cxp 5112   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   NN0cn0 11292   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   Mndcmnd 17294  ._gcmg 17540  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547   LModclmod 18863  algSccascl 19311  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548   Mat cmat 20213   matToPolyMat cmat2pmat 20509

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  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-pm 7860  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-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-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  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-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  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-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-dsmm 20076  df-frlm 20091  df-mamu 20190  df-mat 20214

This theorem is referenced by:  pmatcollpwscmatlem2  20595