Theorem m2cpminvid2lem 20559

Description: Lemma for m2cpminvid2 20560. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.)

Hypotheses

Ref	Expression
m2cpminvid2lem.s	|- S = ( N ConstPolyMat R )
m2cpminvid2lem.p	|- P = (Poly1 ` R )

Assertion

Ref	Expression
m2cpminvid2lem	|- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> A. n e. NN0 ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) )

Distinct variable groups:   n, M   n, N   P, n   R, n   S, n   x, n   y, n

Allowed substitution hints:   P( x, y)   R( x, y)   S( x, y)   M( x, y)   N( x, y)

Proof of Theorem m2cpminvid2lem

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

Step	Hyp	Ref	Expression
1		m2cpminvid2lem.s	. . . . . . . 8 |- S = ( N ConstPolyMat R )
2		m2cpminvid2lem.p	. . . . . . . 8 |- P = (Poly1 ` R )
3		eqid 2622	. . . . . . . 8 |- ( N Mat P ) = ( N Mat P )
4		eqid 2622	. . . . . . . 8 |- ( Base ` ( N Mat P ) ) = ( Base ` ( N Mat P ) )
5	1, 2, 3, 4	cpmatelimp 20517	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> ( M e. S -> ( M e. ( Base ` ( N Mat P ) ) /\ A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) ) ) )
6	5	3impia 1261	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( M e. ( Base ` ( N Mat P ) ) /\ A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) ) )
7		simpr 477	. . . . . 6 |- ( ( M e. ( Base ` ( N Mat P ) ) /\ A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) ) -> A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) )
8	6, 7	syl 17	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) )
9	8	adantr 481	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) )
10		oveq1 6657	. . . . . . . . . . 11 |- ( i = x -> ( i M j ) = ( x M j ) )
11	10	fveq2d 6195	. . . . . . . . . 10 |- ( i = x -> (coe1 ` ( i M j ) ) = (coe1 ` ( x M j ) ) )
12	11	fveq1d 6193	. . . . . . . . 9 |- ( i = x -> ( (coe1 ` ( i M j ) ) ` k ) = ( (coe1 ` ( x M j ) ) ` k ) )
13	12	eqeq1d 2624	. . . . . . . 8 |- ( i = x -> ( ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) <-> ( (coe1 ` ( x M j ) ) ` k ) = ( 0g ` R ) ) )
14	13	ralbidv 2986	. . . . . . 7 |- ( i = x -> ( A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) <-> A. k e. NN ( (coe1 ` ( x M j ) ) ` k ) = ( 0g ` R ) ) )
15		oveq2 6658	. . . . . . . . . . 11 |- ( j = y -> ( x M j ) = ( x M y ) )
16	15	fveq2d 6195	. . . . . . . . . 10 |- ( j = y -> (coe1 ` ( x M j ) ) = (coe1 ` ( x M y ) ) )
17	16	fveq1d 6193	. . . . . . . . 9 |- ( j = y -> ( (coe1 ` ( x M j ) ) ` k ) = ( (coe1 ` ( x M y ) ) ` k ) )
18	17	eqeq1d 2624	. . . . . . . 8 |- ( j = y -> ( ( (coe1 ` ( x M j ) ) ` k ) = ( 0g ` R ) <-> ( (coe1 ` ( x M y ) ) ` k ) = ( 0g ` R ) ) )
19	18	ralbidv 2986	. . . . . . 7 |- ( j = y -> ( A. k e. NN ( (coe1 ` ( x M j ) ) ` k ) = ( 0g ` R ) <-> A. k e. NN ( (coe1 ` ( x M y ) ) ` k ) = ( 0g ` R ) ) )
20	14, 19	rspc2v 3322	. . . . . 6 |- ( ( x e. N /\ y e. N ) -> ( A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) -> A. k e. NN ( (coe1 ` ( x M y ) ) ` k ) = ( 0g ` R ) ) )
21	20	adantl 482	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) -> A. k e. NN ( (coe1 ` ( x M y ) ) ` k ) = ( 0g ` R ) ) )
22		fveq2 6191	. . . . . . . 8 |- ( k = n -> ( (coe1 ` ( x M y ) ) ` k ) = ( (coe1 ` ( x M y ) ) ` n ) )
23	22	eqeq1d 2624	. . . . . . 7 |- ( k = n -> ( ( (coe1 ` ( x M y ) ) ` k ) = ( 0g ` R ) <-> ( (coe1 ` ( x M y ) ) ` n ) = ( 0g ` R ) ) )
24	23	cbvralv 3171	. . . . . 6 |- ( A. k e. NN ( (coe1 ` ( x M y ) ) ` k ) = ( 0g ` R ) <-> A. n e. NN ( (coe1 ` ( x M y ) ) ` n ) = ( 0g ` R ) )
25		simpl2 1065	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> R e. Ring )
26		eqid 2622	. . . . . . . . . . . . . . . . . . 19 |- ( Base ` P ) = ( Base ` P )
27		simprl 794	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> x e. N )
28		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x e. N /\ y e. N ) -> y e. N )
29	28	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> y e. N )
30	1, 2, 3, 4	cpmatpmat 20515	. . . . . . . . . . . . . . . . . . . 20 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> M e. ( Base ` ( N Mat P ) ) )
31	30	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> M e. ( Base ` ( N Mat P ) ) )
32	3, 26, 4, 27, 29, 31	matecld 20232	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( x M y ) e. ( Base ` P ) )
33		0nn0 11307	. . . . . . . . . . . . . . . . . 18 |- 0 e. NN0
34		eqid 2622	. . . . . . . . . . . . . . . . . . 19 |- (coe1 ` ( x M y ) ) = (coe1 ` ( x M y ) )
35		eqid 2622	. . . . . . . . . . . . . . . . . . 19 |- ( Base ` R ) = ( Base ` R )
36	34, 26, 2, 35	coe1fvalcl 19582	. . . . . . . . . . . . . . . . . 18 |- ( ( ( x M y ) e. ( Base ` P ) /\ 0 e. NN0 ) -> ( (coe1 ` ( x M y ) ) ` 0 ) e. ( Base ` R ) )
37	32, 33, 36	sylancl 694	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( (coe1 ` ( x M y ) ) ` 0 ) e. ( Base ` R ) )
38	25, 37	jca 554	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( R e. Ring /\ ( (coe1 ` ( x M y ) ) ` 0 ) e. ( Base ` R ) ) )
39	38	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) -> ( R e. Ring /\ ( (coe1 ` ( x M y ) ) ` 0 ) e. ( Base ` R ) ) )
40		eqid 2622	. . . . . . . . . . . . . . . 16 |- (algSc ` P ) = (algSc ` P )
41		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( 0g ` R ) = ( 0g ` R )
42	2, 40, 35, 41	coe1scl 19657	. . . . . . . . . . . . . . 15 |- ( ( R e. Ring /\ ( (coe1 ` ( x M y ) ) ` 0 ) e. ( Base ` R ) ) -> (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) = ( l e. NN0 |-> if ( l = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) ) )
43	39, 42	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) -> (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) = ( l e. NN0 |-> if ( l = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) ) )
44	43	fveq1d 6193	. . . . . . . . . . . . 13 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) -> ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( ( l e. NN0 |-> if ( l = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) ) ` n ) )
45		eqidd 2623	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) -> ( l e. NN0 |-> if ( l = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) ) = ( l e. NN0 |-> if ( l = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) ) )
46		eqeq1 2626	. . . . . . . . . . . . . . . 16 |- ( l = n -> ( l = 0 <-> n = 0 ) )
47	46	ifbid 4108	. . . . . . . . . . . . . . 15 |- ( l = n -> if ( l = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) = if ( n = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) )
48	47	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) /\ l = n ) -> if ( l = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) = if ( n = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) )
49		nnnn0 11299	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n e. NN0 )
50	49	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) -> n e. NN0 )
51		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( (coe1 ` ( x M y ) ) ` 0 ) e. _V
52		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( 0g ` R ) e. _V
53	51, 52	ifex 4156	. . . . . . . . . . . . . . 15 |- if ( n = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) e. _V
54	53	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) -> if ( n = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) e. _V )
55	45, 48, 50, 54	fvmptd 6288	. . . . . . . . . . . . 13 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) -> ( ( l e. NN0 |-> if ( l = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) ) ` n ) = if ( n = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) )
56		nnne0 11053	. . . . . . . . . . . . . . . 16 |- ( n e. NN -> n =/= 0 )
57	56	neneqd 2799	. . . . . . . . . . . . . . 15 |- ( n e. NN -> -. n = 0 )
58	57	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) -> -. n = 0 )
59	58	iffalsed 4097	. . . . . . . . . . . . 13 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) -> if ( n = 0 , ( (coe1 ` ( x M y ) ) ` 0 ) , ( 0g ` R ) ) = ( 0g ` R ) )
60	44, 55, 59	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) -> ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( 0g ` R ) )
61		eqcom 2629	. . . . . . . . . . . . 13 |- ( ( (coe1 ` ( x M y ) ) ` n ) = ( 0g ` R ) <-> ( 0g ` R ) = ( (coe1 ` ( x M y ) ) ` n ) )
62	61	biimpi 206	. . . . . . . . . . . 12 |- ( ( (coe1 ` ( x M y ) ) ` n ) = ( 0g ` R ) -> ( 0g ` R ) = ( (coe1 ` ( x M y ) ) ` n ) )
63	60, 62	sylan9eq 2676	. . . . . . . . . . 11 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) /\ ( (coe1 ` ( x M y ) ) ` n ) = ( 0g ` R ) ) -> ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) )
64	63	ex 450	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ n e. NN ) -> ( ( (coe1 ` ( x M y ) ) ` n ) = ( 0g ` R ) -> ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) ) )
65	64	ralimdva 2962	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( A. n e. NN ( (coe1 ` ( x M y ) ) ` n ) = ( 0g ` R ) -> A. n e. NN ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) ) )
66	65	imp 445	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ A. n e. NN ( (coe1 ` ( x M y ) ) ` n ) = ( 0g ` R ) ) -> A. n e. NN ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) )
67	38	adantr 481	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ A. n e. NN ( (coe1 ` ( x M y ) ) ` n ) = ( 0g ` R ) ) -> ( R e. Ring /\ ( (coe1 ` ( x M y ) ) ` 0 ) e. ( Base ` R ) ) )
68	2, 40, 35	ply1sclid 19658	. . . . . . . . . 10 |- ( ( R e. Ring /\ ( (coe1 ` ( x M y ) ) ` 0 ) e. ( Base ` R ) ) -> ( (coe1 ` ( x M y ) ) ` 0 ) = ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` 0 ) )
69	68	eqcomd 2628	. . . . . . . . 9 |- ( ( R e. Ring /\ ( (coe1 ` ( x M y ) ) ` 0 ) e. ( Base ` R ) ) -> ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` 0 ) = ( (coe1 ` ( x M y ) ) ` 0 ) )
70	67, 69	syl 17	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ A. n e. NN ( (coe1 ` ( x M y ) ) ` n ) = ( 0g ` R ) ) -> ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` 0 ) = ( (coe1 ` ( x M y ) ) ` 0 ) )
71	66, 70	jca 554	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) /\ A. n e. NN ( (coe1 ` ( x M y ) ) ` n ) = ( 0g ` R ) ) -> ( A. n e. NN ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) /\ ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` 0 ) = ( (coe1 ` ( x M y ) ) ` 0 ) ) )
72	71	ex 450	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( A. n e. NN ( (coe1 ` ( x M y ) ) ` n ) = ( 0g ` R ) -> ( A. n e. NN ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) /\ ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` 0 ) = ( (coe1 ` ( x M y ) ) ` 0 ) ) ) )
73	24, 72	syl5bi 232	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( A. k e. NN ( (coe1 ` ( x M y ) ) ` k ) = ( 0g ` R ) -> ( A. n e. NN ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) /\ ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` 0 ) = ( (coe1 ` ( x M y ) ) ` 0 ) ) ) )
74	21, 73	syld 47	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) -> ( A. n e. NN ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) /\ ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` 0 ) = ( (coe1 ` ( x M y ) ) ` 0 ) ) ) )
75	9, 74	mpd 15	. . 3 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( A. n e. NN ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) /\ ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` 0 ) = ( (coe1 ` ( x M y ) ) ` 0 ) ) )
76		c0ex 10034	. . . 4 |- 0 e. _V
77		fveq2 6191	. . . . . 6 |- ( n = 0 -> ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` 0 ) )
78		fveq2 6191	. . . . . 6 |- ( n = 0 -> ( (coe1 ` ( x M y ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` 0 ) )
79	77, 78	eqeq12d 2637	. . . . 5 |- ( n = 0 -> ( ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) <-> ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` 0 ) = ( (coe1 ` ( x M y ) ) ` 0 ) ) )
80	79	ralunsn 4422	. . . 4 |- ( 0 e. _V -> ( A. n e. ( NN u. { 0 } ) ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) <-> ( A. n e. NN ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) /\ ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` 0 ) = ( (coe1 ` ( x M y ) ) ` 0 ) ) ) )
81	76, 80	mp1i 13	. . 3 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( A. n e. ( NN u. { 0 } ) ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) <-> ( A. n e. NN ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) /\ ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` 0 ) = ( (coe1 ` ( x M y ) ) ` 0 ) ) ) )
82	75, 81	mpbird 247	. 2 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> A. n e. ( NN u. { 0 } ) ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) )
83		df-n0 11293	. . 3 |- NN0 = ( NN u. { 0 } )
84	83	raleqi 3142	. 2 |- ( A. n e. NN0 ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) <-> A. n e. ( NN u. { 0 } ) ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) )
85	82, 84	sylibr 224	1 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> A. n e. NN0 ( (coe1 ` ( (algSc ` P ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   ifcif 4086   {csn 4177   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   NNcn 11020   NN0cn0 11292   Basecbs 15857   0gc0g 16100   Ringcrg 18547  algSccascl 19311  Poly₁cpl1 19547  coe₁cco1 19548   Mat cmat 20213   ConstPolyMat ccpmat 20508

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-mat 20214  df-cpmat 20511

This theorem is referenced by:  m2cpminvid2  20560