Theorem pmatcollpwscmatlem1 21846

Description: Lemma 1 for pmatcollpwscmat 21848. (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 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘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 7268	. . . . . . 7 ⊢ (𝑎𝑀𝑏) = (𝑎(𝑄 ∗ 1 )𝑏)
3		pmatcollpwscmat.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
4	3	ply1ring 21329	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5	4	anim2i 616	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
6		simpr 484	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸) → 𝑄 ∈ 𝐸)
7	5, 6	anim12i 612	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ 𝑄 ∈ 𝐸))
8		df-3an 1087	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ 𝑄 ∈ 𝐸) ↔ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ 𝑄 ∈ 𝐸))
9	7, 8	sylibr 233	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ 𝑄 ∈ 𝐸))
10		pmatcollpwscmat.c	. . . . . . . . 9 ⊢ 𝐶 = (𝑁 Mat 𝑃)
11		pmatcollpwscmat.e2	. . . . . . . . 9 ⊢ 𝐸 = (Base‘𝑃)
12		eqid 2738	. . . . . . . . 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 21564	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ 𝑄 ∈ 𝐸) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝑄 ∗ 1 )𝑏) = if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))
16	9, 15	sylan 579	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝑄 ∗ 1 )𝑏) = if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))
17	2, 16	eqtrid 2790	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎𝑀𝑏) = if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))
18	17	fveq2d 6760	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (coe1‘(𝑎𝑀𝑏)) = (coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃))))
19	18	fveq1d 6758	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → ((coe1‘(𝑎𝑀𝑏))‘𝐿) = ((coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))‘𝐿))
20		fvif 6772	. . . . . 6 ⊢ (coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃))) = if(𝑎 = 𝑏, (coe1‘𝑄), (coe1‘(0g‘𝑃)))
21	20	fveq1i 6757	. . . . 5 ⊢ ((coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))‘𝐿) = (if(𝑎 = 𝑏, (coe1‘𝑄), (coe1‘(0g‘𝑃)))‘𝐿)
22		iffv 6773	. . . . 5 ⊢ (if(𝑎 = 𝑏, (coe1‘𝑄), (coe1‘(0g‘𝑃)))‘𝐿) = if(𝑎 = 𝑏, ((coe1‘𝑄)‘𝐿), ((coe1‘(0g‘𝑃))‘𝐿))
23	21, 22	eqtri 2766	. . . 4 ⊢ ((coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))‘𝐿) = if(𝑎 = 𝑏, ((coe1‘𝑄)‘𝐿), ((coe1‘(0g‘𝑃))‘𝐿))
24	19, 23	eqtrdi 2795	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → ((coe1‘(𝑎𝑀𝑏))‘𝐿) = if(𝑎 = 𝑏, ((coe1‘𝑄)‘𝐿), ((coe1‘(0g‘𝑃))‘𝐿)))
25	24	oveq1d 7270	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘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 7350	. . 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 2738	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
28	3, 12, 27	coe1z 21344	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
29	28	ad2antlr 723	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
30	29	fveq1d 6758	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘(0g‘𝑃))‘𝐿) = ((ℕ0 × {(0g‘𝑅)})‘𝐿))
31		fvexd 6771	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝑅) ∈ V)
32		simpl 482	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸) → 𝐿 ∈ ℕ0)
33	31, 32	anim12i 612	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((0g‘𝑅) ∈ V ∧ 𝐿 ∈ ℕ0))
34		fvconst2g 7059	. . . . . . . . 9 ⊢ (((0g‘𝑅) ∈ V ∧ 𝐿 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝐿) = (0g‘𝑅))
35	33, 34	syl 17	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((ℕ0 × {(0g‘𝑅)})‘𝐿) = (0g‘𝑅))
36	30, 35	eqtrd 2778	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘(0g‘𝑃))‘𝐿) = (0g‘𝑅))
37	36	oveq1d 7270	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 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 21333	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
39	38	ad2antlr 723	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → 𝑃 ∈ LMod)
40		eqid 2738	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
41	40	ringmgp 19704	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
42	4, 41	syl 17	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
43		0nn0 12178	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
44	43	a1i 11	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 0 ∈ ℕ0)
45		eqid 2738	. . . . . . . . . . 11 ⊢ (var1‘𝑅) = (var1‘𝑅)
46	45, 3, 11	vr1cl 21298	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (var1‘𝑅) ∈ 𝐸)
47	40, 11	mgpbas 19641	. . . . . . . . . . 11 ⊢ 𝐸 = (Base‘(mulGrp‘𝑃))
48		eqid 2738	. . . . . . . . . . 11 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
49	47, 48	mulgnn0cl 18635	. . . . . . . . . 10 ⊢ (((mulGrp‘𝑃) ∈ Mnd ∧ 0 ∈ ℕ0 ∧ (var1‘𝑅) ∈ 𝐸) → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ 𝐸)
50	42, 44, 46, 49	syl3anc 1369	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ 𝐸)
51	50	ad2antlr 723	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ 𝐸)
52		eqid 2738	. . . . . . . . 9 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
53		eqid 2738	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
54		eqid 2738	. . . . . . . . 9 ⊢ (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
55	11, 52, 53, 54, 12	lmod0vs 20071	. . . . . . . 8 ⊢ ((𝑃 ∈ LMod ∧ (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ 𝐸) → ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (0g‘𝑃))
56	39, 51, 55	syl2anc 583	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (0g‘𝑃))
57	3	ply1sca 21334	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
58	57	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃))
59	58	fveq2d 6760	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
60	59	oveq1d 7270	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 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 2740	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 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 480	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 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 256	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((0g‘𝑅)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (0g‘𝑃))
64	37, 63	eqtrd 2778	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (((coe1‘(0g‘𝑃))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (0g‘𝑃))
65	64	ifeq2d 4476	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 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 480	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 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	eqtrid 2790	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 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 484	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸))
69	68	ancomd 461	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑄 ∈ 𝐸 ∧ 𝐿 ∈ ℕ0))
70		eqid 2738	. . . . . . . . 9 ⊢ (coe1‘𝑄) = (coe1‘𝑄)
71		pmatcollpwscmat.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑅)
72	70, 11, 3, 71	coe1fvalcl 21293	. . . . . . . 8 ⊢ ((𝑄 ∈ 𝐸 ∧ 𝐿 ∈ ℕ0) → ((coe1‘𝑄)‘𝐿) ∈ 𝐾)
73	69, 72	syl 17	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘𝑄)‘𝐿) ∈ 𝐾)
74	57	eqcomd 2744	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
75	74	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝑃) = 𝑅)
76	75	fveq2d 6760	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
77	76, 71	eqtr4di 2797	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑃)) = 𝐾)
78	77	eleq2d 2824	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (((coe1‘𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)) ↔ ((coe1‘𝑄)‘𝐿) ∈ 𝐾))
79	78	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (((coe1‘𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)) ↔ ((coe1‘𝑄)‘𝐿) ∈ 𝐾))
80	73, 79	mpbird 256	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)))
81		pmatcollpwscmat.u	. . . . . . 7 ⊢ 𝑈 = (algSc‘𝑃)
82		eqid 2738	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
83		eqid 2738	. . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃)
84	81, 52, 82, 53, 83	asclval 20994	. . . . . 6 ⊢ (((coe1‘𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)) → (𝑈‘((coe1‘𝑄)‘𝐿)) = (((coe1‘𝑄)‘𝐿)( ·𝑠 ‘𝑃)(1r‘𝑃)))
85	80, 84	syl 17	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑈‘((coe1‘𝑄)‘𝐿)) = (((coe1‘𝑄)‘𝐿)( ·𝑠 ‘𝑃)(1r‘𝑃)))
86	3, 45, 40, 48	ply1idvr1 21374	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) = (1r‘𝑃))
87	86	eqcomd 2744	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑃) = (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))
88	87	ad2antlr 723	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (1r‘𝑃) = (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))
89	88	oveq2d 7271	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (((coe1‘𝑄)‘𝐿)( ·𝑠 ‘𝑃)(1r‘𝑃)) = (((coe1‘𝑄)‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))
90	85, 89	eqtr2d 2779	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (((coe1‘𝑄)‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (𝑈‘((coe1‘𝑄)‘𝐿)))
91	90	ifeq1d 4475	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 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 480	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 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 2782	1 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = if(𝑎 = 𝑏, (𝑈‘((coe1‘𝑄)‘𝐿)), (0g‘𝑃)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ifcif 4456  {csn 4558   × cxp 5578  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  0cc0 10802  ℕ₀cn0 12163  Basecbs 16840  Scalarcsca 16891   ·_𝑠 cvsca 16892  0_gc0g 17067  Mndcmnd 18300  ._gcmg 18615  mulGrpcmgp 19635  1_rcur 19652  Ringcrg 19698  LModclmod 20038  algSccascl 20969  var₁cv1 21257  Poly₁cpl1 21258  coe₁cco1 21259   Mat cmat 21464   matToPolyMat cmat2pmat 21761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-sup 9131  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-hom 16912  df-cco 16913  df-0g 17069  df-gsum 17070  df-prds 17075  df-pws 17077  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-mulg 18616  df-subg 18667  df-ghm 18747  df-cntz 18838  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-subrg 19937  df-lmod 20040  df-lss 20109  df-sra 20349  df-rgmod 20350  df-dsmm 20849  df-frlm 20864  df-ascl 20972  df-psr 21022  df-mvr 21023  df-mpl 21024  df-opsr 21026  df-psr1 21261  df-vr1 21262  df-ply1 21263  df-coe1 21264  df-mamu 21443  df-mat 21465

This theorem is referenced by:  pmatcollpwscmatlem2  21847