Theorem pmatcollpwscmatlem1 22927

Description: Lemma 1 for pmatcollpwscmat 22929. (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 7425	. . . . . . 7 ⊢ (𝑎𝑀𝑏) = (𝑎(𝑄 ∗ 1 )𝑏)
3		pmatcollpwscmat.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
4	3	ply1ring 22388	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5	4	anim2i 628	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
6		simpr 489	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸) → 𝑄 ∈ 𝐸)
7	5, 6	anim12i 624	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ 𝑄 ∈ 𝐸))
8		df-3an 1105	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ 𝑄 ∈ 𝐸) ↔ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ 𝑄 ∈ 𝐸))
9	7, 8	sylibr 237	. . . . . . . 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 2763	. . . . . . . . 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 22645	. . . . . . . 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 591	. . . . . . 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 2810	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎𝑀𝑏) = if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))
18	17	fveq2d 6887	. . . . 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 6885	. . . 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 6899	. . . . . 6 ⊢ (coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃))) = if(𝑎 = 𝑏, (coe1‘𝑄), (coe1‘(0g‘𝑃)))
21	20	fveq1i 6884	. . . . 5 ⊢ ((coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))‘𝐿) = (if(𝑎 = 𝑏, (coe1‘𝑄), (coe1‘(0g‘𝑃)))‘𝐿)
22		iffv 6900	. . . . 5 ⊢ (if(𝑎 = 𝑏, (coe1‘𝑄), (coe1‘(0g‘𝑃)))‘𝐿) = if(𝑎 = 𝑏, ((coe1‘𝑄)‘𝐿), ((coe1‘(0g‘𝑃))‘𝐿))
23	21, 22	eqtri 2786	. . . 4 ⊢ ((coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))‘𝐿) = if(𝑎 = 𝑏, ((coe1‘𝑄)‘𝐿), ((coe1‘(0g‘𝑃))‘𝐿))
24	19, 23	eqtrdi 2814	. . 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 7427	. 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 7510	. . 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 2763	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
28	3, 12, 27	coe1z 22405	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
29	28	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
30	29	fveq1d 6885	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘(0g‘𝑃))‘𝐿) = ((ℕ0 × {(0g‘𝑅)})‘𝐿))
31		fvexd 6898	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝑅) ∈ V)
32		simpl 487	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸) → 𝐿 ∈ ℕ0)
33	31, 32	anim12i 624	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((0g‘𝑅) ∈ V ∧ 𝐿 ∈ ℕ0))
34		fvconst2g 7202	. . . . . . . . 9 ⊢ (((0g‘𝑅) ∈ V ∧ 𝐿 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝐿) = (0g‘𝑅))
35	33, 34	syl 18	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((ℕ0 × {(0g‘𝑅)})‘𝐿) = (0g‘𝑅))
36	30, 35	eqtrd 2798	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘(0g‘𝑃))‘𝐿) = (0g‘𝑅))
37	36	oveq1d 7427	. . . . . 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 22392	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
39	38	ad2antlr 739	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → 𝑃 ∈ LMod)
40		eqid 2763	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
41	40, 11	mgpbas 20222	. . . . . . . . . 10 ⊢ 𝐸 = (Base‘(mulGrp‘𝑃))
42		eqid 2763	. . . . . . . . . 10 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
43	40	ringmgp 20322	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
44	4, 43	syl 18	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
45		0nn0 12520	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
46	45	a1i 11	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 0 ∈ ℕ0)
47		eqid 2763	. . . . . . . . . . 11 ⊢ (var1‘𝑅) = (var1‘𝑅)
48	47, 3, 11	vr1cl 22358	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (var1‘𝑅) ∈ 𝐸)
49	41, 42, 44, 46, 48	mulgnn0cld 19162	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ 𝐸)
50	49	ad2antlr 739	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ 𝐸)
51		eqid 2763	. . . . . . . . 9 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
52		eqid 2763	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
53		eqid 2763	. . . . . . . . 9 ⊢ (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
54	11, 51, 52, 53, 12	lmod0vs 20997	. . . . . . . 8 ⊢ ((𝑃 ∈ LMod ∧ (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ 𝐸) → ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (0g‘𝑃))
55	39, 50, 54	syl2anc 595	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (0g‘𝑃))
56	3	ply1sca 22393	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
57	56	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃))
58	57	fveq2d 6887	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
59	58	oveq1d 7427	. . . . . . . . 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))))
60	59	eqeq1d 2765	. . . . . . . 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)))
61	60	adantr 485	. . . . . . 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)))
62	55, 61	mpbird 260	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((0g‘𝑅)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (0g‘𝑃))
63	37, 62	eqtrd 2798	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (((coe1‘(0g‘𝑃))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (0g‘𝑃))
64	63	ifeq2d 4509	. . . 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)))
65	64	adantr 485	. . 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)))
66	26, 65	eqtrid 2810	. 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)))
67		simpr 489	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸))
68	67	ancomd 466	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑄 ∈ 𝐸 ∧ 𝐿 ∈ ℕ0))
69		eqid 2763	. . . . . . . . 9 ⊢ (coe1‘𝑄) = (coe1‘𝑄)
70		pmatcollpwscmat.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑅)
71	69, 11, 3, 70	coe1fvalcl 22353	. . . . . . . 8 ⊢ ((𝑄 ∈ 𝐸 ∧ 𝐿 ∈ ℕ0) → ((coe1‘𝑄)‘𝐿) ∈ 𝐾)
72	68, 71	syl 18	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘𝑄)‘𝐿) ∈ 𝐾)
73	56	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
74	73	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝑃) = 𝑅)
75	74	fveq2d 6887	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
76	75, 70	eqtr4di 2816	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑃)) = 𝐾)
77	76	eleq2d 2849	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (((coe1‘𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)) ↔ ((coe1‘𝑄)‘𝐿) ∈ 𝐾))
78	77	adantr 485	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (((coe1‘𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)) ↔ ((coe1‘𝑄)‘𝐿) ∈ 𝐾))
79	72, 78	mpbird 260	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)))
80		pmatcollpwscmat.u	. . . . . . 7 ⊢ 𝑈 = (algSc‘𝑃)
81		eqid 2763	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
82		eqid 2763	. . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃)
83	80, 51, 81, 52, 82	asclval 22010	. . . . . 6 ⊢ (((coe1‘𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)) → (𝑈‘((coe1‘𝑄)‘𝐿)) = (((coe1‘𝑄)‘𝐿)( ·𝑠 ‘𝑃)(1r‘𝑃)))
84	79, 83	syl 18	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑈‘((coe1‘𝑄)‘𝐿)) = (((coe1‘𝑄)‘𝐿)( ·𝑠 ‘𝑃)(1r‘𝑃)))
85	3, 47, 40, 42	ply1idvr1 22436	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) = (1r‘𝑃))
86	85	eqcomd 2769	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑃) = (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))
87	86	ad2antlr 739	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (1r‘𝑃) = (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))
88	87	oveq2d 7428	. . . . 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))))
89	84, 88	eqtr2d 2799	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (((coe1‘𝑄)‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (𝑈‘((coe1‘𝑄)‘𝐿)))
90	89	ifeq1d 4508	. . 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)))
91	90	adantr 485	. 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)))
92	25, 66, 91	3eqtrd 2802	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 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ifcif 4488  {csn 4590   × cxp 5661  ‘cfv 6538  (class class class)co 7412  Fincfn 8944  0cc0 11101  ℕ₀cn0 12505  Basecbs 17270  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17493  Mndcmnd 18793  ._gcmg 19134  mulGrpcmgp 20217  1_rcur 20264  Ringcrg 20316  LModclmod 20962  algSccascl 21983  var₁cv1 22317  Poly₁cpl1 22318  coe₁cco1 22319   Mat cmat 22545   matToPolyMat cmat2pmat 22842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-ofr 7677  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8158  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-pm 8828  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-fsupp 9323  df-sup 9403  df-oi 9473  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-dec 12713  df-uz 12864  df-fz 13537  df-fzo 13685  df-seq 14040  df-hash 14369  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-hom 17335  df-cco 17336  df-0g 17495  df-gsum 17496  df-prds 17501  df-pws 17503  df-mre 17639  df-mrc 17640  df-acs 17642  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-submnd 18843  df-grp 19004  df-minusg 19005  df-sbg 19006  df-mulg 19135  df-subg 19190  df-ghm 19285  df-cntz 19388  df-cmn 19853  df-abl 19854  df-mgp 20218  df-rng 20232  df-ur 20265  df-ring 20318  df-subrng 20632  df-subrg 20656  df-lmod 20964  df-lss 21034  df-sra 21275  df-rgmod 21276  df-dsmm 21863  df-frlm 21878  df-ascl 21986  df-psr 22040  df-mvr 22041  df-mpl 22042  df-opsr 22044  df-psr1 22321  df-vr1 22322  df-ply1 22323  df-coe1 22324  df-mamu 22529  df-mat 22546

This theorem is referenced by:  pmatcollpwscmatlem2  22928