Theorem pmatcollpwscmatlem1 23100

Description: Lemma 1 for pmatcollpwscmat 23102. (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 7431	. . . . . . 7 ⊢ (𝑎𝑀𝑏) = (𝑎(𝑄 ∗ 1 )𝑏)
3		pmatcollpwscmat.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
4	3	ply1ring 22558	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5	4	anim2i 629	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
6		simpr 490	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸) → 𝑄 ∈ 𝐸)
7	5, 6	anim12i 625	. . . . . . . . 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 2761	. . . . . . . . 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 22815	. . . . . . . 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 592	. . . . . . 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 2808	. . . . . 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 2784	. . . 4 ⊢ ((coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))‘𝐿) = if(𝑎 = 𝑏, ((coe1‘𝑄)‘𝐿), ((coe1‘(0g‘𝑃))‘𝐿))
24	19, 23	eqtrdi 2812	. . 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 7433	. 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 7516	. . 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 2761	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
28	3, 12, 27	coe1z 22575	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
29	28	ad2antlr 740	. . . . . . . . 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 488	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸) → 𝐿 ∈ ℕ0)
33	31, 32	anim12i 625	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((0g‘𝑅) ∈ V ∧ 𝐿 ∈ ℕ0))
34		fvconst2g 7206	. . . . . . . . 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 2796	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘(0g‘𝑃))‘𝐿) = (0g‘𝑅))
37	36	oveq1d 7433	. . . . . 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 22562	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
39	38	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → 𝑃 ∈ LMod)
40		eqid 2761	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
41	40, 11	mgpbas 20358	. . . . . . . . . 10 ⊢ 𝐸 = (Base‘(mulGrp‘𝑃))
42		eqid 2761	. . . . . . . . . 10 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
43	40	ringmgp 20458	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
44	4, 43	syl 18	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
45		0nn0 12614	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
46	45	a1i 11	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 0 ∈ ℕ0)
47		eqid 2761	. . . . . . . . . . 11 ⊢ (var1‘𝑅) = (var1‘𝑅)
48	47, 3, 11	vr1cl 22528	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (var1‘𝑅) ∈ 𝐸)
49	41, 42, 44, 46, 48	mulgnn0cld 19298	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ 𝐸)
50	49	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ 𝐸)
51		eqid 2761	. . . . . . . . 9 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
52		eqid 2761	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
53		eqid 2761	. . . . . . . . 9 ⊢ (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
54	11, 51, 52, 53, 12	lmod0vs 21163	. . . . . . . 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 596	. . . . . . 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 22563	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
57	56	adantl 487	. . . . . . . . . . 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 7433	. . . . . . . . 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 2763	. . . . . . . 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 486	. . . . . . 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 2796	. . . . 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 4503	. . . 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 486	. . 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 2808	. 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 490	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸))
68	67	ancomd 467	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑄 ∈ 𝐸 ∧ 𝐿 ∈ ℕ0))
69		eqid 2761	. . . . . . . . 9 ⊢ (coe1‘𝑄) = (coe1‘𝑄)
70		pmatcollpwscmat.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑅)
71	69, 11, 3, 70	coe1fvalcl 22523	. . . . . . . 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 2767	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
74	73	adantl 487	. . . . . . . . . . 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 2814	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑃)) = 𝐾)
77	76	eleq2d 2847	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (((coe1‘𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)) ↔ ((coe1‘𝑄)‘𝐿) ∈ 𝐾))
78	77	adantr 486	. . . . . . 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 2761	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
82		eqid 2761	. . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃)
83	80, 51, 81, 52, 82	asclval 22180	. . . . . 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 22606	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) = (1r‘𝑃))
86	85	eqcomd 2767	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑃) = (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))
87	86	ad2antlr 740	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (1r‘𝑃) = (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))
88	87	oveq2d 7434	. . . . 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 2797	. . . 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 4502	. . 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 486	. 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 2800	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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ifcif 4482  {csn 4584   × cxp 5649  ‘cfv 6537  (class class class)co 7418  Fincfn 8966  0cc0 11193  ℕ₀cn0 12599  Basecbs 17380  Scalarcsca 17424   ·_𝑠 cvsca 17425  0_gc0g 17603  Mndcmnd 18916  ._gcmg 19270  mulGrpcmgp 20353  1_rcur 20400  Ringcrg 20452  LModclmod 21128  algSccascl 22153  var₁cv1 22487  Poly₁cpl1 22488  coe₁cco1 22489   Mat cmat 22715   matToPolyMat cmat2pmat 23015

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7691  df-ofr 7692  df-om 7876  df-1st 7999  df-2nd 8000  df-supp 8171  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-map 8842  df-pm 8843  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9347  df-sup 9427  df-oi 9497  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-fzo 13782  df-seq 14138  df-hash 14468  df-struct 17318  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-mulr 17435  df-sca 17437  df-vsca 17438  df-ip 17439  df-tset 17440  df-ple 17441  df-ds 17443  df-hom 17445  df-cco 17446  df-0g 17605  df-gsum 17606  df-prds 17611  df-pws 17613  df-mre 17749  df-mrc 17750  df-acs 17752  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-mhm 18971  df-submnd 18972  df-grp 19140  df-minusg 19141  df-sbg 19142  df-mulg 19271  df-subg 19326  df-ghm 19421  df-cntz 19524  df-cmn 19989  df-abl 19990  df-mgp 20354  df-rng 20368  df-ur 20401  df-ring 20454  df-subrng 20791  df-subrg 20815  df-lmod 21130  df-lss 21200  df-sra 21441  df-rgmod 21442  df-dsmm 22031  df-frlm 22046  df-ascl 22156  df-psr 22210  df-mvr 22211  df-mpl 22212  df-opsr 22214  df-psr1 22491  df-vr1 22492  df-ply1 22493  df-coe1 22494  df-mamu 22699  df-mat 22716

This theorem is used by:  pmatcollpwscmatlem2  23101