Theorem pmatcollpwscmatlem1 22837

Description: Lemma 1 for pmatcollpwscmat 22839. (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 7404	. . . . . . 7 ⊢ (𝑎𝑀𝑏) = (𝑎(𝑄 ∗ 1 )𝑏)
3		pmatcollpwscmat.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
4	3	ply1ring 22297	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5	4	anim2i 626	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
6		simpr 488	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸) → 𝑄 ∈ 𝐸)
7	5, 6	anim12i 622	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ 𝑄 ∈ 𝐸))
8		df-3an 1099	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ 𝑄 ∈ 𝐸) ↔ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ 𝑄 ∈ 𝐸))
9	7, 8	sylibr 236	. . . . . . . 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 22555	. . . . . . . 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 589	. . . . . . 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 6866	. . . . 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 6864	. . . 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 6878	. . . . . 6 ⊢ (coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃))) = if(𝑎 = 𝑏, (coe1‘𝑄), (coe1‘(0g‘𝑃)))
21	20	fveq1i 6863	. . . . 5 ⊢ ((coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))‘𝐿) = (if(𝑎 = 𝑏, (coe1‘𝑄), (coe1‘(0g‘𝑃)))‘𝐿)
22		iffv 6879	. . . . 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 7406	. 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 7489	. . 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 22314	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
29	28	ad2antlr 737	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
30	29	fveq1d 6864	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘(0g‘𝑃))‘𝐿) = ((ℕ0 × {(0g‘𝑅)})‘𝐿))
31		fvexd 6877	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝑅) ∈ V)
32		simpl 486	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸) → 𝐿 ∈ ℕ0)
33	31, 32	anim12i 622	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((0g‘𝑅) ∈ V ∧ 𝐿 ∈ ℕ0))
34		fvconst2g 7181	. . . . . . . . 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 2796	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘(0g‘𝑃))‘𝐿) = (0g‘𝑅))
37	36	oveq1d 7406	. . . . . 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 22301	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
39	38	ad2antlr 737	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → 𝑃 ∈ LMod)
40		eqid 2761	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
41	40, 11	mgpbas 20182	. . . . . . . . . 10 ⊢ 𝐸 = (Base‘(mulGrp‘𝑃))
42		eqid 2761	. . . . . . . . . 10 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
43	40	ringmgp 20276	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
44	4, 43	syl 17	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
45		0nn0 12490	. . . . . . . . . . 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 22267	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (var1‘𝑅) ∈ 𝐸)
49	41, 42, 44, 46, 48	mulgnn0cld 19128	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ 𝐸)
50	49	ad2antlr 737	. . . . . . . 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 20950	. . . . . . . 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 593	. . . . . . 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 22302	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
57	56	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃))
58	57	fveq2d 6866	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
59	58	oveq1d 7406	. . . . . . . . 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 484	. . . . . . 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 259	. . . . . 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 4498	. . . 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 484	. . 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 488	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸))
68	67	ancomd 465	. . . . . . . 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 22262	. . . . . . . 8 ⊢ ((𝑄 ∈ 𝐸 ∧ 𝐿 ∈ ℕ0) → ((coe1‘𝑄)‘𝐿) ∈ 𝐾)
72	68, 71	syl 17	. . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝑃) = 𝑅)
75	74	fveq2d 6866	. . . . . . . . . 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 484	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (((coe1‘𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)) ↔ ((coe1‘𝑄)‘𝐿) ∈ 𝐾))
79	72, 78	mpbird 259	. . . . . 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 21919	. . . . . 6 ⊢ (((coe1‘𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)) → (𝑈‘((coe1‘𝑄)‘𝐿)) = (((coe1‘𝑄)‘𝐿)( ·𝑠 ‘𝑃)(1r‘𝑃)))
84	79, 83	syl 17	. . . . 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 22345	. . . . . . . 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 737	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (1r‘𝑃) = (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))
88	87	oveq2d 7407	. . . . 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 4497	. . 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 484	. 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)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ifcif 4477  {csn 4579   × cxp 5641  ‘cfv 6516  (class class class)co 7391  Fincfn 8921  0cc0 11067  ℕ₀cn0 12475  Basecbs 17236  Scalarcsca 17280   ·_𝑠 cvsca 17281  0_gc0g 17459  Mndcmnd 18759  ._gcmg 19100  mulGrpcmgp 20177  1_rcur 20218  Ringcrg 20270  LModclmod 20915  algSccascl 21892  var₁cv1 22226  Poly₁cpl1 22227  coe₁cco1 22228   Mat cmat 22455   matToPolyMat cmat2pmat 22752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-isom 6525  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-of 7655  df-ofr 7656  df-om 7842  df-1st 7965  df-2nd 7966  df-supp 8135  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-2o 8432  df-er 8672  df-map 8804  df-pm 8805  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9302  df-sup 9382  df-oi 9452  df-card 9891  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12476  df-z 12563  df-dec 12683  df-uz 12834  df-fz 13507  df-fzo 13654  df-seq 14009  df-hash 14338  df-struct 17174  df-sets 17191  df-slot 17209  df-ndx 17221  df-base 17237  df-ress 17258  df-plusg 17290  df-mulr 17291  df-sca 17293  df-vsca 17294  df-ip 17295  df-tset 17296  df-ple 17297  df-ds 17299  df-hom 17301  df-cco 17302  df-0g 17461  df-gsum 17462  df-prds 17467  df-pws 17469  df-mre 17605  df-mrc 17606  df-acs 17608  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18969  df-minusg 18970  df-sbg 18971  df-mulg 19101  df-subg 19156  df-ghm 19245  df-cntz 19348  df-cmn 19813  df-abl 19814  df-mgp 20178  df-rng 20190  df-ur 20219  df-ring 20272  df-subrng 20583  df-subrg 20607  df-lmod 20917  df-lss 20987  df-sra 21228  df-rgmod 21229  df-dsmm 21772  df-frlm 21787  df-ascl 21895  df-psr 21949  df-mvr 21950  df-mpl 21951  df-opsr 21953  df-psr1 22230  df-vr1 22231  df-ply1 22232  df-coe1 22233  df-mamu 22439  df-mat 22456

This theorem is referenced by:  pmatcollpwscmatlem2  22838