Theorem pmatcollpwscmatlem1 22976

Description: Lemma 1 for pmatcollpwscmat 22978. (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 7429	. . . . . . 7 ⊢ (𝑎𝑀𝑏) = (𝑎(𝑄 ∗ 1 )𝑏)
3		pmatcollpwscmat.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
4	3	ply1ring 22437	. . . . . . . . . . 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 2765	. . . . . . . . 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 22694	. . . . . . . 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 2812	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎𝑀𝑏) = if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))
18	17	fveq2d 6889	. . . . 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 6887	. . . 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 6901	. . . . . 6 ⊢ (coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃))) = if(𝑎 = 𝑏, (coe1‘𝑄), (coe1‘(0g‘𝑃)))
21	20	fveq1i 6886	. . . . 5 ⊢ ((coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))‘𝐿) = (if(𝑎 = 𝑏, (coe1‘𝑄), (coe1‘(0g‘𝑃)))‘𝐿)
22		iffv 6902	. . . . 5 ⊢ (if(𝑎 = 𝑏, (coe1‘𝑄), (coe1‘(0g‘𝑃)))‘𝐿) = if(𝑎 = 𝑏, ((coe1‘𝑄)‘𝐿), ((coe1‘(0g‘𝑃))‘𝐿))
23	21, 22	eqtri 2788	. . . 4 ⊢ ((coe1‘if(𝑎 = 𝑏, 𝑄, (0g‘𝑃)))‘𝐿) = if(𝑎 = 𝑏, ((coe1‘𝑄)‘𝐿), ((coe1‘(0g‘𝑃))‘𝐿))
24	19, 23	eqtrdi 2816	. . 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 7431	. 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 7514	. . 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 2765	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
28	3, 12, 27	coe1z 22454	. . . . . . . . . 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 6887	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘(0g‘𝑃))‘𝐿) = ((ℕ0 × {(0g‘𝑅)})‘𝐿))
31		fvexd 6900	. . . . . . . . . 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 7204	. . . . . . . . 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 2800	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘(0g‘𝑃))‘𝐿) = (0g‘𝑅))
37	36	oveq1d 7431	. . . . . 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 22441	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
39	38	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → 𝑃 ∈ LMod)
40		eqid 2765	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
41	40, 11	mgpbas 20245	. . . . . . . . . 10 ⊢ 𝐸 = (Base‘(mulGrp‘𝑃))
42		eqid 2765	. . . . . . . . . 10 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
43	40	ringmgp 20345	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
44	4, 43	syl 18	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
45		0nn0 12530	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
46	45	a1i 11	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 0 ∈ ℕ0)
47		eqid 2765	. . . . . . . . . . 11 ⊢ (var1‘𝑅) = (var1‘𝑅)
48	47, 3, 11	vr1cl 22407	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (var1‘𝑅) ∈ 𝐸)
49	41, 42, 44, 46, 48	mulgnn0cld 19185	. . . . . . . . 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 2765	. . . . . . . . 9 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
52		eqid 2765	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
53		eqid 2765	. . . . . . . . 9 ⊢ (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
54	11, 51, 52, 53, 12	lmod0vs 21046	. . . . . . . 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 22442	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
57	56	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃))
58	57	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
59	58	oveq1d 7431	. . . . . . . . 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 2767	. . . . . . . 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 2800	. . . . 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 4510	. . . 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 2812	. 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 2765	. . . . . . . . 9 ⊢ (coe1‘𝑄) = (coe1‘𝑄)
70		pmatcollpwscmat.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑅)
71	69, 11, 3, 70	coe1fvalcl 22402	. . . . . . . 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 2771	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
74	73	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝑃) = 𝑅)
75	74	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
76	75, 70	eqtr4di 2818	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑃)) = 𝐾)
77	76	eleq2d 2851	. . . . . . . 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 2765	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
82		eqid 2765	. . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃)
83	80, 51, 81, 52, 82	asclval 22059	. . . . . 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 22485	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) = (1r‘𝑃))
86	85	eqcomd 2771	. . . . . . 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 7432	. . . . 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 2801	. . . 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 4509	. . 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 2804	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 2146  Vcvv 3457  ifcif 4489  {csn 4591   × cxp 5661  ‘cfv 6540  (class class class)co 7416  Fincfn 8945  0cc0 11111  ℕ₀cn0 12515  Basecbs 17287  Scalarcsca 17331   ·_𝑠 cvsca 17332  0_gc0g 17510  Mndcmnd 18814  ._gcmg 19157  mulGrpcmgp 20240  1_rcur 20287  Ringcrg 20339  LModclmod 21011  algSccascl 22032  var₁cv1 22366  Poly₁cpl1 22367  coe₁cco1 22368   Mat cmat 22594   matToPolyMat cmat2pmat 22891

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  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 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7680  df-ofr 7681  df-om 7865  df-1st 7988  df-2nd 7989  df-supp 8159  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-pm 8829  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9325  df-sup 9405  df-oi 9475  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-3 12315  df-4 12316  df-5 12317  df-6 12318  df-7 12319  df-8 12320  df-9 12321  df-n0 12516  df-z 12603  df-dec 12724  df-uz 12875  df-fz 13548  df-fzo 13696  df-seq 14052  df-hash 14381  df-struct 17225  df-sets 17242  df-slot 17260  df-ndx 17272  df-base 17288  df-ress 17309  df-plusg 17341  df-mulr 17342  df-sca 17344  df-vsca 17345  df-ip 17346  df-tset 17347  df-ple 17348  df-ds 17350  df-hom 17352  df-cco 17353  df-0g 17512  df-gsum 17513  df-prds 17518  df-pws 17520  df-mre 17656  df-mrc 17657  df-acs 17659  df-mgm 18716  df-sgrp 18799  df-mnd 18815  df-mhm 18865  df-submnd 18866  df-grp 19027  df-minusg 19028  df-sbg 19029  df-mulg 19158  df-subg 19213  df-ghm 19308  df-cntz 19411  df-cmn 19876  df-abl 19877  df-mgp 20241  df-rng 20255  df-ur 20288  df-ring 20341  df-subrng 20675  df-subrg 20699  df-lmod 21013  df-lss 21083  df-sra 21324  df-rgmod 21325  df-dsmm 21912  df-frlm 21927  df-ascl 22035  df-psr 22089  df-mvr 22090  df-mpl 22091  df-opsr 22093  df-psr1 22370  df-vr1 22371  df-ply1 22372  df-coe1 22373  df-mamu 22578  df-mat 22595

This theorem is used by:  pmatcollpwscmatlem2  22977