Theorem pmatcollpwscmatlem2 21395

Description: Lemma 2 for pmatcollpwscmat 21396. (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
pmatcollpwscmatlem2	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ))

Proof of Theorem pmatcollpwscmatlem2

Dummy variables 𝑎 𝑏 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 486	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
2		simpr 488	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
3	2	adantr 484	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → 𝑅 ∈ Ring)
4		simpr 488	. . . . . . . 8 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸) → 𝑄 ∈ 𝐸)
5	4	anim2i 619	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑄 ∈ 𝐸))
6		df-3an 1086	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑄 ∈ 𝐸))
7	5, 6	sylibr 237	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸))
8		pmatcollpwscmat.m2	. . . . . . 7 ⊢ 𝑀 = (𝑄 ∗ 1 )
9		pmatcollpwscmat.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
10		pmatcollpwscmat.c	. . . . . . . 8 ⊢ 𝐶 = (𝑁 Mat 𝑃)
11		pmatcollpwscmat.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
12		pmatcollpwscmat.e2	. . . . . . . 8 ⊢ 𝐸 = (Base‘𝑃)
13		pmatcollpwscmat.m1	. . . . . . . 8 ⊢ ∗ = ( ·𝑠 ‘𝐶)
14		pmatcollpwscmat.1	. . . . . . . 8 ⊢ 1 = (1r‘𝐶)
15	9, 10, 11, 12, 13, 14	1pmatscmul 21307	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → (𝑄 ∗ 1 ) ∈ 𝐵)
16	8, 15	eqeltrid 2894	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → 𝑀 ∈ 𝐵)
17	7, 16	syl 17	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → 𝑀 ∈ 𝐵)
18		simprl 770	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → 𝐿 ∈ ℕ0)
19		pmatcollpwscmat.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
20		pmatcollpwscmat.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐴)
21	9, 10, 11, 19, 20	decpmatcl 21372	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝐿 ∈ ℕ0) → (𝑀 decompPMat 𝐿) ∈ 𝐷)
22	3, 17, 18, 21	syl3anc 1368	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑀 decompPMat 𝐿) ∈ 𝐷)
23		df-3an 1086	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷))
24	1, 22, 23	sylanbrc 586	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷))
25		pmatcollpwscmat.t	. . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
26		eqid 2798	. . . 4 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
27	25, 19, 20, 9, 26	mat2pmatval 21329	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷) → (𝑇‘(𝑀 decompPMat 𝐿)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗))))
28	24, 27	syl 17	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗))))
29	3, 17, 18	3jca 1125	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝐿 ∈ ℕ0))
30	29	3ad2ant1 1130	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝐿 ∈ ℕ0))
31		3simpc 1147	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
32	9, 10, 11	decpmate 21371	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝐿 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑀 decompPMat 𝐿)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝐿))
33	30, 31, 32	syl2anc 587	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑀 decompPMat 𝐿)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝐿))
34	33	fveq2d 6649	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗)) = ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)))
35	34	mpoeq3dva 7210	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿))))
36		simp1lr 1234	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
37		simp2 1134	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
38		simp3 1135	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
39	17	3ad2ant1 1130	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ 𝐵)
40	10, 12, 11, 37, 38, 39	matecld 21031	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ 𝐸)
41	18	3ad2ant1 1130	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐿 ∈ ℕ0)
42		eqid 2798	. . . . . . 7 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
43		pmatcollpwscmat.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝑅)
44	42, 12, 9, 43	coe1fvalcl 20841	. . . . . 6 ⊢ (((𝑖𝑀𝑗) ∈ 𝐸 ∧ 𝐿 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾)
45	40, 41, 44	syl2anc 587	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾)
46		eqid 2798	. . . . . 6 ⊢ (var1‘𝑅) = (var1‘𝑅)
47		eqid 2798	. . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
48		eqid 2798	. . . . . 6 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
49		eqid 2798	. . . . . 6 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
50	43, 9, 46, 47, 48, 49, 26	ply1scltm 20910	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾) → ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)) = (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))
51	36, 45, 50	syl2anc 587	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)) = (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))
52	51	mpoeq3dva 7210	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))))
53		pmatcollpwscmat.e1	. . . . . . 7 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
54		pmatcollpwscmat.x	. . . . . . 7 ⊢ 𝑋 = (var1‘𝑅)
55		pmatcollpwscmat.u	. . . . . . 7 ⊢ 𝑈 = (algSc‘𝑃)
56		pmatcollpwscmat.s	. . . . . . 7 ⊢ 𝑆 = (algSc‘𝑃)
57	9, 10, 11, 13, 53, 54, 25, 19, 20, 55, 43, 12, 56, 14, 8	pmatcollpwscmatlem1 21394	. . . . . 6 ⊢ ((((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)))
58		eqidd 2799	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))))
59		oveq12 7144	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → (𝑖𝑀𝑗) = (𝑎𝑀𝑏))
60	59	fveq2d 6649	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑎𝑀𝑏)))
61	60	fveq1d 6647	. . . . . . . . 9 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) = ((coe1‘(𝑎𝑀𝑏))‘𝐿))
62	61	oveq1d 7150	. . . . . . . 8 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))
63	62	adantl 485	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ (𝑖 = 𝑎 ∧ 𝑗 = 𝑏)) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))
64		simprl 770	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑎 ∈ 𝑁)
65		simprr 772	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑏 ∈ 𝑁)
66		ovexd 7170	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) ∈ V)
67	58, 63, 64, 65, 66	ovmpod 7281	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))𝑏) = (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))
68		simpll 766	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → 𝑁 ∈ Fin)
69	9	ply1ring 20877	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
70	69	adantl 485	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
71	70	adantr 484	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → 𝑃 ∈ Ring)
72		pm3.22 463	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸) → (𝑄 ∈ 𝐸 ∧ 𝐿 ∈ ℕ0))
73	72	adantl 485	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑄 ∈ 𝐸 ∧ 𝐿 ∈ ℕ0))
74		eqid 2798	. . . . . . . . . . 11 ⊢ (coe1‘𝑄) = (coe1‘𝑄)
75	74, 12, 9, 43	coe1fvalcl 20841	. . . . . . . . . 10 ⊢ ((𝑄 ∈ 𝐸 ∧ 𝐿 ∈ ℕ0) → ((coe1‘𝑄)‘𝐿) ∈ 𝐾)
76	73, 75	syl 17	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((coe1‘𝑄)‘𝐿) ∈ 𝐾)
77	9, 55, 43, 12	ply1sclcl 20915	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝑄)‘𝐿) ∈ 𝐾) → (𝑈‘((coe1‘𝑄)‘𝐿)) ∈ 𝐸)
78	3, 76, 77	syl2anc 587	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑈‘((coe1‘𝑄)‘𝐿)) ∈ 𝐸)
79	68, 71, 78	3jca 1125	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ (𝑈‘((coe1‘𝑄)‘𝐿)) ∈ 𝐸))
80		eqid 2798	. . . . . . . 8 ⊢ (0g‘𝑃) = (0g‘𝑃)
81	10, 12, 80, 14, 13	scmatscmide 21112	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ (𝑈‘((coe1‘𝑄)‘𝐿)) ∈ 𝐸) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 )𝑏) = if(𝑎 = 𝑏, (𝑈‘((coe1‘𝑄)‘𝐿)), (0g‘𝑃)))
82	79, 81	sylan 583	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 )𝑏) = if(𝑎 = 𝑏, (𝑈‘((coe1‘𝑄)‘𝐿)), (0g‘𝑃)))
83	57, 67, 82	3eqtr4d 2843	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))𝑏) = (𝑎((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 )𝑏))
84	83	ralrimivva 3156	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))𝑏) = (𝑎((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 )𝑏))
85		0nn0 11900	. . . . . . . 8 ⊢ 0 ∈ ℕ0
86	85	a1i 11	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 0 ∈ ℕ0)
87	43, 9, 46, 47, 48, 49, 12	ply1tmcl 20901	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾 ∧ 0 ∈ ℕ0) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) ∈ 𝐸)
88	36, 45, 86, 87	syl3anc 1368	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) ∈ 𝐸)
89	10, 12, 11, 68, 71, 88	matbas2d 21028	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) ∈ 𝐵)
90	9, 10, 11, 12, 13, 14	1pmatscmul 21307	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑈‘((coe1‘𝑄)‘𝐿)) ∈ 𝐸) → ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ) ∈ 𝐵)
91	68, 3, 78, 90	syl3anc 1368	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ) ∈ 𝐵)
92	10, 11	eqmat 21029	. . . . 5 ⊢ (((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) ∈ 𝐵 ∧ ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ) ∈ 𝐵) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) = ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ) ↔ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))𝑏) = (𝑎((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 )𝑏)))
93	89, 91, 92	syl2anc 587	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) = ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ) ↔ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))𝑏) = (𝑎((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 )𝑏)))
94	84, 93	mpbird 260	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) = ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ))
95	52, 94	eqtrd 2833	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿))) = ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ))
96	28, 35, 95	3eqtrd 2837	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  ifcif 4425  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  Fincfn 8492  0cc0 10526  ℕ₀cn0 11885  Basecbs 16475   ·_𝑠 cvsca 16561  0_gc0g 16705  ._gcmg 18216  mulGrpcmgp 19232  1_rcur 19244  Ringcrg 19290  algSccascl 20541  var₁cv1 20805  Poly₁cpl1 20806  coe₁cco1 20807   Mat cmat 21012   matToPolyMat cmat2pmat 21309   decompPMat cdecpmat 21367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-ot 4534  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-ofr 7390  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-sup 8890  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-hom 16581  df-cco 16582  df-0g 16707  df-gsum 16708  df-prds 16713  df-pws 16715  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-mulg 18217  df-subg 18268  df-ghm 18348  df-cntz 18439  df-cmn 18900  df-abl 18901  df-mgp 19233  df-ur 19245  df-ring 19292  df-subrg 19526  df-lmod 19629  df-lss 19697  df-sra 19937  df-rgmod 19938  df-dsmm 20421  df-frlm 20436  df-ascl 20544  df-psr 20594  df-mvr 20595  df-mpl 20596  df-opsr 20598  df-psr1 20809  df-vr1 20810  df-ply1 20811  df-coe1 20812  df-mamu 20991  df-mat 21013  df-mat2pmat 21312  df-decpmat 21368

This theorem is referenced by:  pmatcollpwscmat  21396