Theorem pmatcollpwscmatlem2 22706

Description: Lemma 2 for pmatcollpwscmat 22707. (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 482	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
2		simpr 484	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
3	2	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → 𝑅 ∈ Ring)
4		simpr 484	. . . . . . . 8 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸) → 𝑄 ∈ 𝐸)
5	4	anim2i 617	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑄 ∈ 𝐸))
6		df-3an 1088	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑄 ∈ 𝐸))
7	5, 6	sylibr 234	. . . . . 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 22618	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → (𝑄 ∗ 1 ) ∈ 𝐵)
16	8, 15	eqeltrid 2837	. . . . . 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 22683	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝐿 ∈ ℕ0) → (𝑀 decompPMat 𝐿) ∈ 𝐷)
22	3, 17, 18, 21	syl3anc 1373	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑀 decompPMat 𝐿) ∈ 𝐷)
23		df-3an 1088	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷))
24	1, 22, 23	sylanbrc 583	. . 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 2733	. . . 4 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
27	25, 19, 20, 9, 26	mat2pmatval 22640	. . 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 1128	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝐿 ∈ ℕ0))
30	29	3ad2ant1 1133	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝐿 ∈ ℕ0))
31		3simpc 1150	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
32	9, 10, 11	decpmate 22682	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝐿 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑀 decompPMat 𝐿)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝐿))
33	30, 31, 32	syl2anc 584	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑀 decompPMat 𝐿)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝐿))
34	33	fveq2d 6832	. . 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 7429	. 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 1238	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
37		simp2 1137	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
38		simp3 1138	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
39	17	3ad2ant1 1133	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ 𝐵)
40	10, 12, 11, 37, 38, 39	matecld 22342	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ 𝐸)
41	18	3ad2ant1 1133	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐿 ∈ ℕ0)
42		eqid 2733	. . . . . . 7 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
43		pmatcollpwscmat.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝑅)
44	42, 12, 9, 43	coe1fvalcl 22126	. . . . . 6 ⊢ (((𝑖𝑀𝑗) ∈ 𝐸 ∧ 𝐿 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾)
45	40, 41, 44	syl2anc 584	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾)
46		eqid 2733	. . . . . 6 ⊢ (var1‘𝑅) = (var1‘𝑅)
47		eqid 2733	. . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
48		eqid 2733	. . . . . 6 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
49		eqid 2733	. . . . . 6 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
50	43, 9, 46, 47, 48, 49, 26	ply1scltm 22196	. . . . 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 584	. . . 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 7429	. . 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 22705	. . . . . 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 2734	. . . . . . 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 7361	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → (𝑖𝑀𝑗) = (𝑎𝑀𝑏))
60	59	fveq2d 6832	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑎𝑀𝑏)))
61	60	fveq1d 6830	. . . . . . . . 9 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) = ((coe1‘(𝑎𝑀𝑏))‘𝐿))
62	61	oveq1d 7367	. . . . . . . 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 481	. . . . . . 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 7387	. . . . . . 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 7504	. . . . . 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 22161	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
70	69	adantl 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
71	70	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → 𝑃 ∈ Ring)
72		pm3.22 459	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸) → (𝑄 ∈ 𝐸 ∧ 𝐿 ∈ ℕ0))
73	72	adantl 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑄 ∈ 𝐸 ∧ 𝐿 ∈ ℕ0))
74		eqid 2733	. . . . . . . . . . 11 ⊢ (coe1‘𝑄) = (coe1‘𝑄)
75	74, 12, 9, 43	coe1fvalcl 22126	. . . . . . . . . 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 22201	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝑄)‘𝐿) ∈ 𝐾) → (𝑈‘((coe1‘𝑄)‘𝐿)) ∈ 𝐸)
78	3, 76, 77	syl2anc 584	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑈‘((coe1‘𝑄)‘𝐿)) ∈ 𝐸)
79	68, 71, 78	3jca 1128	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ (𝑈‘((coe1‘𝑄)‘𝐿)) ∈ 𝐸))
80		eqid 2733	. . . . . . . 8 ⊢ (0g‘𝑃) = (0g‘𝑃)
81	10, 12, 80, 14, 13	scmatscmide 22423	. . . . . . 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 580	. . . . . 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 2778	. . . . 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 3176	. . . 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 12403	. . . . . . . 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 22187	. . . . . . 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 1373	. . . . . 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 22339	. . . . 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 22618	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑈‘((coe1‘𝑄)‘𝐿)) ∈ 𝐸) → ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ) ∈ 𝐵)
91	68, 3, 78, 90	syl3anc 1373	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ) ∈ 𝐵)
92	10, 11	eqmat 22340	. . . . 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 584	. . . 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 257	. . 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 2768	. 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 2772	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  Vcvv 3437  ifcif 4474  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354  Fincfn 8875  0cc0 11013  ℕ₀cn0 12388  Basecbs 17122   ·_𝑠 cvsca 17167  0_gc0g 17345  ._gcmg 18982  mulGrpcmgp 20060  1_rcur 20101  Ringcrg 20153  algSccascl 21791  var₁cv1 22089  Poly₁cpl1 22090  coe₁cco1 22091   Mat cmat 22323   matToPolyMat cmat2pmat 22620   decompPMat cdecpmat 22678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-ot 4584  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-ofr 7617  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-map 8758  df-pm 8759  df-ixp 8828  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fsupp 9253  df-sup 9333  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-z 12476  df-dec 12595  df-uz 12739  df-fz 13410  df-fzo 13557  df-seq 13911  df-hash 14240  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-mulr 17177  df-sca 17179  df-vsca 17180  df-ip 17181  df-tset 17182  df-ple 17183  df-ds 17185  df-hom 17187  df-cco 17188  df-0g 17347  df-gsum 17348  df-prds 17353  df-pws 17355  df-mre 17490  df-mrc 17491  df-acs 17493  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-mhm 18693  df-submnd 18694  df-grp 18851  df-minusg 18852  df-sbg 18853  df-mulg 18983  df-subg 19038  df-ghm 19127  df-cntz 19231  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-ring 20155  df-subrng 20463  df-subrg 20487  df-lmod 20797  df-lss 20867  df-sra 21109  df-rgmod 21110  df-dsmm 21671  df-frlm 21686  df-ascl 21794  df-psr 21848  df-mvr 21849  df-mpl 21850  df-opsr 21852  df-psr1 22093  df-vr1 22094  df-ply1 22095  df-coe1 22096  df-mamu 22307  df-mat 22324  df-mat2pmat 22623  df-decpmat 22679

This theorem is referenced by:  pmatcollpwscmat  22707