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Theorem pmatcollpwscmatlem2 21392
 Description: Lemma 2 for pmatcollpwscmat 21393. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
Hypotheses
Ref Expression
pmatcollpwscmat.p 𝑃 = (Poly1𝑅)
pmatcollpwscmat.c 𝐶 = (𝑁 Mat 𝑃)
pmatcollpwscmat.b 𝐵 = (Base‘𝐶)
pmatcollpwscmat.m1 = ( ·𝑠𝐶)
pmatcollpwscmat.e1 = (.g‘(mulGrp‘𝑃))
pmatcollpwscmat.x 𝑋 = (var1𝑅)
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 = (1r𝐶)
pmatcollpwscmat.m2 𝑀 = (𝑄 1 )
Assertion
Ref Expression
pmatcollpwscmatlem2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))

Proof of Theorem pmatcollpwscmatlem2
Dummy variables 𝑎 𝑏 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 485 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
2 simpr 487 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
32adantr 483 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑅 ∈ Ring)
4 simpr 487 . . . . . . . 8 ((𝐿 ∈ ℕ0𝑄𝐸) → 𝑄𝐸)
54anim2i 618 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑄𝐸))
6 df-3an 1085 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑄𝐸))
75, 6sylibr 236 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸))
8 pmatcollpwscmat.m2 . . . . . . 7 𝑀 = (𝑄 1 )
9 pmatcollpwscmat.p . . . . . . . 8 𝑃 = (Poly1𝑅)
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 = ( ·𝑠𝐶)
14 pmatcollpwscmat.1 . . . . . . . 8 1 = (1r𝐶)
159, 10, 11, 12, 13, 141pmatscmul 21304 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) → (𝑄 1 ) ∈ 𝐵)
168, 15eqeltrid 2917 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) → 𝑀𝐵)
177, 16syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑀𝐵)
18 simprl 769 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝐿 ∈ ℕ0)
19 pmatcollpwscmat.a . . . . . 6 𝐴 = (𝑁 Mat 𝑅)
20 pmatcollpwscmat.d . . . . . 6 𝐷 = (Base‘𝐴)
219, 10, 11, 19, 20decpmatcl 21369 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0) → (𝑀 decompPMat 𝐿) ∈ 𝐷)
223, 17, 18, 21syl3anc 1367 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑀 decompPMat 𝐿) ∈ 𝐷)
23 df-3an 1085 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷))
241, 22, 23sylanbrc 585 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷))
25 pmatcollpwscmat.t . . . 4 𝑇 = (𝑁 matToPolyMat 𝑅)
26 eqid 2821 . . . 4 (algSc‘𝑃) = (algSc‘𝑃)
2725, 19, 20, 9, 26mat2pmatval 21326 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷) → (𝑇‘(𝑀 decompPMat 𝐿)) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗))))
2824, 27syl 17 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗))))
293, 17, 183jca 1124 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0))
30293ad2ant1 1129 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0))
31 3simpc 1146 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
329, 10, 11decpmate 21368 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝐿)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝐿))
3330, 31, 32syl2anc 586 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑀 decompPMat 𝐿)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝐿))
3433fveq2d 6668 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗)) = ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)))
3534mpoeq3dva 7225 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿))))
36 simp1lr 1233 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
37 simp2 1133 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
38 simp3 1134 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
39173ad2ant1 1129 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑀𝐵)
4010, 12, 11, 37, 38, 39matecld 21029 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ 𝐸)
41183ad2ant1 1129 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝐿 ∈ ℕ0)
42 eqid 2821 . . . . . . 7 (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
43 pmatcollpwscmat.k . . . . . . 7 𝐾 = (Base‘𝑅)
4442, 12, 9, 43coe1fvalcl 20374 . . . . . 6 (((𝑖𝑀𝑗) ∈ 𝐸𝐿 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾)
4540, 41, 44syl2anc 586 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾)
46 eqid 2821 . . . . . 6 (var1𝑅) = (var1𝑅)
47 eqid 2821 . . . . . 6 ( ·𝑠𝑃) = ( ·𝑠𝑃)
48 eqid 2821 . . . . . 6 (mulGrp‘𝑃) = (mulGrp‘𝑃)
49 eqid 2821 . . . . . 6 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
5043, 9, 46, 47, 48, 49, 26ply1scltm 20443 . . . . 5 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾) → ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)) = (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
5136, 45, 50syl2anc 586 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)) = (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
5251mpoeq3dva 7225 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿))) = (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))))
53 pmatcollpwscmat.e1 . . . . . . 7 = (.g‘(mulGrp‘𝑃))
54 pmatcollpwscmat.x . . . . . . 7 𝑋 = (var1𝑅)
55 pmatcollpwscmat.u . . . . . . 7 𝑈 = (algSc‘𝑃)
56 pmatcollpwscmat.s . . . . . . 7 𝑆 = (algSc‘𝑃)
579, 10, 11, 13, 53, 54, 25, 19, 20, 55, 43, 12, 56, 14, 8pmatcollpwscmatlem1 21391 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))
58 eqidd 2822 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))))
59 oveq12 7159 . . . . . . . . . . 11 ((𝑖 = 𝑎𝑗 = 𝑏) → (𝑖𝑀𝑗) = (𝑎𝑀𝑏))
6059fveq2d 6668 . . . . . . . . . 10 ((𝑖 = 𝑎𝑗 = 𝑏) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑎𝑀𝑏)))
6160fveq1d 6666 . . . . . . . . 9 ((𝑖 = 𝑎𝑗 = 𝑏) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) = ((coe1‘(𝑎𝑀𝑏))‘𝐿))
6261oveq1d 7165 . . . . . . . 8 ((𝑖 = 𝑎𝑗 = 𝑏) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
6362adantl 484 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) ∧ (𝑖 = 𝑎𝑗 = 𝑏)) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
64 simprl 769 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑎𝑁)
65 simprr 771 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑏𝑁)
66 ovexd 7185 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ V)
6758, 63, 64, 65, 66ovmpod 7296 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))𝑏) = (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
68 simpll 765 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑁 ∈ Fin)
699ply1ring 20410 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
7069adantl 484 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
7170adantr 483 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑃 ∈ Ring)
72 pm3.22 462 . . . . . . . . . . 11 ((𝐿 ∈ ℕ0𝑄𝐸) → (𝑄𝐸𝐿 ∈ ℕ0))
7372adantl 484 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑄𝐸𝐿 ∈ ℕ0))
74 eqid 2821 . . . . . . . . . . 11 (coe1𝑄) = (coe1𝑄)
7574, 12, 9, 43coe1fvalcl 20374 . . . . . . . . . 10 ((𝑄𝐸𝐿 ∈ ℕ0) → ((coe1𝑄)‘𝐿) ∈ 𝐾)
7673, 75syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((coe1𝑄)‘𝐿) ∈ 𝐾)
779, 55, 43, 12ply1sclcl 20448 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1𝑄)‘𝐿) ∈ 𝐾) → (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸)
783, 76, 77syl2anc 586 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸)
7968, 71, 783jca 1124 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸))
80 eqid 2821 . . . . . . . 8 (0g𝑃) = (0g𝑃)
8110, 12, 80, 14, 13scmatscmide 21110 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))
8279, 81sylan 582 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))
8357, 67, 823eqtr4d 2866 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))𝑏) = (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏))
8483ralrimivva 3191 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ∀𝑎𝑁𝑏𝑁 (𝑎(𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))𝑏) = (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏))
85 0nn0 11906 . . . . . . . 8 0 ∈ ℕ0
8685a1i 11 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 0 ∈ ℕ0)
8743, 9, 46, 47, 48, 49, 12ply1tmcl 20434 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾 ∧ 0 ∈ ℕ0) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ 𝐸)
8836, 45, 86, 87syl3anc 1367 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ 𝐸)
8910, 12, 11, 68, 71, 88matbas2d 21026 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) ∈ 𝐵)
909, 10, 11, 12, 13, 141pmatscmul 21304 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸) → ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ∈ 𝐵)
9168, 3, 78, 90syl3anc 1367 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ∈ 𝐵)
9210, 11eqmat 21027 . . . . 5 (((𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) ∈ 𝐵 ∧ ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ∈ 𝐵) → ((𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ↔ ∀𝑎𝑁𝑏𝑁 (𝑎(𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))𝑏) = (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏)))
9389, 91, 92syl2anc 586 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ↔ ∀𝑎𝑁𝑏𝑁 (𝑎(𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))𝑏) = (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏)))
9484, 93mpbird 259 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))
9552, 94eqtrd 2856 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿))) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))
9628, 35, 953eqtrd 2860 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494  ifcif 4466  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  Fincfn 8503  0cc0 10531  ℕ0cn0 11891  Basecbs 16477   ·𝑠 cvsca 16563  0gc0g 16707  .gcmg 18218  mulGrpcmgp 19233  1rcur 19245  Ringcrg 19291  algSccascl 20078  var1cv1 20338  Poly1cpl1 20339  coe1cco1 20340   Mat cmat 21010   matToPolyMat cmat2pmat 21306   decompPMat cdecpmat 21364 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-ot 4569  df-uni 4832  df-int 4869  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-sup 8900  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12887  df-fzo 13028  df-seq 13364  df-hash 13685  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-ip 16577  df-tset 16578  df-ple 16579  df-ds 16581  df-hom 16583  df-cco 16584  df-0g 16709  df-gsum 16710  df-prds 16715  df-pws 16717  df-mre 16851  df-mrc 16852  df-acs 16854  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-mhm 17950  df-submnd 17951  df-grp 18100  df-minusg 18101  df-sbg 18102  df-mulg 18219  df-subg 18270  df-ghm 18350  df-cntz 18441  df-cmn 18902  df-abl 18903  df-mgp 19234  df-ur 19246  df-ring 19293  df-subrg 19527  df-lmod 19630  df-lss 19698  df-sra 19938  df-rgmod 19939  df-ascl 20081  df-psr 20130  df-mvr 20131  df-mpl 20132  df-opsr 20134  df-psr1 20342  df-vr1 20343  df-ply1 20344  df-coe1 20345  df-dsmm 20870  df-frlm 20885  df-mamu 20989  df-mat 21011  df-mat2pmat 21309  df-decpmat 21365 This theorem is referenced by:  pmatcollpwscmat  21393
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