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Theorem pmatcollpwscmatlem2 20414
 Description: Lemma 2 for pmatcollpwscmat 20415. (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 472 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
2 simpr 476 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
32adantr 480 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑅 ∈ Ring)
4 simpr 476 . . . . . . . 8 ((𝐿 ∈ ℕ0𝑄𝐸) → 𝑄𝐸)
54anim2i 591 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑄𝐸))
6 df-3an 1033 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑄𝐸))
75, 6sylibr 223 . . . . . 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 20326 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) → (𝑄 1 ) ∈ 𝐵)
168, 15syl5eqel 2692 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) → 𝑀𝐵)
177, 16syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑀𝐵)
18 simprl 790 . . . . 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 20391 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0) → (𝑀 decompPMat 𝐿) ∈ 𝐷)
223, 17, 18, 21syl3anc 1318 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑀 decompPMat 𝐿) ∈ 𝐷)
23 df-3an 1033 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷))
241, 22, 23sylanbrc 695 . . 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 2610 . . . 4 (algSc‘𝑃) = (algSc‘𝑃)
2725, 19, 20, 9, 26mat2pmatval 20348 . . 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 1235 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0))
30293ad2ant1 1075 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0))
31 3simpc 1053 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
329, 10, 11decpmate 20390 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝐿)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝐿))
3330, 31, 32syl2anc 691 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑀 decompPMat 𝐿)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝐿))
3433fveq2d 6107 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗)) = ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)))
3534mpt2eq3dva 6617 . 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 1118 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
37 simp2 1055 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
38 simp3 1056 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
39173ad2ant1 1075 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑀𝐵)
4010, 12, 11, 37, 38, 39matecld 20051 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ 𝐸)
41183ad2ant1 1075 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝐿 ∈ ℕ0)
42 eqid 2610 . . . . . . 7 (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
43 pmatcollpwscmat.k . . . . . . 7 𝐾 = (Base‘𝑅)
4442, 12, 9, 43coe1fvalcl 19403 . . . . . 6 (((𝑖𝑀𝑗) ∈ 𝐸𝐿 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾)
4540, 41, 44syl2anc 691 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾)
46 eqid 2610 . . . . . 6 (var1𝑅) = (var1𝑅)
47 eqid 2610 . . . . . 6 ( ·𝑠𝑃) = ( ·𝑠𝑃)
48 eqid 2610 . . . . . 6 (mulGrp‘𝑃) = (mulGrp‘𝑃)
49 eqid 2610 . . . . . 6 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
5043, 9, 46, 47, 48, 49, 26ply1scltm 19472 . . . . 5 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾) → ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)) = (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
5136, 45, 50syl2anc 691 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)) = (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
5251mpt2eq3dva 6617 . . 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 20413 . . . . . 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 2611 . . . . . . 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 6558 . . . . . . . . . . 11 ((𝑖 = 𝑎𝑗 = 𝑏) → (𝑖𝑀𝑗) = (𝑎𝑀𝑏))
6059fveq2d 6107 . . . . . . . . . 10 ((𝑖 = 𝑎𝑗 = 𝑏) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑎𝑀𝑏)))
6160fveq1d 6105 . . . . . . . . 9 ((𝑖 = 𝑎𝑗 = 𝑏) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) = ((coe1‘(𝑎𝑀𝑏))‘𝐿))
6261oveq1d 6564 . . . . . . . 8 ((𝑖 = 𝑎𝑗 = 𝑏) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
6362adantl 481 . . . . . . 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 790 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑎𝑁)
65 simprr 792 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑏𝑁)
66 ovex 6577 . . . . . . . 8 (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ V
6766a1i 11 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ V)
6858, 63, 64, 65, 67ovmpt2d 6686 . . . . . 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))))
69 simpll 786 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑁 ∈ Fin)
709ply1ring 19439 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
7170adantl 481 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
7271adantr 480 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑃 ∈ Ring)
73 pm3.22 464 . . . . . . . . . . 11 ((𝐿 ∈ ℕ0𝑄𝐸) → (𝑄𝐸𝐿 ∈ ℕ0))
7473adantl 481 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑄𝐸𝐿 ∈ ℕ0))
75 eqid 2610 . . . . . . . . . . 11 (coe1𝑄) = (coe1𝑄)
7675, 12, 9, 43coe1fvalcl 19403 . . . . . . . . . 10 ((𝑄𝐸𝐿 ∈ ℕ0) → ((coe1𝑄)‘𝐿) ∈ 𝐾)
7774, 76syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((coe1𝑄)‘𝐿) ∈ 𝐾)
789, 55, 43, 12ply1sclcl 19477 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1𝑄)‘𝐿) ∈ 𝐾) → (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸)
793, 77, 78syl2anc 691 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸)
8069, 72, 793jca 1235 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸))
81 eqid 2610 . . . . . . . 8 (0g𝑃) = (0g𝑃)
8210, 12, 81, 14, 13scmatscmide 20132 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))
8380, 82sylan 487 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))
8457, 68, 833eqtr4d 2654 . . . . 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))
8584ralrimivva 2954 . . . 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))
86 0nn0 11184 . . . . . . . 8 0 ∈ ℕ0
8786a1i 11 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 0 ∈ ℕ0)
8843, 9, 46, 47, 48, 49, 12ply1tmcl 19463 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾 ∧ 0 ∈ ℕ0) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ 𝐸)
8936, 45, 87, 88syl3anc 1318 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ 𝐸)
9010, 12, 11, 69, 72, 89matbas2d 20048 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) ∈ 𝐵)
919, 10, 11, 12, 13, 141pmatscmul 20326 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸) → ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ∈ 𝐵)
9269, 3, 79, 91syl3anc 1318 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ∈ 𝐵)
9310, 11eqmat 20049 . . . . 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)))
9490, 92, 93syl2anc 691 . . . 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)))
9585, 94mpbird 246 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))
9652, 95eqtrd 2644 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿))) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))
9728, 35, 963eqtrd 2648 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ifcif 4036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  0cc0 9815  ℕ0cn0 11169  Basecbs 15695   ·𝑠 cvsca 15772  0gc0g 15923  .gcmg 17363  mulGrpcmgp 18312  1rcur 18324  Ringcrg 18370  algSccascl 19132  var1cv1 19367  Poly1cpl1 19368  coe1cco1 19369   Mat cmat 20032   matToPolyMat cmat2pmat 20328   decompPMat cdecpmat 20386 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-subrg 18601  df-lmod 18688  df-lss 18754  df-sra 18993  df-rgmod 18994  df-ascl 19135  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-vr1 19372  df-ply1 19373  df-coe1 19374  df-dsmm 19895  df-frlm 19910  df-mamu 20009  df-mat 20033  df-mat2pmat 20331  df-decpmat 20387 This theorem is referenced by:  pmatcollpwscmat  20415
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