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Theorem pmatcollpwscmatlem2 20523
Description: Lemma 2 for pmatcollpwscmat 20524. (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 473 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
2 simpr 477 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
32adantr 481 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑅 ∈ Ring)
4 simpr 477 . . . . . . . 8 ((𝐿 ∈ ℕ0𝑄𝐸) → 𝑄𝐸)
54anim2i 592 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑄𝐸))
6 df-3an 1038 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑄𝐸))
75, 6sylibr 224 . . . . . 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 20435 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) → (𝑄 1 ) ∈ 𝐵)
168, 15syl5eqel 2702 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) → 𝑀𝐵)
177, 16syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑀𝐵)
18 simprl 793 . . . . 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 20500 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0) → (𝑀 decompPMat 𝐿) ∈ 𝐷)
223, 17, 18, 21syl3anc 1323 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑀 decompPMat 𝐿) ∈ 𝐷)
23 df-3an 1038 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷))
241, 22, 23sylanbrc 697 . . 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 2621 . . . 4 (algSc‘𝑃) = (algSc‘𝑃)
2725, 19, 20, 9, 26mat2pmatval 20457 . . 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 1240 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0))
30293ad2ant1 1080 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0))
31 3simpc 1058 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
329, 10, 11decpmate 20499 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝐿)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝐿))
3330, 31, 32syl2anc 692 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑀 decompPMat 𝐿)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝐿))
3433fveq2d 6157 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗)) = ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)))
3534mpt2eq3dva 6679 . 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 1123 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
37 simp2 1060 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
38 simp3 1061 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
39173ad2ant1 1080 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑀𝐵)
4010, 12, 11, 37, 38, 39matecld 20160 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ 𝐸)
41183ad2ant1 1080 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝐿 ∈ ℕ0)
42 eqid 2621 . . . . . . 7 (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
43 pmatcollpwscmat.k . . . . . . 7 𝐾 = (Base‘𝑅)
4442, 12, 9, 43coe1fvalcl 19510 . . . . . 6 (((𝑖𝑀𝑗) ∈ 𝐸𝐿 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾)
4540, 41, 44syl2anc 692 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾)
46 eqid 2621 . . . . . 6 (var1𝑅) = (var1𝑅)
47 eqid 2621 . . . . . 6 ( ·𝑠𝑃) = ( ·𝑠𝑃)
48 eqid 2621 . . . . . 6 (mulGrp‘𝑃) = (mulGrp‘𝑃)
49 eqid 2621 . . . . . 6 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
5043, 9, 46, 47, 48, 49, 26ply1scltm 19579 . . . . 5 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾) → ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)) = (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
5136, 45, 50syl2anc 692 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)) = (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
5251mpt2eq3dva 6679 . . 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 20522 . . . . . 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 2622 . . . . . . 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 6619 . . . . . . . . . . 11 ((𝑖 = 𝑎𝑗 = 𝑏) → (𝑖𝑀𝑗) = (𝑎𝑀𝑏))
6059fveq2d 6157 . . . . . . . . . 10 ((𝑖 = 𝑎𝑗 = 𝑏) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑎𝑀𝑏)))
6160fveq1d 6155 . . . . . . . . 9 ((𝑖 = 𝑎𝑗 = 𝑏) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) = ((coe1‘(𝑎𝑀𝑏))‘𝐿))
6261oveq1d 6625 . . . . . . . 8 ((𝑖 = 𝑎𝑗 = 𝑏) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
6362adantl 482 . . . . . . 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 793 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑎𝑁)
65 simprr 795 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑏𝑁)
66 ovex 6638 . . . . . . . 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 6748 . . . . . 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 789 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑁 ∈ Fin)
709ply1ring 19546 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
7170adantl 482 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
7271adantr 481 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑃 ∈ Ring)
73 pm3.22 465 . . . . . . . . . . 11 ((𝐿 ∈ ℕ0𝑄𝐸) → (𝑄𝐸𝐿 ∈ ℕ0))
7473adantl 482 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑄𝐸𝐿 ∈ ℕ0))
75 eqid 2621 . . . . . . . . . . 11 (coe1𝑄) = (coe1𝑄)
7675, 12, 9, 43coe1fvalcl 19510 . . . . . . . . . 10 ((𝑄𝐸𝐿 ∈ ℕ0) → ((coe1𝑄)‘𝐿) ∈ 𝐾)
7774, 76syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((coe1𝑄)‘𝐿) ∈ 𝐾)
789, 55, 43, 12ply1sclcl 19584 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1𝑄)‘𝐿) ∈ 𝐾) → (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸)
793, 77, 78syl2anc 692 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸)
8069, 72, 793jca 1240 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸))
81 eqid 2621 . . . . . . . 8 (0g𝑃) = (0g𝑃)
8210, 12, 81, 14, 13scmatscmide 20241 . . . . . . 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 488 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))
8457, 68, 833eqtr4d 2665 . . . . 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 2966 . . . 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 11258 . . . . . . . 8 0 ∈ ℕ0
8786a1i 11 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 0 ∈ ℕ0)
8843, 9, 46, 47, 48, 49, 12ply1tmcl 19570 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾 ∧ 0 ∈ ℕ0) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ 𝐸)
8936, 45, 87, 88syl3anc 1323 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ 𝐸)
9010, 12, 11, 69, 72, 89matbas2d 20157 . . . . 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 20435 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸) → ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ∈ 𝐵)
9269, 3, 79, 91syl3anc 1323 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ∈ 𝐵)
9310, 11eqmat 20158 . . . . 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 692 . . . 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 247 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))
9652, 95eqtrd 2655 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿))) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))
9728, 35, 963eqtrd 2659 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  ifcif 4063  cfv 5852  (class class class)co 6610  cmpt2 6612  Fincfn 7906  0cc0 9887  0cn0 11243  Basecbs 15788   ·𝑠 cvsca 15873  0gc0g 16028  .gcmg 17468  mulGrpcmgp 18417  1rcur 18429  Ringcrg 18475  algSccascl 19239  var1cv1 19474  Poly1cpl1 19475  coe1cco1 19476   Mat cmat 20141   matToPolyMat cmat2pmat 20437   decompPMat cdecpmat 20495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-ofr 6858  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7860  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-fsupp 8227  df-sup 8299  df-oi 8366  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-5 11033  df-6 11034  df-7 11035  df-8 11036  df-9 11037  df-n0 11244  df-z 11329  df-dec 11445  df-uz 11639  df-fz 12276  df-fzo 12414  df-seq 12749  df-hash 13065  df-struct 15790  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-mulr 15883  df-sca 15885  df-vsca 15886  df-ip 15887  df-tset 15888  df-ple 15889  df-ds 15892  df-hom 15894  df-cco 15895  df-0g 16030  df-gsum 16031  df-prds 16036  df-pws 16038  df-mre 16174  df-mrc 16175  df-acs 16177  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-mhm 17263  df-submnd 17264  df-grp 17353  df-minusg 17354  df-sbg 17355  df-mulg 17469  df-subg 17519  df-ghm 17586  df-cntz 17678  df-cmn 18123  df-abl 18124  df-mgp 18418  df-ur 18430  df-ring 18477  df-subrg 18706  df-lmod 18793  df-lss 18861  df-sra 19100  df-rgmod 19101  df-ascl 19242  df-psr 19284  df-mvr 19285  df-mpl 19286  df-opsr 19288  df-psr1 19478  df-vr1 19479  df-ply1 19480  df-coe1 19481  df-dsmm 20004  df-frlm 20019  df-mamu 20118  df-mat 20142  df-mat2pmat 20440  df-decpmat 20496
This theorem is referenced by:  pmatcollpwscmat  20524
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