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Theorem mp2pm2mp 21512
 Description: A polynomial over matrices transformed into a polynomial matrix transformed back into the polynomial over matrices. (Contributed by AV, 12-Oct-2019.)
Hypotheses
Ref Expression
mp2pm2mp.a 𝐴 = (𝑁 Mat 𝑅)
mp2pm2mp.q 𝑄 = (Poly1𝐴)
mp2pm2mp.l 𝐿 = (Base‘𝑄)
mp2pm2mp.m · = ( ·𝑠𝑃)
mp2pm2mp.e 𝐸 = (.g‘(mulGrp‘𝑃))
mp2pm2mp.y 𝑌 = (var1𝑅)
mp2pm2mp.i 𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
mp2pm2mplem2.p 𝑃 = (Poly1𝑅)
mp2pm2mp.t 𝑇 = (𝑁 pMatToMatPoly 𝑅)
Assertion
Ref Expression
mp2pm2mp ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑇‘(𝐼𝑂)) = 𝑂)
Distinct variable groups:   𝐸,𝑝   𝐿,𝑝   𝑖,𝑁,𝑗,𝑝   𝑖,𝑂,𝑗,𝑝,𝑘   𝑃,𝑝   𝑅,𝑝   𝑌,𝑝   · ,𝑝   𝑘,𝐿   𝑃,𝑖,𝑗,𝑘   𝑅,𝑘   · ,𝑘   𝑖,𝐸,𝑗   𝑖,𝐿,𝑗   𝑘,𝑁   𝑅,𝑖,𝑗   𝑖,𝑌,𝑗   · ,𝑖,𝑗   𝐴,𝑖,𝑗,𝑘   𝑘,𝐸   𝑘,𝑌
Allowed substitution hints:   𝐴(𝑝)   𝑄(𝑖,𝑗,𝑘,𝑝)   𝑇(𝑖,𝑗,𝑘,𝑝)   𝐼(𝑖,𝑗,𝑘,𝑝)

Proof of Theorem mp2pm2mp
Dummy variables 𝑛 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mp2pm2mp.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
2 mp2pm2mp.q . . . 4 𝑄 = (Poly1𝐴)
3 mp2pm2mp.l . . . 4 𝐿 = (Base‘𝑄)
4 mp2pm2mplem2.p . . . 4 𝑃 = (Poly1𝑅)
5 mp2pm2mp.m . . . 4 · = ( ·𝑠𝑃)
6 mp2pm2mp.e . . . 4 𝐸 = (.g‘(mulGrp‘𝑃))
7 mp2pm2mp.y . . . 4 𝑌 = (var1𝑅)
8 mp2pm2mp.i . . . 4 𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
9 eqid 2759 . . . 4 (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
10 eqid 2759 . . . 4 (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10mply1topmatcl 21506 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐼𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
12 eqid 2759 . . . 4 ( ·𝑠𝑄) = ( ·𝑠𝑄)
13 eqid 2759 . . . 4 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
14 eqid 2759 . . . 4 (var1𝐴) = (var1𝐴)
15 mp2pm2mp.t . . . 4 𝑇 = (𝑁 pMatToMatPoly 𝑅)
164, 9, 10, 12, 13, 14, 1, 2, 15pm2mpfval 21497 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐼𝑂) ∈ (Base‘(𝑁 Mat 𝑃))) → (𝑇‘(𝐼𝑂)) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))
1711, 16syld3an3 1407 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑇‘(𝐼𝑂)) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))
181matring 21144 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19183adant3 1130 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝐴 ∈ Ring)
20 eqid 2759 . . . . 5 (0g𝑄) = (0g𝑄)
212ply1ring 20973 . . . . . . 7 (𝐴 ∈ Ring → 𝑄 ∈ Ring)
22 ringcmn 19403 . . . . . . 7 (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
2318, 21, 223syl 18 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
24233adant3 1130 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝑄 ∈ CMnd)
25 nn0ex 11941 . . . . . 6 0 ∈ V
2625a1i 11 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → ℕ0 ∈ V)
2719adantr 485 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
28 simpl2 1190 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
2911adantr 485 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → (𝐼𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
30 simpr 489 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
31 eqid 2759 . . . . . . . . 9 (Base‘𝐴) = (Base‘𝐴)
324, 9, 10, 1, 31decpmatcl 21468 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐼𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑛 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
3328, 29, 30, 32syl3anc 1369 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
34 eqid 2759 . . . . . . . 8 (mulGrp‘𝑄) = (mulGrp‘𝑄)
3531, 2, 14, 12, 34, 13, 3ply1tmcl 20997 . . . . . . 7 ((𝐴 ∈ Ring ∧ ((𝐼𝑂) decompPMat 𝑛) ∈ (Base‘𝐴) ∧ 𝑛 ∈ ℕ0) → (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ 𝐿)
3627, 33, 30, 35syl3anc 1369 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ 𝐿)
3736fmpttd 6871 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0𝐿)
38 fveq2 6659 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((coe1𝑝)‘𝑘) = ((coe1𝑝)‘𝑛))
3938oveqd 7168 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑖((coe1𝑝)‘𝑘)𝑗) = (𝑖((coe1𝑝)‘𝑛)𝑗))
40 oveq1 7158 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑘𝐸𝑌) = (𝑛𝐸𝑌))
4139, 40oveq12d 7169 . . . . . . . . . . . 12 (𝑘 = 𝑛 → ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))
4241cbvmptv 5136 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))
4342a1i 11 . . . . . . . . . 10 ((𝑖𝑁𝑗𝑁) → (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌))))
4443oveq2d 7167 . . . . . . . . 9 ((𝑖𝑁𝑗𝑁) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))))
4544mpoeq3ia 7227 . . . . . . . 8 (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))))
4645mpteq2i 5125 . . . . . . 7 (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌))))))
478, 46eqtri 2782 . . . . . 6 𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌))))))
481, 2, 3, 5, 6, 7, 47, 4, 12, 13, 14mp2pm2mplem5 21511 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
493, 20, 24, 26, 37, 48gsumcl 19104 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ 𝐿)
50 simp3 1136 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝑂𝐿)
5119, 49, 503jca 1126 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐴 ∈ Ring ∧ (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ 𝐿𝑂𝐿))
521, 2, 3, 5, 6, 7, 8, 4mp2pm2mplem4 21510 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝑛) = ((coe1𝑂)‘𝑛))
5352oveq1d 7166 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))) = (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))
5453adantlr 715 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))) = (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))
5554mpteq2dva 5128 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))) = (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))))
5655oveq2d 7167 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))
5756fveq2d 6663 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))))) = (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))))))
5857fveq1d 6661 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙) = ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙))
5919, 50jca 516 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐴 ∈ Ring ∧ 𝑂𝐿))
6059adantr 485 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (𝐴 ∈ Ring ∧ 𝑂𝐿))
61 eqid 2759 . . . . . . . . . 10 (coe1𝑂) = (coe1𝑂)
622, 14, 3, 12, 34, 13, 61ply1coe 21021 . . . . . . . . 9 ((𝐴 ∈ Ring ∧ 𝑂𝐿) → 𝑂 = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))
6360, 62syl 17 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → 𝑂 = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))
6463eqcomd 2765 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) = 𝑂)
6564fveq2d 6663 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))))) = (coe1𝑂))
6665fveq1d 6661 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙) = ((coe1𝑂)‘𝑙))
6758, 66eqtrd 2794 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙) = ((coe1𝑂)‘𝑙))
6867ralrimiva 3114 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → ∀𝑙 ∈ ℕ0 ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙) = ((coe1𝑂)‘𝑙))
69 eqid 2759 . . . 4 (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))))) = (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))
702, 3, 69, 61eqcoe1ply1eq 21022 . . 3 ((𝐴 ∈ Ring ∧ (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ 𝐿𝑂𝐿) → (∀𝑙 ∈ ℕ0 ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙) = ((coe1𝑂)‘𝑙) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) = 𝑂))
7151, 68, 70sylc 65 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) = 𝑂)
7217, 71eqtrd 2794 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑇‘(𝐼𝑂)) = 𝑂)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  Vcvv 3410   ↦ cmpt 5113  ‘cfv 6336  (class class class)co 7151   ∈ cmpo 7153  Fincfn 8528  ℕ0cn0 11935  Basecbs 16542   ·𝑠 cvsca 16628  0gc0g 16772   Σg cgsu 16773  .gcmg 18292  CMndccmn 18974  mulGrpcmgp 19308  Ringcrg 19366  var1cv1 20901  Poly1cpl1 20902  coe1cco1 20903   Mat cmat 21108   decompPMat cdecpmat 21463   pMatToMatPoly cpm2mp 21493 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10632  ax-resscn 10633  ax-1cn 10634  ax-icn 10635  ax-addcl 10636  ax-addrcl 10637  ax-mulcl 10638  ax-mulrcl 10639  ax-mulcom 10640  ax-addass 10641  ax-mulass 10642  ax-distr 10643  ax-i2m1 10644  ax-1ne0 10645  ax-1rid 10646  ax-rnegex 10647  ax-rrecex 10648  ax-cnre 10649  ax-pre-lttri 10650  ax-pre-lttrn 10651  ax-pre-ltadd 10652  ax-pre-mulgt0 10653 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-ot 4532  df-uni 4800  df-int 4840  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-se 5485  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-isom 6345  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7406  df-ofr 7407  df-om 7581  df-1st 7694  df-2nd 7695  df-supp 7837  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-1o 8113  df-er 8300  df-map 8419  df-pm 8420  df-ixp 8481  df-en 8529  df-dom 8530  df-sdom 8531  df-fin 8532  df-fsupp 8868  df-sup 8940  df-oi 9008  df-card 9402  df-pnf 10716  df-mnf 10717  df-xr 10718  df-ltxr 10719  df-le 10720  df-sub 10911  df-neg 10912  df-nn 11676  df-2 11738  df-3 11739  df-4 11740  df-5 11741  df-6 11742  df-7 11743  df-8 11744  df-9 11745  df-n0 11936  df-z 12022  df-dec 12139  df-uz 12284  df-fz 12941  df-fzo 13084  df-seq 13420  df-hash 13742  df-struct 16544  df-ndx 16545  df-slot 16546  df-base 16548  df-sets 16549  df-ress 16550  df-plusg 16637  df-mulr 16638  df-sca 16640  df-vsca 16641  df-ip 16642  df-tset 16643  df-ple 16644  df-ds 16646  df-hom 16648  df-cco 16649  df-0g 16774  df-gsum 16775  df-prds 16780  df-pws 16782  df-mre 16916  df-mrc 16917  df-acs 16919  df-mgm 17919  df-sgrp 17968  df-mnd 17979  df-mhm 18023  df-submnd 18024  df-grp 18173  df-minusg 18174  df-sbg 18175  df-mulg 18293  df-subg 18344  df-ghm 18424  df-cntz 18515  df-cmn 18976  df-abl 18977  df-mgp 19309  df-ur 19321  df-srg 19325  df-ring 19368  df-subrg 19602  df-lmod 19705  df-lss 19773  df-sra 20013  df-rgmod 20014  df-dsmm 20498  df-frlm 20513  df-psr 20672  df-mvr 20673  df-mpl 20674  df-opsr 20676  df-psr1 20905  df-vr1 20906  df-ply1 20907  df-coe1 20908  df-mamu 21087  df-mat 21109  df-decpmat 21464  df-pm2mp 21494 This theorem is referenced by:  pm2mpfo  21515
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