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Theorem mp2pm2mp 21103
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 2795 . . . 4 (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
10 eqid 2795 . . . 4 (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10mply1topmatcl 21097 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐼𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
12 eqid 2795 . . . 4 ( ·𝑠𝑄) = ( ·𝑠𝑄)
13 eqid 2795 . . . 4 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
14 eqid 2795 . . . 4 (var1𝐴) = (var1𝐴)
15 mp2pm2mp.t . . . 4 𝑇 = (𝑁 pMatToMatPoly 𝑅)
164, 9, 10, 12, 13, 14, 1, 2, 15pm2mpfval 21088 . . 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 1402 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑇‘(𝐼𝑂)) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))
181matring 20736 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19183adant3 1125 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝐴 ∈ Ring)
20 eqid 2795 . . . . 5 (0g𝑄) = (0g𝑄)
212ply1ring 20099 . . . . . . 7 (𝐴 ∈ Ring → 𝑄 ∈ Ring)
22 ringcmn 19021 . . . . . . 7 (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
2318, 21, 223syl 18 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
24233adant3 1125 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝑄 ∈ CMnd)
25 nn0ex 11751 . . . . . 6 0 ∈ V
2625a1i 11 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → ℕ0 ∈ V)
2719adantr 481 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
28 simpl2 1185 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
2911adantr 481 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → (𝐼𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
30 simpr 485 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
31 eqid 2795 . . . . . . . . 9 (Base‘𝐴) = (Base‘𝐴)
324, 9, 10, 1, 31decpmatcl 21059 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐼𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑛 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
3328, 29, 30, 32syl3anc 1364 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
34 eqid 2795 . . . . . . . 8 (mulGrp‘𝑄) = (mulGrp‘𝑄)
3531, 2, 14, 12, 34, 13, 3ply1tmcl 20123 . . . . . . 7 ((𝐴 ∈ Ring ∧ ((𝐼𝑂) decompPMat 𝑛) ∈ (Base‘𝐴) ∧ 𝑛 ∈ ℕ0) → (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ 𝐿)
3627, 33, 30, 35syl3anc 1364 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ 𝐿)
3736fmpttd 6742 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0𝐿)
38 fveq2 6538 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((coe1𝑝)‘𝑘) = ((coe1𝑝)‘𝑛))
3938oveqd 7033 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑖((coe1𝑝)‘𝑘)𝑗) = (𝑖((coe1𝑝)‘𝑛)𝑗))
40 oveq1 7023 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑘𝐸𝑌) = (𝑛𝐸𝑌))
4139, 40oveq12d 7034 . . . . . . . . . . . 12 (𝑘 = 𝑛 → ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))
4241cbvmptv 5061 . . . . . . . . . . 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 7032 . . . . . . . . 9 ((𝑖𝑁𝑗𝑁) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))))
4544mpoeq3ia 7090 . . . . . . . 8 (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))))
4645mpteq2i 5052 . . . . . . 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 2819 . . . . . 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 21102 . . . . 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 18756 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ 𝐿)
50 simp3 1131 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝑂𝐿)
5119, 49, 503jca 1121 . . 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 21101 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝑛) = ((coe1𝑂)‘𝑛))
5352oveq1d 7031 . . . . . . . . . 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 711 . . . . . . . . 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 5055 . . . . . . . 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 7032 . . . . . . 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 6542 . . . . . 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 6540 . . . . 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 512 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐴 ∈ Ring ∧ 𝑂𝐿))
6059adantr 481 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (𝐴 ∈ Ring ∧ 𝑂𝐿))
61 eqid 2795 . . . . . . . . . 10 (coe1𝑂) = (coe1𝑂)
622, 14, 3, 12, 34, 13, 61ply1coe 20147 . . . . . . . . 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 2801 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) = 𝑂)
6564fveq2d 6542 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))))) = (coe1𝑂))
6665fveq1d 6540 . . . . 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 2831 . . . 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 3149 . . 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 2795 . . . 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 20148 . . 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 2831 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑇‘(𝐼𝑂)) = 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  Vcvv 3437  cmpt 5041  cfv 6225  (class class class)co 7016  cmpo 7018  Fincfn 8357  0cn0 11745  Basecbs 16312   ·𝑠 cvsca 16398  0gc0g 16542   Σg cgsu 16543  .gcmg 17981  CMndccmn 18633  mulGrpcmgp 18929  Ringcrg 18987  var1cv1 20027  Poly1cpl1 20028  coe1cco1 20029   Mat cmat 20700   decompPMat cdecpmat 21054   pMatToMatPoly cpm2mp 21084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-ot 4481  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-ofr 7268  df-om 7437  df-1st 7545  df-2nd 7546  df-supp 7682  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fsupp 8680  df-sup 8752  df-oi 8820  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-fz 12743  df-fzo 12884  df-seq 13220  df-hash 13541  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-sca 16410  df-vsca 16411  df-ip 16412  df-tset 16413  df-ple 16414  df-ds 16416  df-hom 16418  df-cco 16419  df-0g 16544  df-gsum 16545  df-prds 16550  df-pws 16552  df-mre 16686  df-mrc 16687  df-acs 16689  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-mhm 17774  df-submnd 17775  df-grp 17864  df-minusg 17865  df-sbg 17866  df-mulg 17982  df-subg 18030  df-ghm 18097  df-cntz 18188  df-cmn 18635  df-abl 18636  df-mgp 18930  df-ur 18942  df-srg 18946  df-ring 18989  df-subrg 19223  df-lmod 19326  df-lss 19394  df-sra 19634  df-rgmod 19635  df-psr 19824  df-mvr 19825  df-mpl 19826  df-opsr 19828  df-psr1 20031  df-vr1 20032  df-ply1 20033  df-coe1 20034  df-dsmm 20558  df-frlm 20573  df-mamu 20677  df-mat 20701  df-decpmat 21055  df-pm2mp 21085
This theorem is referenced by:  pm2mpfo  21106
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