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Theorem mp2pm2mp 22833
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 2735 . . . 4 (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
10 eqid 2735 . . . 4 (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10mply1topmatcl 22827 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐼𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
12 eqid 2735 . . . 4 ( ·𝑠𝑄) = ( ·𝑠𝑄)
13 eqid 2735 . . . 4 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
14 eqid 2735 . . . 4 (var1𝐴) = (var1𝐴)
15 mp2pm2mp.t . . . 4 𝑇 = (𝑁 pMatToMatPoly 𝑅)
164, 9, 10, 12, 13, 14, 1, 2, 15pm2mpfval 22818 . . 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 1408 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑇‘(𝐼𝑂)) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))
181matring 22465 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19183adant3 1131 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝐴 ∈ Ring)
20 eqid 2735 . . . . 5 (0g𝑄) = (0g𝑄)
212ply1ring 22265 . . . . . . 7 (𝐴 ∈ Ring → 𝑄 ∈ Ring)
22 ringcmn 20296 . . . . . . 7 (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
2318, 21, 223syl 18 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
24233adant3 1131 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝑄 ∈ CMnd)
25 nn0ex 12530 . . . . . 6 0 ∈ V
2625a1i 11 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → ℕ0 ∈ V)
2719adantr 480 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
28 simpl2 1191 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
2911adantr 480 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → (𝐼𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
30 simpr 484 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
31 eqid 2735 . . . . . . . . 9 (Base‘𝐴) = (Base‘𝐴)
324, 9, 10, 1, 31decpmatcl 22789 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐼𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑛 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
3328, 29, 30, 32syl3anc 1370 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
34 eqid 2735 . . . . . . . 8 (mulGrp‘𝑄) = (mulGrp‘𝑄)
3531, 2, 14, 12, 34, 13, 3ply1tmcl 22291 . . . . . . 7 ((𝐴 ∈ Ring ∧ ((𝐼𝑂) decompPMat 𝑛) ∈ (Base‘𝐴) ∧ 𝑛 ∈ ℕ0) → (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ 𝐿)
3627, 33, 30, 35syl3anc 1370 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ 𝐿)
3736fmpttd 7135 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0𝐿)
38 fveq2 6907 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((coe1𝑝)‘𝑘) = ((coe1𝑝)‘𝑛))
3938oveqd 7448 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑖((coe1𝑝)‘𝑘)𝑗) = (𝑖((coe1𝑝)‘𝑛)𝑗))
40 oveq1 7438 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑘𝐸𝑌) = (𝑛𝐸𝑌))
4139, 40oveq12d 7449 . . . . . . . . . . . 12 (𝑘 = 𝑛 → ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))
4241cbvmptv 5261 . . . . . . . . . . 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 7447 . . . . . . . . 9 ((𝑖𝑁𝑗𝑁) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))))
4544mpoeq3ia 7511 . . . . . . . 8 (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))))
4645mpteq2i 5253 . . . . . . 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 2763 . . . . . 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 22832 . . . . 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 19948 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ 𝐿)
50 simp3 1137 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝑂𝐿)
5119, 49, 503jca 1127 . . 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 22831 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝑛) = ((coe1𝑂)‘𝑛))
5352oveq1d 7446 . . . . . . . . . 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 5248 . . . . . . . 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 7447 . . . . . . 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 6911 . . . . . 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 6909 . . . . 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 511 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐴 ∈ Ring ∧ 𝑂𝐿))
6059adantr 480 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (𝐴 ∈ Ring ∧ 𝑂𝐿))
61 eqid 2735 . . . . . . . . . 10 (coe1𝑂) = (coe1𝑂)
622, 14, 3, 12, 34, 13, 61ply1coe 22318 . . . . . . . . 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 2741 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) = 𝑂)
6564fveq2d 6911 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))))) = (coe1𝑂))
6665fveq1d 6909 . . . . 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 2775 . . . 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 3144 . . 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 2735 . . . 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 22319 . . 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 2775 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑇‘(𝐼𝑂)) = 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cmpt 5231  cfv 6563  (class class class)co 7431  cmpo 7433  Fincfn 8984  0cn0 12524  Basecbs 17245   ·𝑠 cvsca 17302  0gc0g 17486   Σg cgsu 17487  .gcmg 19098  CMndccmn 19813  mulGrpcmgp 20152  Ringcrg 20251  var1cv1 22193  Poly1cpl1 22194  coe1cco1 22195   Mat cmat 22427   decompPMat cdecpmat 22784   pMatToMatPoly cpm2mp 22814
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-srg 20205  df-ring 20253  df-subrng 20563  df-subrg 20587  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-dsmm 21770  df-frlm 21785  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-psr1 22197  df-vr1 22198  df-ply1 22199  df-coe1 22200  df-mamu 22411  df-mat 22428  df-decpmat 22785  df-pm2mp 22815
This theorem is referenced by:  pm2mpfo  22836
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