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Theorem mp2pm2mp 20538
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 2621 . . . 4 (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
10 eqid 2621 . . . 4 (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10mply1topmatcl 20532 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐼𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
12 eqid 2621 . . . 4 ( ·𝑠𝑄) = ( ·𝑠𝑄)
13 eqid 2621 . . . 4 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
14 eqid 2621 . . . 4 (var1𝐴) = (var1𝐴)
15 mp2pm2mp.t . . . 4 𝑇 = (𝑁 pMatToMatPoly 𝑅)
164, 9, 10, 12, 13, 14, 1, 2, 15pm2mpfval 20523 . . 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 1368 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑇‘(𝐼𝑂)) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))
181matring 20171 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19183adant3 1079 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝐴 ∈ Ring)
20 eqid 2621 . . . . 5 (0g𝑄) = (0g𝑄)
212ply1ring 19540 . . . . . . 7 (𝐴 ∈ Ring → 𝑄 ∈ Ring)
22 ringcmn 18505 . . . . . . 7 (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
2318, 21, 223syl 18 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
24233adant3 1079 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝑄 ∈ CMnd)
25 nn0ex 11245 . . . . . 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 1063 . . . . . . . 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 477 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
31 eqid 2621 . . . . . . . . 9 (Base‘𝐴) = (Base‘𝐴)
324, 9, 10, 1, 31decpmatcl 20494 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐼𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑛 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
3328, 29, 30, 32syl3anc 1323 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
34 eqid 2621 . . . . . . . 8 (mulGrp‘𝑄) = (mulGrp‘𝑄)
3531, 2, 14, 12, 34, 13, 3ply1tmcl 19564 . . . . . . 7 ((𝐴 ∈ Ring ∧ ((𝐼𝑂) decompPMat 𝑛) ∈ (Base‘𝐴) ∧ 𝑛 ∈ ℕ0) → (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ 𝐿)
3627, 33, 30, 35syl3anc 1323 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ 𝐿)
37 eqid 2621 . . . . . 6 (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))) = (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))
3836, 37fmptd 6343 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0𝐿)
39 fveq2 6150 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((coe1𝑝)‘𝑘) = ((coe1𝑝)‘𝑛))
4039oveqd 6624 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑖((coe1𝑝)‘𝑘)𝑗) = (𝑖((coe1𝑝)‘𝑛)𝑗))
41 oveq1 6614 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑘𝐸𝑌) = (𝑛𝐸𝑌))
4240, 41oveq12d 6625 . . . . . . . . . . . 12 (𝑘 = 𝑛 → ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))
4342cbvmptv 4712 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))
4443a1i 11 . . . . . . . . . 10 ((𝑖𝑁𝑗𝑁) → (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌))))
4544oveq2d 6623 . . . . . . . . 9 ((𝑖𝑁𝑗𝑁) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))))
4645mpt2eq3ia 6676 . . . . . . . 8 (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))))
4746mpteq2i 4703 . . . . . . 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))))))
488, 47eqtri 2643 . . . . . 6 𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌))))))
491, 2, 3, 5, 6, 7, 48, 4, 12, 13, 14mp2pm2mplem5 20537 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
503, 20, 24, 26, 38, 49gsumcl 18240 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ 𝐿)
51 simp3 1061 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → 𝑂𝐿)
5219, 50, 513jca 1240 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐴 ∈ Ring ∧ (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ 𝐿𝑂𝐿))
531, 2, 3, 5, 6, 7, 8, 4mp2pm2mplem4 20536 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝑛) = ((coe1𝑂)‘𝑛))
5453oveq1d 6622 . . . . . . . . . 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))))
5554adantlr 750 . . . . . . . . 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))))
5655mpteq2dva 4706 . . . . . . . 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)))))
5756oveq2d 6623 . . . . . . 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))))))
5857fveq2d 6154 . . . . . 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)))))))
5958fveq1d 6152 . . . . 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))
6019, 51jca 554 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐴 ∈ Ring ∧ 𝑂𝐿))
6160adantr 481 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (𝐴 ∈ Ring ∧ 𝑂𝐿))
62 eqid 2621 . . . . . . . . . 10 (coe1𝑂) = (coe1𝑂)
632, 14, 3, 12, 34, 13, 62ply1coe 19588 . . . . . . . . 9 ((𝐴 ∈ Ring ∧ 𝑂𝐿) → 𝑂 = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))
6461, 63syl 17 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → 𝑂 = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))
6564eqcomd 2627 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) = 𝑂)
6665fveq2d 6154 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴)))))) = (coe1𝑂))
6766fveq1d 6152 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝑙 ∈ ℕ0) → ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝑂)‘𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙) = ((coe1𝑂)‘𝑙))
6859, 67eqtrd 2655 . . . 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))
6968ralrimiva 2960 . . 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))
70 eqid 2621 . . . 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))))))
712, 3, 70, 62eqcoe1ply1eq 19589 . . 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))
7252, 69, 71sylc 65 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑛)( ·𝑠𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1𝐴))))) = 𝑂)
7317, 72eqtrd 2655 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑇‘(𝐼𝑂)) = 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cmpt 4675  cfv 5849  (class class class)co 6607  cmpt2 6609  Fincfn 7902  0cn0 11239  Basecbs 15784   ·𝑠 cvsca 15869  0gc0g 16024   Σg cgsu 16025  .gcmg 17464  CMndccmn 18117  mulGrpcmgp 18413  Ringcrg 18471  var1cv1 19468  Poly1cpl1 19469  coe1cco1 19470   Mat cmat 20135   decompPMat cdecpmat 20489   pMatToMatPoly cpm2mp 20519
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 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  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 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-ot 4159  df-uni 4405  df-int 4443  df-iun 4489  df-iin 4490  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-of 6853  df-ofr 6854  df-om 7016  df-1st 7116  df-2nd 7117  df-supp 7244  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-ixp 7856  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-fsupp 8223  df-sup 8295  df-oi 8362  df-card 8712  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-3 11027  df-4 11028  df-5 11029  df-6 11030  df-7 11031  df-8 11032  df-9 11033  df-n0 11240  df-z 11325  df-dec 11441  df-uz 11635  df-fz 12272  df-fzo 12410  df-seq 12745  df-hash 13061  df-struct 15786  df-ndx 15787  df-slot 15788  df-base 15789  df-sets 15790  df-ress 15791  df-plusg 15878  df-mulr 15879  df-sca 15881  df-vsca 15882  df-ip 15883  df-tset 15884  df-ple 15885  df-ds 15888  df-hom 15890  df-cco 15891  df-0g 16026  df-gsum 16027  df-prds 16032  df-pws 16034  df-mre 16170  df-mrc 16171  df-acs 16173  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-mhm 17259  df-submnd 17260  df-grp 17349  df-minusg 17350  df-sbg 17351  df-mulg 17465  df-subg 17515  df-ghm 17582  df-cntz 17674  df-cmn 18119  df-abl 18120  df-mgp 18414  df-ur 18426  df-srg 18430  df-ring 18473  df-subrg 18702  df-lmod 18789  df-lss 18855  df-sra 19094  df-rgmod 19095  df-psr 19278  df-mvr 19279  df-mpl 19280  df-opsr 19282  df-psr1 19472  df-vr1 19473  df-ply1 19474  df-coe1 19475  df-dsmm 19998  df-frlm 20013  df-mamu 20112  df-mat 20136  df-decpmat 20490  df-pm2mp 20520
This theorem is referenced by:  pm2mpfo  20541
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