MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm2mpmhmlem1 Structured version   Visualization version   GIF version

Theorem pm2mpmhmlem1 22707
Description: Lemma 1 for pm2mpmhm 22709. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 6-Dec-2019.)
Hypotheses
Ref Expression
pm2mpfo.p 𝑃 = (Poly1β€˜π‘…)
pm2mpfo.c 𝐢 = (𝑁 Mat 𝑃)
pm2mpfo.b 𝐡 = (Baseβ€˜πΆ)
pm2mpfo.m βˆ— = ( ·𝑠 β€˜π‘„)
pm2mpfo.e ↑ = (.gβ€˜(mulGrpβ€˜π‘„))
pm2mpfo.x 𝑋 = (var1β€˜π΄)
pm2mpfo.a 𝐴 = (𝑁 Mat 𝑅)
pm2mpfo.q 𝑄 = (Poly1β€˜π΄)
pm2mpfo.l 𝐿 = (Baseβ€˜π‘„)
pm2mpfo.t 𝑇 = (𝑁 pMatToMatPoly 𝑅)
Assertion
Ref Expression
pm2mpmhmlem1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (𝑙 ∈ β„•0 ↦ ((𝐴 Ξ£g (π‘˜ ∈ (0...𝑙) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑙 βˆ’ π‘˜))))) βˆ— (𝑙 ↑ 𝑋))) finSupp (0gβ€˜π‘„))
Distinct variable groups:   𝐴,π‘˜   𝐡,π‘˜,𝑙   𝐢,π‘˜,𝑙   π‘˜,𝐿   π‘˜,𝑁,𝑙   𝑄,π‘˜   𝑅,π‘˜   βˆ— ,π‘˜   𝐴,𝑙   𝑃,π‘˜   𝑅,𝑙   𝑋,𝑙   βˆ— ,𝑙   ↑ ,𝑙   π‘₯,𝑦,π‘˜,𝑙
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐡(π‘₯,𝑦)   𝐢(π‘₯,𝑦)   𝑃(π‘₯,𝑦,𝑙)   𝑄(π‘₯,𝑦,𝑙)   𝑅(π‘₯,𝑦)   𝑇(π‘₯,𝑦,π‘˜,𝑙)   ↑ (π‘₯,𝑦,π‘˜)   βˆ— (π‘₯,𝑦)   𝐿(π‘₯,𝑦,𝑙)   𝑁(π‘₯,𝑦)   𝑋(π‘₯,𝑦,π‘˜)

Proof of Theorem pm2mpmhmlem1
Dummy variables π‘Ž 𝑏 𝑖 𝑗 𝑛 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6906 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (0gβ€˜π‘„) ∈ V)
2 ovexd 7449 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑙 ∈ β„•0) β†’ ((𝐴 Ξ£g (π‘˜ ∈ (0...𝑙) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑙 βˆ’ π‘˜))))) βˆ— (𝑙 ↑ 𝑋)) ∈ V)
3 oveq2 7422 . . . . 5 (𝑙 = 𝑛 β†’ (0...𝑙) = (0...𝑛))
4 oveq1 7421 . . . . . . 7 (𝑙 = 𝑛 β†’ (𝑙 βˆ’ π‘˜) = (𝑛 βˆ’ π‘˜))
54oveq2d 7430 . . . . . 6 (𝑙 = 𝑛 β†’ (𝑦 decompPMat (𝑙 βˆ’ π‘˜)) = (𝑦 decompPMat (𝑛 βˆ’ π‘˜)))
65oveq2d 7430 . . . . 5 (𝑙 = 𝑛 β†’ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑙 βˆ’ π‘˜))) = ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜))))
73, 6mpteq12dv 5233 . . . 4 (𝑙 = 𝑛 β†’ (π‘˜ ∈ (0...𝑙) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑙 βˆ’ π‘˜)))) = (π‘˜ ∈ (0...𝑛) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜)))))
87oveq2d 7430 . . 3 (𝑙 = 𝑛 β†’ (𝐴 Ξ£g (π‘˜ ∈ (0...𝑙) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑙 βˆ’ π‘˜))))) = (𝐴 Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜))))))
9 oveq1 7421 . . 3 (𝑙 = 𝑛 β†’ (𝑙 ↑ 𝑋) = (𝑛 ↑ 𝑋))
108, 9oveq12d 7432 . 2 (𝑙 = 𝑛 β†’ ((𝐴 Ξ£g (π‘˜ ∈ (0...𝑙) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑙 βˆ’ π‘˜))))) βˆ— (𝑙 ↑ 𝑋)) = ((𝐴 Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜))))) βˆ— (𝑛 ↑ 𝑋)))
11 simpll 766 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ 𝑁 ∈ Fin)
12 simplr 768 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ 𝑅 ∈ Ring)
13 pm2mpfo.p . . . . . . . . . 10 𝑃 = (Poly1β€˜π‘…)
14 pm2mpfo.c . . . . . . . . . 10 𝐢 = (𝑁 Mat 𝑃)
1513, 14pmatring 22581 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝐢 ∈ Ring)
1615anim1i 614 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (𝐢 ∈ Ring ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)))
17 3anass 1093 . . . . . . . 8 ((𝐢 ∈ Ring ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ↔ (𝐢 ∈ Ring ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)))
1816, 17sylibr 233 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (𝐢 ∈ Ring ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡))
19 pm2mpfo.b . . . . . . . 8 𝐡 = (Baseβ€˜πΆ)
20 eqid 2727 . . . . . . . 8 (.rβ€˜πΆ) = (.rβ€˜πΆ)
2119, 20ringcl 20181 . . . . . . 7 ((𝐢 ∈ Ring ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯(.rβ€˜πΆ)𝑦) ∈ 𝐡)
2218, 21syl 17 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΆ)𝑦) ∈ 𝐡)
23 eqid 2727 . . . . . . 7 (0gβ€˜π‘…) = (0gβ€˜π‘…)
2413, 14, 19, 23pmatcoe1fsupp 22590 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (π‘₯(.rβ€˜πΆ)𝑦) ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)))
2511, 12, 22, 24syl3anc 1369 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)))
26 fvoveq1 7437 . . . . . . . . . . . . . . . . . . 19 (π‘Ž = 𝑖 β†’ (coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏)) = (coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑏)))
2726fveq1d 6893 . . . . . . . . . . . . . . . . . 18 (π‘Ž = 𝑖 β†’ ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›))
2827eqeq1d 2729 . . . . . . . . . . . . . . . . 17 (π‘Ž = 𝑖 β†’ (((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…) ↔ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)))
29 oveq2 7422 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑗 β†’ (𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑏) = (𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))
3029fveq2d 6895 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑗 β†’ (coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑏)) = (coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗)))
3130fveq1d 6893 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑗 β†’ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›))
3231eqeq1d 2729 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑗 β†’ (((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…) ↔ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›) = (0gβ€˜π‘…)))
3328, 32rspc2va 3619 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›) = (0gβ€˜π‘…))
3433expcom 413 . . . . . . . . . . . . . . 15 (βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…) β†’ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) β†’ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›) = (0gβ€˜π‘…)))
3534adantl 481 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) ∧ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) β†’ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›) = (0gβ€˜π‘…)))
36353impib 1114 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) ∧ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) β†’ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›) = (0gβ€˜π‘…))
3736mpoeq3dva 7491 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) ∧ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0gβ€˜π‘…)))
38 pm2mpfo.a . . . . . . . . . . . . . 14 𝐴 = (𝑁 Mat 𝑅)
3938, 23mat0op 22308 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (0gβ€˜π΄) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0gβ€˜π‘…)))
4039ad3antrrr 729 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) ∧ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ (0gβ€˜π΄) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0gβ€˜π‘…)))
4138matring 22332 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝐴 ∈ Ring)
42 pm2mpfo.q . . . . . . . . . . . . . . . 16 𝑄 = (Poly1β€˜π΄)
4342ply1sca 22158 . . . . . . . . . . . . . . 15 (𝐴 ∈ Ring β†’ 𝐴 = (Scalarβ€˜π‘„))
4441, 43syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝐴 = (Scalarβ€˜π‘„))
4544ad3antrrr 729 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) ∧ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ 𝐴 = (Scalarβ€˜π‘„))
4645fveq2d 6895 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) ∧ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ (0gβ€˜π΄) = (0gβ€˜(Scalarβ€˜π‘„)))
4737, 40, 463eqtr2d 2773 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) ∧ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) = (0gβ€˜(Scalarβ€˜π‘„)))
4847oveq1d 7429 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) ∧ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)) = ((0gβ€˜(Scalarβ€˜π‘„)) βˆ— (𝑛 ↑ 𝑋)))
4942ply1lmod 22157 . . . . . . . . . . . . . 14 (𝐴 ∈ Ring β†’ 𝑄 ∈ LMod)
5041, 49syl 17 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝑄 ∈ LMod)
5150adantr 480 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ 𝑄 ∈ LMod)
5241adantr 480 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ 𝐴 ∈ Ring)
53 pm2mpfo.x . . . . . . . . . . . . . 14 𝑋 = (var1β€˜π΄)
54 eqid 2727 . . . . . . . . . . . . . 14 (mulGrpβ€˜π‘„) = (mulGrpβ€˜π‘„)
55 pm2mpfo.e . . . . . . . . . . . . . 14 ↑ = (.gβ€˜(mulGrpβ€˜π‘„))
56 pm2mpfo.l . . . . . . . . . . . . . 14 𝐿 = (Baseβ€˜π‘„)
5742, 53, 54, 55, 56ply1moncl 22177 . . . . . . . . . . . . 13 ((𝐴 ∈ Ring ∧ 𝑛 ∈ β„•0) β†’ (𝑛 ↑ 𝑋) ∈ 𝐿)
5852, 57sylan 579 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ (𝑛 ↑ 𝑋) ∈ 𝐿)
59 eqid 2727 . . . . . . . . . . . . 13 (Scalarβ€˜π‘„) = (Scalarβ€˜π‘„)
60 pm2mpfo.m . . . . . . . . . . . . 13 βˆ— = ( ·𝑠 β€˜π‘„)
61 eqid 2727 . . . . . . . . . . . . 13 (0gβ€˜(Scalarβ€˜π‘„)) = (0gβ€˜(Scalarβ€˜π‘„))
62 eqid 2727 . . . . . . . . . . . . 13 (0gβ€˜π‘„) = (0gβ€˜π‘„)
6356, 59, 60, 61, 62lmod0vs 20767 . . . . . . . . . . . 12 ((𝑄 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ 𝐿) β†’ ((0gβ€˜(Scalarβ€˜π‘„)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))
6451, 58, 63syl2an2r 684 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ ((0gβ€˜(Scalarβ€˜π‘„)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))
6564adantr 480 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) ∧ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ ((0gβ€˜(Scalarβ€˜π‘„)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))
6648, 65eqtrd 2767 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) ∧ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))
6766ex 412 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ (βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…) β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„)))
6867imim2d 57 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ ((𝑠 < 𝑛 β†’ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ (𝑠 < 𝑛 β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))))
6968ralimdva 3162 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))))
7069reximdv 3165 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ βˆ€π‘Ž ∈ 𝑁 βˆ€π‘ ∈ 𝑁 ((coe1β€˜(π‘Ž(π‘₯(.rβ€˜πΆ)𝑦)𝑏))β€˜π‘›) = (0gβ€˜π‘…)) β†’ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))))
7125, 70mpd 15 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„)))
7214, 19decpmatval 22654 . . . . . . . . . 10 (((π‘₯(.rβ€˜πΆ)𝑦) ∈ 𝐡 ∧ 𝑛 ∈ β„•0) β†’ ((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)))
7322, 72sylan 579 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ ((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)))
7473oveq1d 7429 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ (((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) βˆ— (𝑛 ↑ 𝑋)) = ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)))
7574eqeq1d 2729 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ ((((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„) ↔ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„)))
7675imbi2d 340 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ ((𝑠 < 𝑛 β†’ (((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„)) ↔ (𝑠 < 𝑛 β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))))
7776ralbidva 3170 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„)) ↔ βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))))
7877rexbidv 3173 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„)) ↔ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1β€˜(𝑖(π‘₯(.rβ€˜πΆ)𝑦)𝑗))β€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))))
7971, 78mpbird 257 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„)))
8013, 14, 19, 38decpmatmul 22661 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ 𝑛 ∈ β„•0) β†’ ((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) = (𝐴 Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜))))))
8180ad4ant234 1173 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ ((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) = (𝐴 Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜))))))
8281eqcomd 2733 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ (𝐴 Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜))))) = ((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛))
8382oveq1d 7429 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ ((𝐴 Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜))))) βˆ— (𝑛 ↑ 𝑋)) = (((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) βˆ— (𝑛 ↑ 𝑋)))
8483eqeq1d 2729 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ (((𝐴 Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜))))) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„) ↔ (((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„)))
8584imbi2d 340 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ 𝑛 ∈ β„•0) β†’ ((𝑠 < 𝑛 β†’ ((𝐴 Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜))))) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„)) ↔ (𝑠 < 𝑛 β†’ (((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))))
8685ralbidva 3170 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ ((𝐴 Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜))))) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„)) ↔ βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))))
8786rexbidv 3173 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ ((𝐴 Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜))))) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„)) ↔ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (((π‘₯(.rβ€˜πΆ)𝑦) decompPMat 𝑛) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„))))
8879, 87mpbird 257 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ ((𝐴 Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑛 βˆ’ π‘˜))))) βˆ— (𝑛 ↑ 𝑋)) = (0gβ€˜π‘„)))
891, 2, 10, 88mptnn0fsuppd 13987 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (𝑙 ∈ β„•0 ↦ ((𝐴 Ξ£g (π‘˜ ∈ (0...𝑙) ↦ ((π‘₯ decompPMat π‘˜)(.rβ€˜π΄)(𝑦 decompPMat (𝑙 βˆ’ π‘˜))))) βˆ— (𝑙 ↑ 𝑋))) finSupp (0gβ€˜π‘„))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  βˆƒwrex 3065  Vcvv 3469   class class class wbr 5142   ↦ cmpt 5225  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  Fincfn 8955   finSupp cfsupp 9377  0cc0 11130   < clt 11270   βˆ’ cmin 11466  β„•0cn0 12494  ...cfz 13508  Basecbs 17171  .rcmulr 17225  Scalarcsca 17227   ·𝑠 cvsca 17228  0gc0g 17412   Ξ£g cgsu 17413  .gcmg 19014  mulGrpcmgp 20065  Ringcrg 20164  LModclmod 20732  var1cv1 22082  Poly1cpl1 22083  coe1cco1 22084   Mat cmat 22294   decompPMat cdecpmat 22651   pMatToMatPoly cpm2mp 22681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-ot 4633  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-ofr 7680  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8838  df-pm 8839  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9378  df-sup 9457  df-oi 9525  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-fz 13509  df-fzo 13652  df-seq 13991  df-hash 14314  df-struct 17107  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-mulr 17238  df-sca 17240  df-vsca 17241  df-ip 17242  df-tset 17243  df-ple 17244  df-ds 17246  df-hom 17248  df-cco 17249  df-0g 17414  df-gsum 17415  df-prds 17420  df-pws 17422  df-mre 17557  df-mrc 17558  df-acs 17560  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-mhm 18731  df-submnd 18732  df-grp 18884  df-minusg 18885  df-sbg 18886  df-mulg 19015  df-subg 19069  df-ghm 19159  df-cntz 19259  df-cmn 19728  df-abl 19729  df-mgp 20066  df-rng 20084  df-ur 20113  df-ring 20166  df-subrng 20472  df-subrg 20497  df-lmod 20734  df-lss 20805  df-sra 21047  df-rgmod 21048  df-dsmm 21653  df-frlm 21668  df-psr 21829  df-mvr 21830  df-mpl 21831  df-opsr 21833  df-psr1 22086  df-vr1 22087  df-ply1 22088  df-coe1 22089  df-mamu 22273  df-mat 22295  df-decpmat 22652
This theorem is referenced by:  pm2mpmhmlem2  22708
  Copyright terms: Public domain W3C validator