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| Mirrors > Home > MPE Home > Th. List > decpmatcl | Structured version Visualization version GIF version | ||
| Description: Closure of the decomposition of a polynomial matrix: The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix for the same power is a matrix. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 2-Dec-2019.) |
| Ref | Expression |
|---|---|
| decpmate.p | ⊢ 𝑃 = (Poly1‘𝑅) |
| decpmate.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
| decpmate.b | ⊢ 𝐵 = (Base‘𝐶) |
| decpmatcl.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
| decpmatcl.d | ⊢ 𝐷 = (Base‘𝐴) |
| Ref | Expression |
|---|---|
| decpmatcl | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) ∈ 𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | decpmate.c | . . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
| 2 | decpmate.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
| 3 | 1, 2 | decpmatval 22771 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾))) |
| 4 | 3 | 3adant1 1131 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾))) |
| 5 | decpmatcl.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
| 6 | eqid 2737 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 7 | decpmatcl.d | . . 3 ⊢ 𝐷 = (Base‘𝐴) | |
| 8 | 1, 2 | matrcl 22416 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑃 ∈ V)) |
| 9 | 8 | simpld 494 | . . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
| 10 | 9 | 3ad2ant2 1135 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → 𝑁 ∈ Fin) |
| 11 | simp1 1137 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ 𝑉) | |
| 12 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
| 13 | simp2 1138 | . . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁) | |
| 14 | simp3 1139 | . . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁) | |
| 15 | simp2 1138 | . . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → 𝑀 ∈ 𝐵) | |
| 16 | 15 | 3ad2ant1 1134 | . . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ 𝐵) |
| 17 | 1, 12, 2, 13, 14, 16 | matecld 22432 | . . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑃)) |
| 18 | simp3 1139 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0) | |
| 19 | 18 | 3ad2ant1 1134 | . . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐾 ∈ ℕ0) |
| 20 | eqid 2737 | . . . . 5 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗)) | |
| 21 | decpmate.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 22 | 20, 12, 21, 6 | coe1fvalcl 22214 | . . . 4 ⊢ (((𝑖𝑀𝑗) ∈ (Base‘𝑃) ∧ 𝐾 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘𝐾) ∈ (Base‘𝑅)) |
| 23 | 17, 19, 22 | syl2anc 584 | . . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝐾) ∈ (Base‘𝑅)) |
| 24 | 5, 6, 7, 10, 11, 23 | matbas2d 22429 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)) ∈ 𝐷) |
| 25 | 4, 24 | eqeltrd 2841 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) ∈ 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Fincfn 8985 ℕ0cn0 12526 Basecbs 17247 Poly1cpl1 22178 coe1cco1 22179 Mat cmat 22411 decompPMat cdecpmat 22768 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-prds 17492 df-pws 17494 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 df-psr 21929 df-opsr 21933 df-psr1 22181 df-ply1 22183 df-coe1 22184 df-mat 22412 df-decpmat 22769 |
| This theorem is referenced by: decpmataa0 22774 decpmatmul 22778 pmatcollpw1 22782 pmatcollpw2 22784 pmatcollpwlem 22786 pmatcollpw 22787 pmatcollpwfi 22788 pmatcollpwscmatlem2 22796 pm2mpf1lem 22800 pm2mpcl 22803 pm2mpcoe1 22806 mp2pm2mplem5 22816 mp2pm2mp 22817 pm2mpghmlem2 22818 pm2mpghmlem1 22819 pm2mpghm 22822 pm2mpmhmlem2 22825 |
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