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| Mirrors > Home > MPE Home > Th. List > chfacfscmulcl | Structured version Visualization version GIF version | ||
| Description: Closure of a scaled value of the "characteristic factor function". (Contributed by AV, 9-Nov-2019.) |
| Ref | Expression |
|---|---|
| chfacfisf.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
| chfacfisf.b | ⊢ 𝐵 = (Base‘𝐴) |
| chfacfisf.p | ⊢ 𝑃 = (Poly1‘𝑅) |
| chfacfisf.y | ⊢ 𝑌 = (𝑁 Mat 𝑃) |
| chfacfisf.r | ⊢ × = (.r‘𝑌) |
| chfacfisf.s | ⊢ − = (-g‘𝑌) |
| chfacfisf.0 | ⊢ 0 = (0g‘𝑌) |
| chfacfisf.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
| chfacfisf.g | ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) |
| chfacfscmulcl.x | ⊢ 𝑋 = (var1‘𝑅) |
| chfacfscmulcl.m | ⊢ · = ( ·𝑠 ‘𝑌) |
| chfacfscmulcl.e | ⊢ ↑ = (.g‘(mulGrp‘𝑃)) |
| Ref | Expression |
|---|---|
| chfacfscmulcl | ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) ∈ (Base‘𝑌)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | crngring 20242 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 2 | chfacfisf.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 3 | chfacfisf.y | . . . . . 6 ⊢ 𝑌 = (𝑁 Mat 𝑃) | |
| 4 | 2, 3 | pmatlmod 22699 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod) |
| 5 | 1, 4 | sylan2 593 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod) |
| 6 | 5 | 3adant3 1133 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ LMod) |
| 7 | 6 | 3ad2ant1 1134 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → 𝑌 ∈ LMod) |
| 8 | eqid 2737 | . . . . 5 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
| 9 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
| 10 | 8, 9 | mgpbas 20142 | . . . 4 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃)) |
| 11 | chfacfscmulcl.e | . . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝑃)) | |
| 12 | 2 | ply1ring 22249 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
| 13 | 1, 12 | syl 17 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring) |
| 14 | 13 | 3ad2ant2 1135 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring) |
| 15 | 8 | ringmgp 20236 | . . . . . 6 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd) |
| 16 | 14, 15 | syl 17 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑃) ∈ Mnd) |
| 17 | 16 | 3ad2ant1 1134 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd) |
| 18 | simp3 1139 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0) | |
| 19 | 1 | 3ad2ant2 1135 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
| 20 | chfacfscmulcl.x | . . . . . . 7 ⊢ 𝑋 = (var1‘𝑅) | |
| 21 | 20, 2, 9 | vr1cl 22219 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
| 22 | 19, 21 | syl 17 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
| 23 | 22 | 3ad2ant1 1134 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → 𝑋 ∈ (Base‘𝑃)) |
| 24 | 10, 11, 17, 18, 23 | mulgnn0cld 19113 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (𝐾 ↑ 𝑋) ∈ (Base‘𝑃)) |
| 25 | 2 | ply1crng 22200 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
| 26 | 25 | anim2i 617 | . . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing)) |
| 27 | 26 | 3adant3 1133 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing)) |
| 28 | 3 | matsca2 22426 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌)) |
| 29 | 27, 28 | syl 17 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 = (Scalar‘𝑌)) |
| 30 | 29 | eqcomd 2743 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑌) = 𝑃) |
| 31 | 30 | fveq2d 6910 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
| 32 | 31 | 3ad2ant1 1134 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
| 33 | 24, 32 | eleqtrrd 2844 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (𝐾 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌))) |
| 34 | chfacfisf.a | . . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
| 35 | chfacfisf.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
| 36 | chfacfisf.r | . . . . . 6 ⊢ × = (.r‘𝑌) | |
| 37 | chfacfisf.s | . . . . . 6 ⊢ − = (-g‘𝑌) | |
| 38 | chfacfisf.0 | . . . . . 6 ⊢ 0 = (0g‘𝑌) | |
| 39 | chfacfisf.t | . . . . . 6 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
| 40 | chfacfisf.g | . . . . . 6 ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) | |
| 41 | 34, 35, 2, 3, 36, 37, 38, 39, 40 | chfacfisf 22860 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌)) |
| 42 | 1, 41 | syl3anl2 1415 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌)) |
| 43 | 42 | 3adant3 1133 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → 𝐺:ℕ0⟶(Base‘𝑌)) |
| 44 | 43, 18 | ffvelcdmd 7105 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (𝐺‘𝐾) ∈ (Base‘𝑌)) |
| 45 | eqid 2737 | . . 3 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
| 46 | eqid 2737 | . . 3 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌) | |
| 47 | chfacfscmulcl.m | . . 3 ⊢ · = ( ·𝑠 ‘𝑌) | |
| 48 | eqid 2737 | . . 3 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌)) | |
| 49 | 45, 46, 47, 48 | lmodvscl 20876 | . 2 ⊢ ((𝑌 ∈ LMod ∧ (𝐾 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ∧ (𝐺‘𝐾) ∈ (Base‘𝑌)) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) ∈ (Base‘𝑌)) |
| 50 | 7, 33, 44, 49 | syl3anc 1373 | 1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) ∈ (Base‘𝑌)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Fincfn 8985 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 − cmin 11492 ℕcn 12266 ℕ0cn0 12526 ...cfz 13547 Basecbs 17247 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 Mndcmnd 18747 -gcsg 18953 .gcmg 19085 mulGrpcmgp 20137 Ringcrg 20230 CRingccrg 20231 LModclmod 20858 var1cv1 22177 Poly1cpl1 22178 Mat cmat 22411 matToPolyMat cmat2pmat 22710 |
| 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-rmo 3380 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-int 4947 df-iun 4993 df-iin 4994 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-se 5638 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-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 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-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 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-fzo 13695 df-seq 14043 df-hash 14370 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-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-mamu 22395 df-mat 22412 df-mat2pmat 22713 |
| This theorem is referenced by: chfacfscmulgsum 22866 |
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