Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19302 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | chfacfisf.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | chfacfisf.y | . . . . . 6 ⊢ 𝑌 = (𝑁 Mat 𝑃) | |
4 | 2, 3 | pmatlmod 21296 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod) |
5 | 1, 4 | sylan2 594 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod) |
6 | 5 | 3adant3 1128 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ LMod) |
7 | 6 | 3ad2ant1 1129 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → 𝑌 ∈ LMod) |
8 | 2 | ply1ring 20410 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
9 | 1, 8 | syl 17 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring) |
10 | 9 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring) |
11 | eqid 2821 | . . . . . . 7 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
12 | 11 | ringmgp 19297 | . . . . . 6 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd) |
13 | 10, 12 | syl 17 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑃) ∈ Mnd) |
14 | 13 | 3ad2ant1 1129 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd) |
15 | simp3 1134 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0) | |
16 | 1 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
17 | chfacfscmulcl.x | . . . . . . 7 ⊢ 𝑋 = (var1‘𝑅) | |
18 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
19 | 17, 2, 18 | vr1cl 20379 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
20 | 16, 19 | syl 17 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
21 | 20 | 3ad2ant1 1129 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → 𝑋 ∈ (Base‘𝑃)) |
22 | 11, 18 | mgpbas 19239 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃)) |
23 | chfacfscmulcl.e | . . . . 5 ⊢ ↑ = (.g‘(mulGrp‘𝑃)) | |
24 | 22, 23 | mulgnn0cl 18238 | . . . 4 ⊢ (((mulGrp‘𝑃) ∈ Mnd ∧ 𝐾 ∈ ℕ0 ∧ 𝑋 ∈ (Base‘𝑃)) → (𝐾 ↑ 𝑋) ∈ (Base‘𝑃)) |
25 | 14, 15, 21, 24 | syl3anc 1367 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (𝐾 ↑ 𝑋) ∈ (Base‘𝑃)) |
26 | 2 | ply1crng 20360 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
27 | 26 | anim2i 618 | . . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing)) |
28 | 27 | 3adant3 1128 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing)) |
29 | 3 | matsca2 21023 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌)) |
30 | 28, 29 | syl 17 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 = (Scalar‘𝑌)) |
31 | 30 | eqcomd 2827 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑌) = 𝑃) |
32 | 31 | fveq2d 6668 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
33 | 32 | 3ad2ant1 1129 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
34 | 25, 33 | eleqtrrd 2916 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (𝐾 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌))) |
35 | chfacfisf.a | . . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
36 | chfacfisf.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
37 | chfacfisf.r | . . . . . 6 ⊢ × = (.r‘𝑌) | |
38 | chfacfisf.s | . . . . . 6 ⊢ − = (-g‘𝑌) | |
39 | chfacfisf.0 | . . . . . 6 ⊢ 0 = (0g‘𝑌) | |
40 | chfacfisf.t | . . . . . 6 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
41 | 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)))))))) | |
42 | 35, 36, 2, 3, 37, 38, 39, 40, 41 | chfacfisf 21456 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌)) |
43 | 1, 42 | syl3anl2 1409 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌)) |
44 | 43 | 3adant3 1128 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → 𝐺:ℕ0⟶(Base‘𝑌)) |
45 | 44, 15 | ffvelrnd 6846 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (𝐺‘𝐾) ∈ (Base‘𝑌)) |
46 | eqid 2821 | . . 3 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
47 | eqid 2821 | . . 3 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌) | |
48 | chfacfscmulcl.m | . . 3 ⊢ · = ( ·𝑠 ‘𝑌) | |
49 | eqid 2821 | . . 3 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌)) | |
50 | 46, 47, 48, 49 | lmodvscl 19645 | . 2 ⊢ ((𝑌 ∈ LMod ∧ (𝐾 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ∧ (𝐺‘𝐾) ∈ (Base‘𝑌)) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) ∈ (Base‘𝑌)) |
51 | 7, 34, 45, 50 | syl3anc 1367 | 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 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ifcif 4466 class class class wbr 5058 ↦ cmpt 5138 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 Fincfn 8503 0cc0 10531 1c1 10532 + caddc 10534 < clt 10669 − cmin 10864 ℕcn 11632 ℕ0cn0 11891 ...cfz 12886 Basecbs 16477 .rcmulr 16560 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 Mndcmnd 17905 -gcsg 18099 .gcmg 18218 mulGrpcmgp 19233 Ringcrg 19291 CRingccrg 19292 LModclmod 19628 var1cv1 20338 Poly1cpl1 20339 Mat cmat 21010 matToPolyMat cmat2pmat 21306 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-ot 4569 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-gsum 16710 df-prds 16715 df-pws 16717 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-lmod 19630 df-lss 19698 df-sra 19938 df-rgmod 19939 df-ascl 20081 df-psr 20130 df-mvr 20131 df-mpl 20132 df-opsr 20134 df-psr1 20342 df-vr1 20343 df-ply1 20344 df-dsmm 20870 df-frlm 20885 df-mamu 20989 df-mat 21011 df-mat2pmat 21309 |
This theorem is referenced by: chfacfscmulgsum 21462 |
Copyright terms: Public domain | W3C validator |