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 19300 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | chfacfisf.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | chfacfisf.y | . . . . . 6 ⊢ 𝑌 = (𝑁 Mat 𝑃) | |
4 | 2, 3 | pmatlmod 21294 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod) |
5 | 1, 4 | sylan2 594 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod) |
6 | 5 | 3adant3 1127 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ LMod) |
7 | 6 | 3ad2ant1 1128 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → 𝑌 ∈ LMod) |
8 | 2 | ply1ring 20408 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
9 | 1, 8 | syl 17 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring) |
10 | 9 | 3ad2ant2 1129 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring) |
11 | eqid 2819 | . . . . . . 7 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
12 | 11 | ringmgp 19295 | . . . . . 6 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd) |
13 | 10, 12 | syl 17 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑃) ∈ Mnd) |
14 | 13 | 3ad2ant1 1128 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd) |
15 | simp3 1133 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0) | |
16 | 1 | 3ad2ant2 1129 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
17 | chfacfscmulcl.x | . . . . . . 7 ⊢ 𝑋 = (var1‘𝑅) | |
18 | eqid 2819 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
19 | 17, 2, 18 | vr1cl 20377 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
20 | 16, 19 | syl 17 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
21 | 20 | 3ad2ant1 1128 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → 𝑋 ∈ (Base‘𝑃)) |
22 | 11, 18 | mgpbas 19237 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃)) |
23 | chfacfscmulcl.e | . . . . 5 ⊢ ↑ = (.g‘(mulGrp‘𝑃)) | |
24 | 22, 23 | mulgnn0cl 18236 | . . . 4 ⊢ (((mulGrp‘𝑃) ∈ Mnd ∧ 𝐾 ∈ ℕ0 ∧ 𝑋 ∈ (Base‘𝑃)) → (𝐾 ↑ 𝑋) ∈ (Base‘𝑃)) |
25 | 14, 15, 21, 24 | syl3anc 1366 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (𝐾 ↑ 𝑋) ∈ (Base‘𝑃)) |
26 | 2 | ply1crng 20358 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
27 | 26 | anim2i 618 | . . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing)) |
28 | 27 | 3adant3 1127 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing)) |
29 | 3 | matsca2 21021 | . . . . . . 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 2825 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑌) = 𝑃) |
32 | 31 | fveq2d 6667 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
33 | 32 | 3ad2ant1 1128 | . . 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 2914 | . 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 21454 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌)) |
43 | 1, 42 | syl3anl2 1408 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌)) |
44 | 43 | 3adant3 1127 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → 𝐺:ℕ0⟶(Base‘𝑌)) |
45 | 44, 15 | ffvelrnd 6845 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → (𝐺‘𝐾) ∈ (Base‘𝑌)) |
46 | eqid 2819 | . . 3 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
47 | eqid 2819 | . . 3 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌) | |
48 | chfacfscmulcl.m | . . 3 ⊢ · = ( ·𝑠 ‘𝑌) | |
49 | eqid 2819 | . . 3 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌)) | |
50 | 46, 47, 48, 49 | lmodvscl 19643 | . 2 ⊢ ((𝑌 ∈ LMod ∧ (𝐾 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ∧ (𝐺‘𝐾) ∈ (Base‘𝑌)) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) ∈ (Base‘𝑌)) |
51 | 7, 34, 45, 50 | syl3anc 1366 | 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 1082 = wceq 1531 ∈ wcel 2108 ifcif 4465 class class class wbr 5057 ↦ cmpt 5137 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ↑m cmap 8398 Fincfn 8501 0cc0 10529 1c1 10530 + caddc 10532 < clt 10667 − cmin 10862 ℕcn 11630 ℕ0cn0 11889 ...cfz 12884 Basecbs 16475 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 Mndcmnd 17903 -gcsg 18097 .gcmg 18216 mulGrpcmgp 19231 Ringcrg 19289 CRingccrg 19290 LModclmod 19626 var1cv1 20336 Poly1cpl1 20337 Mat cmat 21008 matToPolyMat cmat2pmat 21304 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-ot 4568 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-ofr 7402 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-sup 8898 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-fz 12885 df-fzo 13026 df-seq 13362 df-hash 13683 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-ring 19291 df-cring 19292 df-subrg 19525 df-lmod 19628 df-lss 19696 df-sra 19936 df-rgmod 19937 df-ascl 20079 df-psr 20128 df-mvr 20129 df-mpl 20130 df-opsr 20132 df-psr1 20340 df-vr1 20341 df-ply1 20342 df-dsmm 20868 df-frlm 20883 df-mamu 20987 df-mat 21009 df-mat2pmat 21307 |
This theorem is referenced by: chfacfscmulgsum 21460 |
Copyright terms: Public domain | W3C validator |