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Theorem chfacfpmmulcl 22010

Description: Closure of the value of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)

**Hypotheses**
Ref	Expression
cayhamlem1.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cayhamlem1.b	⊢ 𝐵 = (Base‘𝐴)
cayhamlem1.p	⊢ 𝑃 = (Poly₁‘𝑅)
cayhamlem1.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cayhamlem1.r	⊢ × = (._r‘𝑌)
cayhamlem1.s	⊢ − = (-_g‘𝑌)
cayhamlem1.0	⊢ 0 = (0_g‘𝑌)
cayhamlem1.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cayhamlem1.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))))))))
cayhamlem1.e	⊢ ↑ = (._g‘(mulGrp‘𝑌))

**Assertion**
Ref	Expression
chfacfpmmulcl	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ ℕ₀) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) ∈ (Base‘𝑌))

Distinct variable groups: 𝐵,𝑛 𝑛,𝑀 𝑛,𝑁 𝑅,𝑛 𝑛,𝑌 𝑛,𝑏 𝑛,𝑠 Allowed substitution hints: 𝐴(𝑛,𝑠,𝑏) 𝐵(𝑠,𝑏) 𝑃(𝑛,𝑠,𝑏) 𝑅(𝑠,𝑏) 𝑇(𝑛,𝑠,𝑏) × (𝑛,𝑠,𝑏) ↑ (𝑛,𝑠,𝑏) 𝐺(𝑛,𝑠,𝑏) 𝐾(𝑛,𝑠,𝑏) 𝑀(𝑠,𝑏) − (𝑛,𝑠,𝑏) 𝑁(𝑠,𝑏) 𝑌(𝑠,𝑏) 0 (𝑛,𝑠,𝑏)

**Proof of Theorem chfacfpmmulcl**
Step	Hyp	Ref	Expression
1		crngring 19795	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		cayhamlem1.p	. . . . . 6 ⊢ 𝑃 = (Poly₁‘𝑅)
3		cayhamlem1.y	. . . . . 6 ⊢ 𝑌 = (𝑁 Mat 𝑃)
4	2, 3	pmatring 21841	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
5	1, 4	sylan2 593	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
6	5	3adant3 1131	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
7	6	3ad2ant1 1132	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ ℕ₀) → 𝑌 ∈ Ring)
8		eqid 2738	. . . . . 6 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
9	8	ringmgp 19789	. . . . 5 ⊢ (𝑌 ∈ Ring → (mulGrp‘𝑌) ∈ Mnd)
10	6, 9	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑌) ∈ Mnd)
11	10	3ad2ant1 1132	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ ℕ₀) → (mulGrp‘𝑌) ∈ Mnd)
12		simp3 1137	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ ℕ₀) → 𝐾 ∈ ℕ₀)
13		cayhamlem1.t	. . . . . 6 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
14		cayhamlem1.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
15		cayhamlem1.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
16	13, 14, 15, 2, 3	mat2pmatbas 21875	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
17	1, 16	syl3an2 1163	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
18	17	3ad2ant1 1132	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ ℕ₀) → (𝑇‘𝑀) ∈ (Base‘𝑌))
19		eqid 2738	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
20	8, 19	mgpbas 19726	. . . 4 ⊢ (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
21		cayhamlem1.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝑌))
22	20, 21	mulgnn0cl 18720	. . 3 ⊢ (((mulGrp‘𝑌) ∈ Mnd ∧ 𝐾 ∈ ℕ₀ ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → (𝐾 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
23	11, 12, 18, 22	syl3anc 1370	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ ℕ₀) → (𝐾 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
24		cayhamlem1.r	. . . . . 6 ⊢ × = (._r‘𝑌)
25		cayhamlem1.s	. . . . . 6 ⊢ − = (-_g‘𝑌)
26		cayhamlem1.0	. . . . . 6 ⊢ 0 = (0_g‘𝑌)
27		cayhamlem1.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))))))))
28	14, 15, 2, 3, 24, 25, 26, 13, 27	chfacfisf 22003	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(Base‘𝑌))
29	1, 28	syl3anl2 1412	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(Base‘𝑌))
30	29	3adant3 1131	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ ℕ₀) → 𝐺:ℕ₀⟶(Base‘𝑌))
31	30, 12	ffvelrnd 6962	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ ℕ₀) → (𝐺‘𝐾) ∈ (Base‘𝑌))
32	19, 24	ringcl 19800	. 2 ⊢ ((𝑌 ∈ Ring ∧ (𝐾 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌) ∧ (𝐺‘𝐾) ∈ (Base‘𝑌)) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) ∈ (Base‘𝑌))
33	7, 23, 31, 32	syl3anc 1370	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ ℕ₀) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) ∈ (Base‘𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ifcif 4459 class class class wbr 5074 ↦ cmpt 5157 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 Fincfn 8733 0cc0 10871 1c1 10872 + caddc 10874 < clt 11009 − cmin 11205 ℕcn 11973 ℕ₀cn0 12233 ...cfz 13239 Basecbs 16912 ._rcmulr 16963 0_gc0g 17150 Mndcmnd 18385 -_gcsg 18579 ._gcmg 18700 mulGrpcmgp 19720 Ringcrg 19783 CRingccrg 19784 Poly₁cpl1 21348 Mat cmat 21554 matToPolyMat cmat2pmat 21853

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-subrg 20022 df-lmod 20125 df-lss 20194 df-sra 20434 df-rgmod 20435 df-dsmm 20939 df-frlm 20954 df-ascl 21062 df-psr 21112 df-mpl 21114 df-opsr 21116 df-psr1 21351 df-ply1 21353 df-mamu 21533 df-mat 21555 df-mat2pmat 21856

This theorem is referenced by: chfacfpmmulgsum 22013

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