pm2mpcl - Metamath Proof Explorer

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Theorem pm2mpcl 21956

Description: The transformation of polynomial matrices into polynomials over matrices maps polynomial matrices to polynomials over matrices. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 5-Dec-2019.)

**Hypotheses**
Ref	Expression
pm2mpval.p	⊢ 𝑃 = (Poly₁‘𝑅)
pm2mpval.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pm2mpval.b	⊢ 𝐵 = (Base‘𝐶)
pm2mpval.m	⊢ ∗ = ( ·_𝑠 ‘𝑄)
pm2mpval.e	⊢ ↑ = (._g‘(mulGrp‘𝑄))
pm2mpval.x	⊢ 𝑋 = (var₁‘𝐴)
pm2mpval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
pm2mpval.q	⊢ 𝑄 = (Poly₁‘𝐴)
pm2mpval.t	⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
pm2mpcl.l	⊢ 𝐿 = (Base‘𝑄)

**Assertion**
Ref	Expression
pm2mpcl	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝐿)

Proof of Theorem pm2mpcl Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm2mpval.p	. . 3 ⊢ 𝑃 = (Poly₁‘𝑅)
2		pm2mpval.c	. . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃)
3		pm2mpval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
4		pm2mpval.m	. . 3 ⊢ ∗ = ( ·_𝑠 ‘𝑄)
5		pm2mpval.e	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑄))
6		pm2mpval.x	. . 3 ⊢ 𝑋 = (var₁‘𝐴)
7		pm2mpval.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
8		pm2mpval.q	. . 3 ⊢ 𝑄 = (Poly₁‘𝐴)
9		pm2mpval.t	. . 3 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	pm2mpfval 21955	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
11		pm2mpcl.l	. . 3 ⊢ 𝐿 = (Base‘𝑄)
12		eqid 2738	. . 3 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
13	7	matring 21602	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
14	8	ply1ring 21429	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
15		ringcmn 19830	. . . . 5 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
16	13, 14, 15	3syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
17	16	3adant3 1131	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ CMnd)
18		nn0ex 12249	. . . 4 ⊢ ℕ₀ ∈ V
19	18	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ℕ₀ ∈ V)
20	13	3adant3 1131	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring)
21	20	adantr 481	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ Ring)
22		simpl2 1191	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
23		simpl3 1192	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑀 ∈ 𝐵)
24		simpr 485	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
25		eqid 2738	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
26	1, 2, 3, 7, 25	decpmatcl 21926	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴))
27	22, 23, 24, 26	syl3anc 1370	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴))
28		eqid 2738	. . . . . 6 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
29	25, 8, 6, 4, 28, 5, 11	ply1tmcl 21453	. . . . 5 ⊢ ((𝐴 ∈ Ring ∧ (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴) ∧ 𝑘 ∈ ℕ₀) → ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) ∈ 𝐿)
30	21, 27, 24, 29	syl3anc 1370	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) ∈ 𝐿)
31	30	fmpttd 6981	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))):ℕ₀⟶𝐿)
32	8	ply1lmod 21433	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
33	20, 32	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ LMod)
34		eqidd 2739	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑄) = (Scalar‘𝑄))
35	8, 6, 28, 5, 11	ply1moncl 21452	. . . . 5 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
36	20, 35	sylan 580	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
37		eqid 2738	. . . 4 ⊢ (0_g‘(Scalar‘𝑄)) = (0_g‘(Scalar‘𝑄))
38		eqid 2738	. . . . . . 7 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
39	1, 2, 3, 7, 38	decpmatfsupp 21928	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑀 decompPMat 𝑘)) finSupp (0_g‘𝐴))
40	39	3adant1 1129	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑀 decompPMat 𝑘)) finSupp (0_g‘𝐴))
41	8	ply1sca 21434	. . . . . . . 8 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
42	41	eqcomd 2744	. . . . . . 7 ⊢ (𝐴 ∈ Ring → (Scalar‘𝑄) = 𝐴)
43	20, 42	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑄) = 𝐴)
44	43	fveq2d 6770	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0_g‘(Scalar‘𝑄)) = (0_g‘𝐴))
45	40, 44	breqtrrd 5101	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑀 decompPMat 𝑘)) finSupp (0_g‘(Scalar‘𝑄)))
46	19, 33, 34, 11, 27, 36, 12, 37, 4, 45	mptscmfsupp0 20198	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) finSupp (0_g‘𝑄))
47	11, 12, 17, 19, 31, 46	gsumcl 19526	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ 𝐿)
48	10, 47	eqeltrd 2839	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝐿)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Vcvv 3429 class class class wbr 5073 ↦ cmpt 5156 ‘cfv 6426 (class class class)co 7267 Fincfn 8720 finSupp cfsupp 9115 ℕ₀cn0 12243 Basecbs 16922 Scalarcsca 16975 ·_𝑠 cvsca 16976 0_gc0g 17160 Σ_g cgsu 17161 ._gcmg 18710 CMndccmn 19396 mulGrpcmgp 19730 Ringcrg 19793 LModclmod 20133 var₁cv1 21357 Poly₁cpl1 21358 Mat cmat 21564 decompPMat cdecpmat 21921 pMatToMatPoly cpm2mp 21951

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 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958

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-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 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 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-ofr 7524 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-pm 8605 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-sup 9188 df-oi 9256 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-fz 13250 df-fzo 13393 df-seq 13732 df-hash 14055 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-hom 16996 df-cco 16997 df-0g 17162 df-gsum 17163 df-prds 17168 df-pws 17170 df-mre 17305 df-mrc 17306 df-acs 17308 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-mhm 18440 df-submnd 18441 df-grp 18590 df-minusg 18591 df-sbg 18592 df-mulg 18711 df-subg 18762 df-ghm 18842 df-cntz 18933 df-cmn 19398 df-abl 19399 df-mgp 19731 df-ur 19748 df-ring 19795 df-subrg 20032 df-lmod 20135 df-lss 20204 df-sra 20444 df-rgmod 20445 df-dsmm 20949 df-frlm 20964 df-psr 21122 df-mvr 21123 df-mpl 21124 df-opsr 21126 df-psr1 21361 df-vr1 21362 df-ply1 21363 df-coe1 21364 df-mamu 21543 df-mat 21565 df-decpmat 21922 df-pm2mp 21952

This theorem is referenced by: pm2mpf 21957 pm2mpf1 21958

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