pm2mpcl - Metamath Proof Explorer

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Theorem pm2mpcl 21009

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 21008	. 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 2778	. . 3 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
13	7	matring 20653	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
14	8	ply1ring 20014	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
15		ringcmn 18968	. . . . 5 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
16	13, 14, 15	3syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
17	16	3adant3 1123	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ CMnd)
18		nn0ex 11649	. . . 4 ⊢ ℕ₀ ∈ V
19	18	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ℕ₀ ∈ V)
20	13	3adant3 1123	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring)
21	20	adantr 474	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ Ring)
22		simpl2 1201	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
23		simpl3 1203	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑀 ∈ 𝐵)
24		simpr 479	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
25		eqid 2778	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
26	1, 2, 3, 7, 25	decpmatcl 20979	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴))
27	22, 23, 24, 26	syl3anc 1439	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴))
28		eqid 2778	. . . . . 6 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
29	25, 8, 6, 4, 28, 5, 11	ply1tmcl 20038	. . . . 5 ⊢ ((𝐴 ∈ Ring ∧ (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴) ∧ 𝑘 ∈ ℕ₀) → ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) ∈ 𝐿)
30	21, 27, 24, 29	syl3anc 1439	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) ∈ 𝐿)
31	30	fmpttd 6649	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))):ℕ₀⟶𝐿)
32	8	ply1lmod 20018	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
33	20, 32	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ LMod)
34		eqidd 2779	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑄) = (Scalar‘𝑄))
35	8, 6, 28, 5, 11	ply1moncl 20037	. . . . 5 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
36	20, 35	sylan 575	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
37		eqid 2778	. . . 4 ⊢ (0_g‘(Scalar‘𝑄)) = (0_g‘(Scalar‘𝑄))
38		eqid 2778	. . . . . . 7 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
39	1, 2, 3, 7, 38	decpmatfsupp 20981	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑀 decompPMat 𝑘)) finSupp (0_g‘𝐴))
40	39	3adant1 1121	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑀 decompPMat 𝑘)) finSupp (0_g‘𝐴))
41	8	ply1sca 20019	. . . . . . . 8 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
42	41	eqcomd 2784	. . . . . . 7 ⊢ (𝐴 ∈ Ring → (Scalar‘𝑄) = 𝐴)
43	20, 42	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑄) = 𝐴)
44	43	fveq2d 6450	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0_g‘(Scalar‘𝑄)) = (0_g‘𝐴))
45	40, 44	breqtrrd 4914	. . . 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 19320	. . 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 18702	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ 𝐿)
48	10, 47	eqeltrd 2859	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝐿)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 Vcvv 3398 class class class wbr 4886 ↦ cmpt 4965 ‘cfv 6135 (class class class)co 6922 Fincfn 8241 finSupp cfsupp 8563 ℕ₀cn0 11642 Basecbs 16255 Scalarcsca 16341 ·_𝑠 cvsca 16342 0_gc0g 16486 Σ_g cgsu 16487 ._gcmg 17927 CMndccmn 18579 mulGrpcmgp 18876 Ringcrg 18934 LModclmod 19255 var₁cv1 19942 Poly₁cpl1 19943 Mat cmat 20617 decompPMat cdecpmat 20974 pMatToMatPoly cpm2mp 21004

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-ofr 7175 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-hom 16362 df-cco 16363 df-0g 16488 df-gsum 16489 df-prds 16494 df-pws 16496 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-ghm 18042 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-subrg 19170 df-lmod 19257 df-lss 19325 df-sra 19569 df-rgmod 19570 df-psr 19753 df-mvr 19754 df-mpl 19755 df-opsr 19757 df-psr1 19946 df-vr1 19947 df-ply1 19948 df-coe1 19949 df-dsmm 20475 df-frlm 20490 df-mamu 20594 df-mat 20618 df-decpmat 20975 df-pm2mp 21005

This theorem is referenced by: pm2mpf 21010 pm2mpf1 21011

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