pm2mpcl - Metamath Proof Explorer

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Theorem pm2mpcl 22684

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 22683	. 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 2729	. . 3 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
13	7	matring 22330	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
14	8	ply1ring 22132	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
15		ringcmn 20191	. . . . 5 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
16	13, 14, 15	3syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
17	16	3adant3 1132	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ CMnd)
18		nn0ex 12448	. . . 4 ⊢ ℕ₀ ∈ V
19	18	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ℕ₀ ∈ V)
20	13	3adant3 1132	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring)
21	20	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ Ring)
22		simpl2 1193	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
23		simpl3 1194	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑀 ∈ 𝐵)
24		simpr 484	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
25		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
26	1, 2, 3, 7, 25	decpmatcl 22654	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴))
27	22, 23, 24, 26	syl3anc 1373	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴))
28		eqid 2729	. . . . . 6 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
29	25, 8, 6, 4, 28, 5, 11	ply1tmcl 22158	. . . . 5 ⊢ ((𝐴 ∈ Ring ∧ (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴) ∧ 𝑘 ∈ ℕ₀) → ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) ∈ 𝐿)
30	21, 27, 24, 29	syl3anc 1373	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) ∈ 𝐿)
31	30	fmpttd 7087	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))):ℕ₀⟶𝐿)
32	8	ply1lmod 22136	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
33	20, 32	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ LMod)
34		eqidd 2730	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑄) = (Scalar‘𝑄))
35	8, 6, 28, 5, 11	ply1moncl 22157	. . . . 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 2729	. . . 4 ⊢ (0_g‘(Scalar‘𝑄)) = (0_g‘(Scalar‘𝑄))
38		eqid 2729	. . . . . . 7 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
39	1, 2, 3, 7, 38	decpmatfsupp 22656	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑀 decompPMat 𝑘)) finSupp (0_g‘𝐴))
40	39	3adant1 1130	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑀 decompPMat 𝑘)) finSupp (0_g‘𝐴))
41	8	ply1sca 22137	. . . . . . . 8 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
42	41	eqcomd 2735	. . . . . . 7 ⊢ (𝐴 ∈ Ring → (Scalar‘𝑄) = 𝐴)
43	20, 42	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑄) = 𝐴)
44	43	fveq2d 6862	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0_g‘(Scalar‘𝑄)) = (0_g‘𝐴))
45	40, 44	breqtrrd 5135	. . . 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 20833	. . 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 19845	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ 𝐿)
48	10, 47	eqeltrd 2828	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝐿)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 Vcvv 3447 class class class wbr 5107 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 Fincfn 8918 finSupp cfsupp 9312 ℕ₀cn0 12442 Basecbs 17179 Scalarcsca 17223 ·_𝑠 cvsca 17224 0_gc0g 17402 Σ_g cgsu 17403 ._gcmg 18999 CMndccmn 19710 mulGrpcmgp 20049 Ringcrg 20142 LModclmod 20766 var₁cv1 22060 Poly₁cpl1 22061 Mat cmat 22294 decompPMat cdecpmat 22649 pMatToMatPoly cpm2mp 22679

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-ot 4598 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-ghm 19145 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-subrng 20455 df-subrg 20479 df-lmod 20768 df-lss 20838 df-sra 21080 df-rgmod 21081 df-dsmm 21641 df-frlm 21656 df-psr 21818 df-mvr 21819 df-mpl 21820 df-opsr 21822 df-psr1 22064 df-vr1 22065 df-ply1 22066 df-coe1 22067 df-mamu 22278 df-mat 22295 df-decpmat 22650 df-pm2mp 22680

This theorem is referenced by: pm2mpf 22685 pm2mpf1 22686

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