pm2mpcl - Metamath Proof Explorer

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Theorem pm2mpcl 22690

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 22689	. 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 2730	. . 3 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
13	7	matring 22336	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
14	8	ply1ring 22138	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
15		ringcmn 20197	. . . . 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 12454	. . . 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 2730	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
26	1, 2, 3, 7, 25	decpmatcl 22660	. . . . . 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 2730	. . . . . 6 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
29	25, 8, 6, 4, 28, 5, 11	ply1tmcl 22164	. . . . 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 7089	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))):ℕ₀⟶𝐿)
32	8	ply1lmod 22142	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
33	20, 32	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ LMod)
34		eqidd 2731	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑄) = (Scalar‘𝑄))
35	8, 6, 28, 5, 11	ply1moncl 22163	. . . . 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 2730	. . . 4 ⊢ (0_g‘(Scalar‘𝑄)) = (0_g‘(Scalar‘𝑄))
38		eqid 2730	. . . . . . 7 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
39	1, 2, 3, 7, 38	decpmatfsupp 22662	. . . . . 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 22143	. . . . . . . 8 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
42	41	eqcomd 2736	. . . . . . 7 ⊢ (𝐴 ∈ Ring → (Scalar‘𝑄) = 𝐴)
43	20, 42	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑄) = 𝐴)
44	43	fveq2d 6864	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0_g‘(Scalar‘𝑄)) = (0_g‘𝐴))
45	40, 44	breqtrrd 5137	. . . 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 20839	. . 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 19851	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ 𝐿)
48	10, 47	eqeltrd 2829	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 3450 class class class wbr 5109 ↦ cmpt 5190 ‘cfv 6513 (class class class)co 7389 Fincfn 8920 finSupp cfsupp 9318 ℕ₀cn0 12448 Basecbs 17185 Scalarcsca 17229 ·_𝑠 cvsca 17230 0_gc0g 17408 Σ_g cgsu 17409 ._gcmg 19005 CMndccmn 19716 mulGrpcmgp 20055 Ringcrg 20148 LModclmod 20772 var₁cv1 22066 Poly₁cpl1 22067 Mat cmat 22300 decompPMat cdecpmat 22655 pMatToMatPoly cpm2mp 22685

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 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-ofr 7656 df-om 7845 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-er 8673 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9319 df-sup 9399 df-oi 9469 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-z 12536 df-dec 12656 df-uz 12800 df-fz 13475 df-fzo 13622 df-seq 13973 df-hash 14302 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-hom 17250 df-cco 17251 df-0g 17410 df-gsum 17411 df-prds 17416 df-pws 17418 df-mre 17553 df-mrc 17554 df-acs 17556 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18716 df-submnd 18717 df-grp 18874 df-minusg 18875 df-sbg 18876 df-mulg 19006 df-subg 19061 df-ghm 19151 df-cntz 19255 df-cmn 19718 df-abl 19719 df-mgp 20056 df-rng 20068 df-ur 20097 df-ring 20150 df-subrng 20461 df-subrg 20485 df-lmod 20774 df-lss 20844 df-sra 21086 df-rgmod 21087 df-dsmm 21647 df-frlm 21662 df-psr 21824 df-mvr 21825 df-mpl 21826 df-opsr 21828 df-psr1 22070 df-vr1 22071 df-ply1 22072 df-coe1 22073 df-mamu 22284 df-mat 22301 df-decpmat 22656 df-pm2mp 22686

This theorem is referenced by: pm2mpf 22691 pm2mpf1 22692

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