pm2mpcoe1 - Metamath Proof Explorer

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Theorem pm2mpcoe1 21949

Description: A coefficient of the polynomial over matrices which is the result of the transformation of a polynomial matrix is the matrix consisting of the coefficients in the polynomial entries of the polynomial matrix. (Contributed by AV, 20-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)

**Assertion**
Ref	Expression
pm2mpcoe1	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ((coe₁‘(𝑇‘𝑀))‘𝐾) = (𝑀 decompPMat 𝐾))

Proof of Theorem pm2mpcoe1 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 764	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → 𝑁 ∈ Fin)
2		simplr 766	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → 𝑅 ∈ Ring)
3		simprl 768	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → 𝑀 ∈ 𝐵)
4		pm2mpval.p	. . . . . 6 ⊢ 𝑃 = (Poly₁‘𝑅)
5		pm2mpval.c	. . . . . 6 ⊢ 𝐶 = (𝑁 Mat 𝑃)
6		pm2mpval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
7		pm2mpval.m	. . . . . 6 ⊢ ∗ = ( ·_𝑠 ‘𝑄)
8		pm2mpval.e	. . . . . 6 ⊢ ↑ = (._g‘(mulGrp‘𝑄))
9		pm2mpval.x	. . . . . 6 ⊢ 𝑋 = (var₁‘𝐴)
10		pm2mpval.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
11		pm2mpval.q	. . . . . 6 ⊢ 𝑄 = (Poly₁‘𝐴)
12		pm2mpval.t	. . . . . 6 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
13	4, 5, 6, 7, 8, 9, 10, 11, 12	pm2mpfval 21945	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
14	1, 2, 3, 13	syl3anc 1370	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → (𝑇‘𝑀) = (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
15	14	fveq2d 6778	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → (coe₁‘(𝑇‘𝑀)) = (coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))))
16	15	fveq1d 6776	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ((coe₁‘(𝑇‘𝑀))‘𝐾) = ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))‘𝐾))
17		eqid 2738	. . 3 ⊢ (Base‘𝑄) = (Base‘𝑄)
18	10	matring 21592	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19	18	adantr 481	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → 𝐴 ∈ Ring)
20		eqid 2738	. . 3 ⊢ (Base‘𝐴) = (Base‘𝐴)
21		eqid 2738	. . 3 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
22	2	adantr 481	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
23	3	adantr 481	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) ∧ 𝑘 ∈ ℕ₀) → 𝑀 ∈ 𝐵)
24		simpr 485	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
25	4, 5, 6, 10, 20	decpmatcl 21916	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴))
26	22, 23, 24, 25	syl3anc 1370	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) ∧ 𝑘 ∈ ℕ₀) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴))
27	26	ralrimiva 3103	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ∀𝑘 ∈ ℕ₀ (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴))
28	4, 5, 6, 10, 21	decpmatfsupp 21918	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑀 decompPMat 𝑘)) finSupp (0_g‘𝐴))
29	28	ad2ant2lr 745	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → (𝑘 ∈ ℕ₀ ↦ (𝑀 decompPMat 𝑘)) finSupp (0_g‘𝐴))
30		simpr 485	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀) → 𝐾 ∈ ℕ₀)
31	30	adantl 482	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → 𝐾 ∈ ℕ₀)
32	11, 17, 9, 8, 19, 20, 7, 21, 27, 29, 31	gsummoncoe1 21475	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))‘𝐾) = ⦋𝐾 / 𝑘⦌(𝑀 decompPMat 𝑘))
33		csbov2g 7321	. . . . 5 ⊢ (𝐾 ∈ ℕ₀ → ⦋𝐾 / 𝑘⦌(𝑀 decompPMat 𝑘) = (𝑀 decompPMat ⦋𝐾 / 𝑘⦌𝑘))
34		csbvarg 4365	. . . . . 6 ⊢ (𝐾 ∈ ℕ₀ → ⦋𝐾 / 𝑘⦌𝑘 = 𝐾)
35	34	oveq2d 7291	. . . . 5 ⊢ (𝐾 ∈ ℕ₀ → (𝑀 decompPMat ⦋𝐾 / 𝑘⦌𝑘) = (𝑀 decompPMat 𝐾))
36	33, 35	eqtrd 2778	. . . 4 ⊢ (𝐾 ∈ ℕ₀ → ⦋𝐾 / 𝑘⦌(𝑀 decompPMat 𝑘) = (𝑀 decompPMat 𝐾))
37	36	adantl 482	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀) → ⦋𝐾 / 𝑘⦌(𝑀 decompPMat 𝑘) = (𝑀 decompPMat 𝐾))
38	37	adantl 482	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ⦋𝐾 / 𝑘⦌(𝑀 decompPMat 𝑘) = (𝑀 decompPMat 𝐾))
39	16, 32, 38	3eqtrd 2782	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ((coe₁‘(𝑇‘𝑀))‘𝐾) = (𝑀 decompPMat 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ⦋csb 3832 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 finSupp cfsupp 9128 ℕ₀cn0 12233 Basecbs 16912 ·_𝑠 cvsca 16966 0_gc0g 17150 Σ_g cgsu 17151 ._gcmg 18700 mulGrpcmgp 19720 Ringcrg 19783 var₁cv1 21347 Poly₁cpl1 21348 coe₁cco1 21349 Mat cmat 21554 decompPMat cdecpmat 21911 pMatToMatPoly cpm2mp 21941

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-subrg 20022 df-lmod 20125 df-lss 20194 df-sra 20434 df-rgmod 20435 df-dsmm 20939 df-frlm 20954 df-psr 21112 df-mvr 21113 df-mpl 21114 df-opsr 21116 df-psr1 21351 df-vr1 21352 df-ply1 21353 df-coe1 21354 df-mamu 21533 df-mat 21555 df-decpmat 21912 df-pm2mp 21942

This theorem is referenced by: (None)

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