pmatcollpw3fi - Metamath Proof Explorer

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Theorem pmatcollpw3fi 22609

Description: Write a polynomial matrix (over a commutative ring) as a finite sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 4-Nov-2019.) (Revised by AV, 4-Dec-2019.) (Proof shortened by AV, 8-Dec-2019.)

**Hypotheses**
Ref	Expression
pmatcollpw.p	⊢ 𝑃 = (Poly₁‘𝑅)
pmatcollpw.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pmatcollpw.b	⊢ 𝐵 = (Base‘𝐶)
pmatcollpw.m	⊢ ∗ = ( ·_𝑠 ‘𝐶)
pmatcollpw.e	⊢ ↑ = (._g‘(mulGrp‘𝑃))
pmatcollpw.x	⊢ 𝑋 = (var₁‘𝑅)
pmatcollpw.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
pmatcollpw3.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
pmatcollpw3.d	⊢ 𝐷 = (Base‘𝐴)

**Assertion**
Ref	Expression
pmatcollpw3fi	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ₀ ∃𝑓 ∈ (𝐷 ↑_m (0...𝑠))𝑀 = (𝐶 Σ_g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))

Distinct variable groups: 𝐵,𝑛 𝑛,𝑀 𝑛,𝑁 𝑃,𝑛 𝑅,𝑛 𝑛,𝑋 ↑ ,𝑛 𝐵,𝑠,𝑛 𝐶,𝑛 𝑀,𝑠 𝑁,𝑠 𝑅,𝑠 𝐵,𝑓 𝐶,𝑓,𝑛 𝐷,𝑓 𝑓,𝑀 𝑓,𝑁 𝑅,𝑓 𝑇,𝑓 𝑓,𝑋 ↑ ,𝑓 ∗ ,𝑓 𝑓,𝑠 Allowed substitution hints: 𝐴(𝑓,𝑛,𝑠) 𝐶(𝑠) 𝐷(𝑛,𝑠) 𝑃(𝑓,𝑠) 𝑇(𝑛,𝑠) ↑ (𝑠) ∗ (𝑛,𝑠) 𝑋(𝑠)

**Proof of Theorem pmatcollpw3fi**
Step	Hyp	Ref	Expression
1		pmatcollpw.p	. . 3 ⊢ 𝑃 = (Poly₁‘𝑅)
2		pmatcollpw.c	. . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃)
3		pmatcollpw.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
4		pmatcollpw.m	. . 3 ⊢ ∗ = ( ·_𝑠 ‘𝐶)
5		pmatcollpw.e	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑃))
6		pmatcollpw.x	. . 3 ⊢ 𝑋 = (var₁‘𝑅)
7		pmatcollpw.t	. . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
8	1, 2, 3, 4, 5, 6, 7	pmatcollpwfi 22606	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ₀ 𝑀 = (𝐶 Σ_g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))
9		elnn0uz 12864	. . . . . 6 ⊢ (𝑠 ∈ ℕ₀ ↔ 𝑠 ∈ (ℤ_≥‘0))
10		fzn0 13512	. . . . . 6 ⊢ ((0...𝑠) ≠ ∅ ↔ 𝑠 ∈ (ℤ_≥‘0))
11	9, 10	sylbb2 237	. . . . 5 ⊢ (𝑠 ∈ ℕ₀ → (0...𝑠) ≠ ∅)
12		fz0ssnn0 13593	. . . . 5 ⊢ (0...𝑠) ⊆ ℕ₀
13	11, 12	jctil 519	. . . 4 ⊢ (𝑠 ∈ ℕ₀ → ((0...𝑠) ⊆ ℕ₀ ∧ (0...𝑠) ≠ ∅))
14		pmatcollpw3.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
15		pmatcollpw3.d	. . . . 5 ⊢ 𝐷 = (Base‘𝐴)
16	1, 2, 3, 4, 5, 6, 7, 14, 15	pmatcollpw3lem 22607	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ ((0...𝑠) ⊆ ℕ₀ ∧ (0...𝑠) ≠ ∅)) → (𝑀 = (𝐶 Σ_g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑓 ∈ (𝐷 ↑_m (0...𝑠))𝑀 = (𝐶 Σ_g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))
17	13, 16	sylan2 592	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ₀) → (𝑀 = (𝐶 Σ_g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑓 ∈ (𝐷 ↑_m (0...𝑠))𝑀 = (𝐶 Σ_g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))
18	17	reximdva 3160	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ ℕ₀ 𝑀 = (𝐶 Σ_g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑠 ∈ ℕ₀ ∃𝑓 ∈ (𝐷 ↑_m (0...𝑠))𝑀 = (𝐶 Σ_g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))
19	8, 18	mpd 15	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ₀ ∃𝑓 ∈ (𝐷 ↑_m (0...𝑠))𝑀 = (𝐶 Σ_g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∃wrex 3062 ⊆ wss 3940 ∅c0 4314 ↦ cmpt 5221 ‘cfv 6533 (class class class)co 7401 ↑_m cmap 8816 Fincfn 8935 0cc0 11106 ℕ₀cn0 12469 ℤ_≥cuz 12819 ...cfz 13481 Basecbs 17143 ·_𝑠 cvsca 17200 Σ_g cgsu 17385 ._gcmg 18985 mulGrpcmgp 20029 CRingccrg 20129 var₁cv1 22018 Poly₁cpl1 22019 Mat cmat 22229 matToPolyMat cmat2pmat 22528 decompPMat cdecpmat 22586

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-ot 4629 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-cur 8247 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-srg 20082 df-ring 20130 df-cring 20131 df-subrng 20436 df-subrg 20461 df-lmod 20698 df-lss 20769 df-sra 21011 df-rgmod 21012 df-dsmm 21595 df-frlm 21610 df-assa 21716 df-ascl 21718 df-psr 21771 df-mvr 21772 df-mpl 21773 df-opsr 21775 df-psr1 22022 df-vr1 22023 df-ply1 22024 df-coe1 22025 df-mamu 22208 df-mat 22230 df-mat2pmat 22531 df-decpmat 22587

This theorem is referenced by: pmatcollpw3fi1 22612

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