pmatcollpw3fi - Metamath Proof Explorer

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Theorem pmatcollpw3fi 22718

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 22715	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ₀ 𝑀 = (𝐶 Σ_g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))
9		elnn0uz 12897	. . . . . 6 ⊢ (𝑠 ∈ ℕ₀ ↔ 𝑠 ∈ (ℤ_≥‘0))
10		fzn0 13547	. . . . . 6 ⊢ ((0...𝑠) ≠ ∅ ↔ 𝑠 ∈ (ℤ_≥‘0))
11	9, 10	sylbb2 237	. . . . 5 ⊢ (𝑠 ∈ ℕ₀ → (0...𝑠) ≠ ∅)
12		fz0ssnn0 13628	. . . . 5 ⊢ (0...𝑠) ⊆ ℕ₀
13	11, 12	jctil 518	. . . 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 22716	. . . 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 591	. . 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 3158	. 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∃wrex 3060 ⊆ wss 3945 ∅c0 4323 ↦ cmpt 5231 ‘cfv 6547 (class class class)co 7417 ↑_m cmap 8843 Fincfn 8962 0cc0 11138 ℕ₀cn0 12502 ℤ_≥cuz 12852 ...cfz 13516 Basecbs 17180 ·_𝑠 cvsca 17237 Σ_g cgsu 17422 ._gcmg 19028 mulGrpcmgp 20079 CRingccrg 20179 var₁cv1 22104 Poly₁cpl1 22105 Mat cmat 22338 matToPolyMat cmat2pmat 22637 decompPMat cdecpmat 22695

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 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-ofr 7684 df-om 7870 df-1st 7992 df-2nd 7993 df-supp 8164 df-cur 8271 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18898 df-minusg 18899 df-sbg 18900 df-mulg 19029 df-subg 19083 df-ghm 19173 df-cntz 19273 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-srg 20132 df-ring 20180 df-cring 20181 df-subrng 20488 df-subrg 20513 df-lmod 20750 df-lss 20821 df-sra 21063 df-rgmod 21064 df-dsmm 21671 df-frlm 21686 df-assa 21792 df-ascl 21794 df-psr 21847 df-mvr 21848 df-mpl 21849 df-opsr 21851 df-psr1 22108 df-vr1 22109 df-ply1 22110 df-coe1 22111 df-mamu 22322 df-mat 22339 df-mat2pmat 22640 df-decpmat 22696

This theorem is referenced by: pmatcollpw3fi1 22721

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