pmatcollpw3fi - Metamath Proof Explorer

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Theorem pmatcollpw3fi 22674

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 22671	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ₀ 𝑀 = (𝐶 Σ_g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))
9		elnn0uz 12889	. . . . . 6 ⊢ (𝑠 ∈ ℕ₀ ↔ 𝑠 ∈ (ℤ_≥‘0))
10		fzn0 13539	. . . . . 6 ⊢ ((0...𝑠) ≠ ∅ ↔ 𝑠 ∈ (ℤ_≥‘0))
11	9, 10	sylbb2 237	. . . . 5 ⊢ (𝑠 ∈ ℕ₀ → (0...𝑠) ≠ ∅)
12		fz0ssnn0 13620	. . . . 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 22672	. . . 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 3163	. 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 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∃wrex 3065 ⊆ wss 3944 ∅c0 4318 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 ↑_m cmap 8836 Fincfn 8955 0cc0 11130 ℕ₀cn0 12494 ℤ_≥cuz 12844 ...cfz 13508 Basecbs 17171 ·_𝑠 cvsca 17228 Σ_g cgsu 17413 ._gcmg 19014 mulGrpcmgp 20065 CRingccrg 20165 var₁cv1 22082 Poly₁cpl1 22083 Mat cmat 22294 matToPolyMat cmat2pmat 22593 decompPMat cdecpmat 22651

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-cur 8266 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-fzo 13652 df-seq 13991 df-hash 14314 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-ghm 19159 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-srg 20118 df-ring 20166 df-cring 20167 df-subrng 20472 df-subrg 20497 df-lmod 20734 df-lss 20805 df-sra 21047 df-rgmod 21048 df-dsmm 21653 df-frlm 21668 df-assa 21774 df-ascl 21776 df-psr 21829 df-mvr 21830 df-mpl 21831 df-opsr 21833 df-psr1 22086 df-vr1 22087 df-ply1 22088 df-coe1 22089 df-mamu 22273 df-mat 22295 df-mat2pmat 22596 df-decpmat 22652

This theorem is referenced by: pmatcollpw3fi1 22677

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