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Theorem cpmadumatpolylem2 20889

Description: Lemma 2 for cpmadumatpoly 20890. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)

**Hypotheses**
Ref	Expression
cpmadumatpoly.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cpmadumatpoly.b	⊢ 𝐵 = (Base‘𝐴)
cpmadumatpoly.p	⊢ 𝑃 = (Poly₁‘𝑅)
cpmadumatpoly.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cpmadumatpoly.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cpmadumatpoly.r	⊢ × = (._r‘𝑌)
cpmadumatpoly.m0	⊢ − = (-_g‘𝑌)
cpmadumatpoly.0	⊢ 0 = (0_g‘𝑌)
cpmadumatpoly.g	⊢ 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
cpmadumatpoly.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmadumatpoly.m1	⊢ · = ( ·_𝑠 ‘𝑌)
cpmadumatpoly.1	⊢ 1 = (1_r‘𝑌)
cpmadumatpoly.z	⊢ 𝑍 = (var₁‘𝑅)
cpmadumatpoly.d	⊢ 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀))
cpmadumatpoly.j	⊢ 𝐽 = (𝑁 maAdju 𝑃)
cpmadumatpoly.w	⊢ 𝑊 = (Base‘𝑌)
cpmadumatpoly.q	⊢ 𝑄 = (Poly₁‘𝐴)
cpmadumatpoly.x	⊢ 𝑋 = (var₁‘𝐴)
cpmadumatpoly.m2	⊢ ∗ = ( ·_𝑠 ‘𝑄)
cpmadumatpoly.e	⊢ ↑ = (._g‘(mulGrp‘𝑄))
cpmadumatpoly.u	⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)

**Assertion**
Ref	Expression
cpmadumatpolylem2	⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))

Distinct variable groups: 𝐵,𝑛 𝑛,𝑀 𝑛,𝑁 𝑅,𝑛 𝑆,𝑛 𝑛,𝑌 𝑛,𝑏 𝑛,𝑠 Allowed substitution hints: 𝐴(𝑛,𝑠,𝑏) 𝐵(𝑠,𝑏) 𝐷(𝑛,𝑠,𝑏) 𝑃(𝑛,𝑠,𝑏) 𝑄(𝑛,𝑠,𝑏) 𝑅(𝑠,𝑏) 𝑆(𝑠,𝑏) 𝑇(𝑛,𝑠,𝑏) · (𝑛,𝑠,𝑏) × (𝑛,𝑠,𝑏) 𝑈(𝑛,𝑠,𝑏) 1 (𝑛,𝑠,𝑏) ↑ (𝑛,𝑠,𝑏) 𝐺(𝑛,𝑠,𝑏) ∗ (𝑛,𝑠,𝑏) 𝐽(𝑛,𝑠,𝑏) 𝑀(𝑠,𝑏) − (𝑛,𝑠,𝑏) 𝑁(𝑠,𝑏) 𝑊(𝑛,𝑠,𝑏) 𝑋(𝑛,𝑠,𝑏) 𝑌(𝑠,𝑏) 0 (𝑛,𝑠,𝑏) 𝑍(𝑛,𝑠,𝑏)

**Proof of Theorem cpmadumatpolylem2**
Step	Hyp	Ref	Expression
1		fvexd 6364	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → (0_g‘𝐴) ∈ V)
2		crngring 18758	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	anim2i 594	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
4	3	3adant3 1127	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
5	4	ad2antrr 764	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
6		cpmadumatpoly.s	. . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
7		cpmadumatpoly.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
8		cpmadumatpoly.y	. . . 4 ⊢ 𝑌 = (𝑁 Mat 𝑃)
9	6, 7, 8	0elcpmat 20729	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0_g‘𝑌) ∈ 𝑆)
10	5, 9	syl 17	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → (0_g‘𝑌) ∈ 𝑆)
11		cpmadumatpoly.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
12		cpmadumatpoly.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
13		cpmadumatpoly.r	. . . . 5 ⊢ × = (._r‘𝑌)
14		cpmadumatpoly.m0	. . . . 5 ⊢ − = (-_g‘𝑌)
15		cpmadumatpoly.0	. . . . 5 ⊢ 0 = (0_g‘𝑌)
16		cpmadumatpoly.t	. . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
17		cpmadumatpoly.g	. . . . 5 ⊢ 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
18	11, 12, 7, 8, 13, 14, 15, 16, 17, 6	chfacfisfcpmat 20862	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠)))) → 𝐺:ℕ₀⟶𝑆)
19	2, 18	syl3anl2 1522	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠)))) → 𝐺:ℕ₀⟶𝑆)
20	19	anassrs 683	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → 𝐺:ℕ₀⟶𝑆)
21		cpmadumatpoly.u	. . . 4 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
22	11, 12, 6, 21	cpm2mf 20759	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:𝑆⟶𝐵)
23	5, 22	syl 17	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → 𝑈:𝑆⟶𝐵)
24		ssid 3765	. . 3 ⊢ 𝑆 ⊆ 𝑆
25	24	a1i 11	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → 𝑆 ⊆ 𝑆)
26		nn0ex 11490	. . 3 ⊢ ℕ₀ ∈ V
27	26	a1i 11	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → ℕ₀ ∈ V)
28		ovex 6841	. . . 4 ⊢ (𝑁 ConstPolyMat 𝑅) ∈ V
29	6, 28	eqeltri 2835	. . 3 ⊢ 𝑆 ∈ V
30	29	a1i 11	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → 𝑆 ∈ V)
31	11, 12, 7, 8, 13, 14, 15, 16, 17	chfacffsupp 20863	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠)))) → 𝐺 finSupp (0_g‘𝑌))
32	31	anassrs 683	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → 𝐺 finSupp (0_g‘𝑌))
33		eqid 2760	. . . . . 6 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
34		eqid 2760	. . . . . 6 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
35	11, 21, 7, 8, 33, 34	m2cpminv0 20768	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
36	2, 35	sylan2 492	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
37	36	3adant3 1127	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
38	37	ad2antrr 764	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
39	1, 10, 20, 23, 25, 27, 30, 32, 38	fsuppcor 8474	1 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ⊆ wss 3715 ifcif 4230 class class class wbr 4804 ↦ cmpt 4881 ∘ ccom 5270 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 ↑_𝑚 cmap 8023 Fincfn 8121 finSupp cfsupp 8440 0cc0 10128 1c1 10129 + caddc 10131 < clt 10266 − cmin 10458 ℕcn 11212 ℕ₀cn0 11484 ...cfz 12519 Basecbs 16059 ._rcmulr 16144 ·_𝑠 cvsca 16147 0_gc0g 16302 -_gcsg 17625 ._gcmg 17741 mulGrpcmgp 18689 1_rcur 18701 Ringcrg 18747 CRingccrg 18748 var₁cv1 19748 Poly₁cpl1 19749 Mat cmat 20415 maAdju cmadu 20640 ConstPolyMat ccpmat 20710 matToPolyMat cmat2pmat 20711 cPolyMatToMat ccpmat2mat 20712

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-ot 4330 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-ofr 7063 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-map 8025 df-pm 8026 df-ixp 8075 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-sup 8513 df-oi 8580 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-fz 12520 df-fzo 12660 df-seq 12996 df-hash 13312 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-hom 16168 df-cco 16169 df-0g 16304 df-gsum 16305 df-prds 16310 df-pws 16312 df-mre 16448 df-mrc 16449 df-acs 16451 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-mhm 17536 df-submnd 17537 df-grp 17626 df-minusg 17627 df-sbg 17628 df-mulg 17742 df-subg 17792 df-ghm 17859 df-cntz 17950 df-cmn 18395 df-abl 18396 df-mgp 18690 df-ur 18702 df-srg 18706 df-ring 18749 df-cring 18750 df-subrg 18980 df-lmod 19067 df-lss 19135 df-sra 19374 df-rgmod 19375 df-ascl 19516 df-psr 19558 df-mvr 19559 df-mpl 19560 df-opsr 19562 df-psr1 19752 df-vr1 19753 df-ply1 19754 df-coe1 19755 df-dsmm 20278 df-frlm 20293 df-mamu 20392 df-mat 20416 df-cpmat 20713 df-mat2pmat 20714 df-cpmat2mat 20715

This theorem is referenced by: cpmadumatpoly 20890 chcoeffeqlem 20892

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