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Theorem cpmadumatpolylem2 20910

Description: Lemma 2 for cpmadumatpoly 20911. (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 6366	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_𝑚 (0...𝑠))) → (0_g‘𝐴) ∈ V)
2		crngring 18779	. . . . . 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 20750	. . 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 20883	. . . 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 20780	. . 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 3766	. . 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 11511	. . 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 6843	. . . 4 ⊢ (𝑁 ConstPolyMat 𝑅) ∈ V
29	6, 28	eqeltri 2836	. . 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 20884	. . 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 2761	. . . . . 6 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
34		eqid 2761	. . . . . 6 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
35	11, 21, 7, 8, 33, 34	m2cpminv0 20789	. . . . 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 8477	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 2140 Vcvv 3341 ⊆ wss 3716 ifcif 4231 class class class wbr 4805 ↦ cmpt 4882 ∘ ccom 5271 ⟶wf 6046 ‘cfv 6050 (class class class)co 6815 ↑_𝑚 cmap 8026 Fincfn 8124 finSupp cfsupp 8443 0cc0 10149 1c1 10150 + caddc 10152 < clt 10287 − cmin 10479 ℕcn 11233 ℕ₀cn0 11505 ...cfz 12540 Basecbs 16080 ._rcmulr 16165 ·_𝑠 cvsca 16168 0_gc0g 16323 -_gcsg 17646 ._gcmg 17762 mulGrpcmgp 18710 1_rcur 18722 Ringcrg 18768 CRingccrg 18769 var₁cv1 19769 Poly₁cpl1 19770 Mat cmat 20436 maAdju cmadu 20661 ConstPolyMat ccpmat 20731 matToPolyMat cmat2pmat 20732 cPolyMatToMat ccpmat2mat 20733

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 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-rep 4924 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-inf2 8714 ax-cnex 10205 ax-resscn 10206 ax-1cn 10207 ax-icn 10208 ax-addcl 10209 ax-addrcl 10210 ax-mulcl 10211 ax-mulrcl 10212 ax-mulcom 10213 ax-addass 10214 ax-mulass 10215 ax-distr 10216 ax-i2m1 10217 ax-1ne0 10218 ax-1rid 10219 ax-rnegex 10220 ax-rrecex 10221 ax-cnre 10222 ax-pre-lttri 10223 ax-pre-lttrn 10224 ax-pre-ltadd 10225 ax-pre-mulgt0 10226

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 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-ot 4331 df-uni 4590 df-int 4629 df-iun 4675 df-iin 4676 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-se 5227 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-isom 6059 df-riota 6776 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-of 7064 df-ofr 7065 df-om 7233 df-1st 7335 df-2nd 7336 df-supp 7466 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-1o 7731 df-2o 7732 df-oadd 7735 df-er 7914 df-map 8028 df-pm 8029 df-ixp 8078 df-en 8125 df-dom 8126 df-sdom 8127 df-fin 8128 df-fsupp 8444 df-sup 8516 df-oi 8583 df-card 8976 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-sub 10481 df-neg 10482 df-nn 11234 df-2 11292 df-3 11293 df-4 11294 df-5 11295 df-6 11296 df-7 11297 df-8 11298 df-9 11299 df-n0 11506 df-z 11591 df-dec 11707 df-uz 11901 df-fz 12541 df-fzo 12681 df-seq 13017 df-hash 13333 df-struct 16082 df-ndx 16083 df-slot 16084 df-base 16086 df-sets 16087 df-ress 16088 df-plusg 16177 df-mulr 16178 df-sca 16180 df-vsca 16181 df-ip 16182 df-tset 16183 df-ple 16184 df-ds 16187 df-hom 16189 df-cco 16190 df-0g 16325 df-gsum 16326 df-prds 16331 df-pws 16333 df-mre 16469 df-mrc 16470 df-acs 16472 df-mgm 17464 df-sgrp 17506 df-mnd 17517 df-mhm 17557 df-submnd 17558 df-grp 17647 df-minusg 17648 df-sbg 17649 df-mulg 17763 df-subg 17813 df-ghm 17880 df-cntz 17971 df-cmn 18416 df-abl 18417 df-mgp 18711 df-ur 18723 df-srg 18727 df-ring 18770 df-cring 18771 df-subrg 19001 df-lmod 19088 df-lss 19156 df-sra 19395 df-rgmod 19396 df-ascl 19537 df-psr 19579 df-mvr 19580 df-mpl 19581 df-opsr 19583 df-psr1 19773 df-vr1 19774 df-ply1 19775 df-coe1 19776 df-dsmm 20299 df-frlm 20314 df-mamu 20413 df-mat 20437 df-cpmat 20734 df-mat2pmat 20735 df-cpmat2mat 20736

This theorem is referenced by: cpmadumatpoly 20911 chcoeffeqlem 20913

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