cpmadumatpolylem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cpmadumatpolylem2		Structured version Visualization version GIF version

Theorem cpmadumatpolylem2 22777

Description: Lemma 2 for cpmadumatpoly 22778. (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 6906	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (0_g‘𝐴) ∈ V)
2		crngring 20178	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	anim2i 616	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
4	3	3adant3 1130	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
5	4	ad2antrr 725	. . 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 22617	. . 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 22750	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶𝑆)
19	2, 18	syl3anl2 1411	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶𝑆)
20	19	anassrs 467	. 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 22647	. . 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		ssidd 4001	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑆 ⊆ 𝑆)
25		nn0ex 12502	. . 3 ⊢ ℕ₀ ∈ V
26	25	a1i 11	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → ℕ₀ ∈ V)
27	6	ovexi 7448	. . 3 ⊢ 𝑆 ∈ V
28	27	a1i 11	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑆 ∈ V)
29	11, 12, 7, 8, 13, 14, 15, 16, 17	chfacffsupp 22751	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺 finSupp (0_g‘𝑌))
30	29	anassrs 467	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝐺 finSupp (0_g‘𝑌))
31		eqid 2728	. . . . . 6 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
32		eqid 2728	. . . . . 6 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
33	11, 21, 7, 8, 31, 32	m2cpminv0 22656	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
34	2, 33	sylan2 592	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
35	34	3adant3 1130	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
36	35	ad2antrr 725	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
37	1, 10, 20, 23, 24, 26, 28, 30, 36	fsuppcor 9421	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 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ifcif 4524 class class class wbr 5142 ↦ cmpt 5225 ∘ ccom 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ↑_m cmap 8838 Fincfn 8957 finSupp cfsupp 9379 0cc0 11132 1c1 11133 + caddc 11135 < clt 11272 − cmin 11468 ℕcn 12236 ℕ₀cn0 12496 ...cfz 13510 Basecbs 17173 ._rcmulr 17227 ·_𝑠 cvsca 17230 0_gc0g 17414 -_gcsg 18885 ._gcmg 19016 mulGrpcmgp 20067 1_rcur 20114 Ringcrg 20166 CRingccrg 20167 var₁cv1 22088 Poly₁cpl1 22089 Mat cmat 22300 maAdju cmadu 22527 ConstPolyMat ccpmat 22598 matToPolyMat cmat2pmat 22599 cPolyMatToMat ccpmat2mat 22600

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 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 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-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-hom 17250 df-cco 17251 df-0g 17416 df-gsum 17417 df-prds 17422 df-pws 17424 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-mulg 19017 df-subg 19071 df-ghm 19161 df-cntz 19261 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-srg 20120 df-ring 20168 df-cring 20169 df-subrng 20476 df-subrg 20501 df-lmod 20738 df-lss 20809 df-sra 21051 df-rgmod 21052 df-dsmm 21659 df-frlm 21674 df-ascl 21782 df-psr 21835 df-mvr 21836 df-mpl 21837 df-opsr 21839 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-coe1 22095 df-mamu 22279 df-mat 22301 df-cpmat 22601 df-mat2pmat 22602 df-cpmat2mat 22603

This theorem is referenced by: cpmadumatpoly 22778 chcoeffeqlem 22780

W3C validator

	
		OSZAR »