m2pmfzgsumcl - Metamath Proof Explorer

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Theorem m2pmfzgsumcl 22670

Description: Closure of the sum of scaled transformed matrices. (Contributed by AV, 4-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.)

**Hypotheses**
Ref	Expression
m2pmfzmap.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
m2pmfzmap.b	⊢ 𝐵 = (Base‘𝐴)
m2pmfzmap.p	⊢ 𝑃 = (Poly₁‘𝑅)
m2pmfzmap.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
m2pmfzmap.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
m2pmfzmapfsupp.x	⊢ 𝑋 = (var₁‘𝑅)
m2pmfzmapfsupp.e	⊢ ↑ = (._g‘(mulGrp‘𝑃))
m2pmfzgsumcl.m	⊢ · = ( ·_𝑠 ‘𝑌)

**Assertion**
Ref	Expression
m2pmfzgsumcl	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) ∈ (Base‘𝑌))

Distinct variable groups: 𝐵,𝑖 𝑖,𝑀 𝑖,𝑁 𝑅,𝑖 𝑖,𝑌 𝑖,𝑏 𝑖,𝑠 Allowed substitution hints: 𝐴(𝑖,𝑠,𝑏) 𝐵(𝑠,𝑏) 𝑃(𝑖,𝑠,𝑏) 𝑅(𝑠,𝑏) 𝑇(𝑖,𝑠,𝑏) · (𝑖,𝑠,𝑏) ↑ (𝑖,𝑠,𝑏) 𝑀(𝑠,𝑏) 𝑁(𝑠,𝑏) 𝑋(𝑖,𝑠,𝑏) 𝑌(𝑠,𝑏)

**Proof of Theorem m2pmfzgsumcl**
Step	Hyp	Ref	Expression
1		eqid 2729	. 2 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		crngring 20167	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3		m2pmfzmap.p	. . . . . . . 8 ⊢ 𝑃 = (Poly₁‘𝑅)
4	3	ply1ring 22167	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5	2, 4	syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
6		m2pmfzmap.y	. . . . . . 7 ⊢ 𝑌 = (𝑁 Mat 𝑃)
7	6	matring 22365	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ Ring)
8	5, 7	sylan2 593	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
9		ringcmn 20204	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
10	8, 9	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ CMnd)
11	10	3adant3 1132	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ CMnd)
12	11	adantr 480	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ CMnd)
13		fzfid 13917	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0...𝑠) ∈ Fin)
14		simpll1 1213	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑁 ∈ Fin)
15	5	3ad2ant2 1134	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
16	15	ad2antrr 726	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑃 ∈ Ring)
17	2	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
18	17	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑅 ∈ Ring)
19		elfznn0 13560	. . . . 5 ⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ₀)
20		m2pmfzmapfsupp.x	. . . . . 6 ⊢ 𝑋 = (var₁‘𝑅)
21		eqid 2729	. . . . . 6 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
22		m2pmfzmapfsupp.e	. . . . . 6 ⊢ ↑ = (._g‘(mulGrp‘𝑃))
23		eqid 2729	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
24	3, 20, 21, 22, 23	ply1moncl 22192	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ ℕ₀) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
25	18, 19, 24	syl2an 596	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
26	2	anim2i 617	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
27	26	3adant3 1132	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
28		simpl 482	. . . . . . 7 ⊢ ((𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑠 ∈ ℕ₀)
29	27, 28	anim12i 613	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ ℕ₀))
30		df-3an 1088	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ₀) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ ℕ₀))
31	29, 30	sylibr 234	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ₀))
32		simprr 772	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))
33	32	anim1i 615	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠)))
34		m2pmfzmap.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
35		m2pmfzmap.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
36		m2pmfzmap.t	. . . . . 6 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
37	34, 35, 3, 6, 36	m2pmfzmap 22669	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ₀) ∧ (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠))) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
38	31, 33, 37	syl2an2r 685	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
39		m2pmfzgsumcl.m	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝑌)
40	23, 6, 1, 39	matvscl 22353	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝑖 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
41	14, 16, 25, 38, 40	syl22anc 838	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
42	41	ralrimiva 3125	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑖 ∈ (0...𝑠)((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
43	1, 12, 13, 42	gsummptcl 19883	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) ∈ (Base‘𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ↦ cmpt 5183 ‘cfv 6500 (class class class)co 7370 ↑_m cmap 8777 Fincfn 8896 0cc0 11047 ℕ₀cn0 12421 ...cfz 13447 Basecbs 17157 ·_𝑠 cvsca 17202 Σ_g cgsu 17381 ._gcmg 18983 CMndccmn 19696 mulGrpcmgp 20062 Ringcrg 20155 CRingccrg 20156 var₁cv1 22095 Poly₁cpl1 22096 Mat cmat 22329 matToPolyMat cmat2pmat 22626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7692 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6453 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-ofr 7635 df-om 7824 df-1st 7948 df-2nd 7949 df-supp 8118 df-frecs 8238 df-wrecs 8269 df-recs 8318 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8649 df-map 8779 df-pm 8780 df-ixp 8849 df-en 8897 df-dom 8898 df-sdom 8899 df-fin 8900 df-fsupp 9290 df-sup 9370 df-oi 9440 df-card 9871 df-pnf 11189 df-mnf 11190 df-xr 11191 df-ltxr 11192 df-le 11193 df-sub 11386 df-neg 11387 df-nn 12166 df-2 12228 df-3 12229 df-4 12230 df-5 12231 df-6 12232 df-7 12233 df-8 12234 df-9 12235 df-n0 12422 df-z 12509 df-dec 12629 df-uz 12773 df-fz 13448 df-fzo 13595 df-seq 13946 df-hash 14275 df-struct 17095 df-sets 17112 df-slot 17130 df-ndx 17142 df-base 17158 df-ress 17179 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-ip 17216 df-tset 17217 df-ple 17218 df-ds 17220 df-hom 17222 df-cco 17223 df-0g 17382 df-gsum 17383 df-prds 17388 df-pws 17390 df-mre 17525 df-mrc 17526 df-acs 17528 df-mgm 18551 df-sgrp 18630 df-mnd 18646 df-mhm 18694 df-submnd 18695 df-grp 18852 df-minusg 18853 df-sbg 18854 df-mulg 18984 df-subg 19039 df-ghm 19129 df-cntz 19233 df-cmn 19698 df-abl 19699 df-mgp 20063 df-rng 20075 df-ur 20104 df-ring 20157 df-cring 20158 df-subrng 20468 df-subrg 20492 df-lmod 20802 df-lss 20872 df-sra 21114 df-rgmod 21115 df-dsmm 21676 df-frlm 21691 df-ascl 21799 df-psr 21853 df-mvr 21854 df-mpl 21855 df-opsr 21857 df-psr1 22099 df-vr1 22100 df-ply1 22101 df-mamu 22313 df-mat 22330 df-mat2pmat 22629

This theorem is referenced by: (None)

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