m2pmfzgsumcl - Metamath Proof Explorer

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Theorem m2pmfzgsumcl 22710

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 2737	. 2 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		crngring 20223	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3		m2pmfzmap.p	. . . . . . . 8 ⊢ 𝑃 = (Poly₁‘𝑅)
4	3	ply1ring 22208	. . . . . . 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 22405	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ Ring)
8	5, 7	sylan2 594	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
9		ringcmn 20260	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
10	8, 9	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ CMnd)
11	10	3adant3 1133	. . 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 13932	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0...𝑠) ∈ Fin)
14		simpll1 1214	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑁 ∈ Fin)
15	5	3ad2ant2 1135	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
16	15	ad2antrr 727	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑃 ∈ Ring)
17	2	3ad2ant2 1135	. . . . . 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 13571	. . . . 5 ⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ₀)
20		m2pmfzmapfsupp.x	. . . . . 6 ⊢ 𝑋 = (var₁‘𝑅)
21		eqid 2737	. . . . . 6 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
22		m2pmfzmapfsupp.e	. . . . . 6 ⊢ ↑ = (._g‘(mulGrp‘𝑃))
23		eqid 2737	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
24	3, 20, 21, 22, 23	ply1moncl 22233	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ ℕ₀) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
25	18, 19, 24	syl2an 597	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
26	2	anim2i 618	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
27	26	3adant3 1133	. . . . . . 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 614	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ ℕ₀))
30		df-3an 1089	. . . . . 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 773	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))
33	32	anim1i 616	. . . . 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 22709	. . . . 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 686	. . . 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 22393	. . . 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 839	. . 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 3130	. 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 19939	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 1087 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5167 ‘cfv 6496 (class class class)co 7364 ↑_m cmap 8770 Fincfn 8890 0cc0 11035 ℕ₀cn0 12434 ...cfz 13458 Basecbs 17176 ·_𝑠 cvsca 17221 Σ_g cgsu 17400 ._gcmg 19040 CMndccmn 19752 mulGrpcmgp 20118 Ringcrg 20211 CRingccrg 20212 var₁cv1 22136 Poly₁cpl1 22137 Mat cmat 22369 matToPolyMat cmat2pmat 22666

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-ot 4577 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-se 5582 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-of 7628 df-ofr 7629 df-om 7815 df-1st 7939 df-2nd 7940 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-sup 9352 df-oi 9422 df-card 9860 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-nn 12172 df-2 12241 df-3 12242 df-4 12243 df-5 12244 df-6 12245 df-7 12246 df-8 12247 df-9 12248 df-n0 12435 df-z 12522 df-dec 12642 df-uz 12786 df-fz 13459 df-fzo 13606 df-seq 13961 df-hash 14290 df-struct 17114 df-sets 17131 df-slot 17149 df-ndx 17161 df-base 17177 df-ress 17198 df-plusg 17230 df-mulr 17231 df-sca 17233 df-vsca 17234 df-ip 17235 df-tset 17236 df-ple 17237 df-ds 17239 df-hom 17241 df-cco 17242 df-0g 17401 df-gsum 17402 df-prds 17407 df-pws 17409 df-mre 17545 df-mrc 17546 df-acs 17548 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18748 df-submnd 18749 df-grp 18909 df-minusg 18910 df-sbg 18911 df-mulg 19041 df-subg 19096 df-ghm 19185 df-cntz 19289 df-cmn 19754 df-abl 19755 df-mgp 20119 df-rng 20131 df-ur 20160 df-ring 20213 df-cring 20214 df-subrng 20520 df-subrg 20544 df-lmod 20854 df-lss 20924 df-sra 21165 df-rgmod 21166 df-dsmm 21709 df-frlm 21724 df-ascl 21832 df-psr 21886 df-mvr 21887 df-mpl 21888 df-opsr 21890 df-psr1 22140 df-vr1 22141 df-ply1 22142 df-mamu 22353 df-mat 22370 df-mat2pmat 22669

This theorem is referenced by: (None)

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