m2pmfzgsumcl - Metamath Proof Explorer

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Theorem m2pmfzgsumcl 22884

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 2761	. 2 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		crngring 20326	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3		m2pmfzmap.p	. . . . . . . 8 ⊢ 𝑃 = (Poly₁‘𝑅)
4	3	ply1ring 22386	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5	2, 4	syl 18	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
6		m2pmfzmap.y	. . . . . . 7 ⊢ 𝑌 = (𝑁 Mat 𝑃)
7	6	matring 22579	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ Ring)
8	5, 7	sylan2 604	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
9		ringcmn 20364	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
10	8, 9	syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ CMnd)
11	10	3adant3 1148	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ CMnd)
12	11	adantr 485	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ CMnd)
13		fzfid 14008	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0...𝑠) ∈ Fin)
14		simpll1 1229	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑁 ∈ Fin)
15	5	3ad2ant2 1150	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
16	15	ad2antrr 738	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑃 ∈ Ring)
17	2	3ad2ant2 1150	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
18	17	adantr 485	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑅 ∈ Ring)
19		elfznn0 13647	. . . . 5 ⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ₀)
20		m2pmfzmapfsupp.x	. . . . . 6 ⊢ 𝑋 = (var₁‘𝑅)
21		eqid 2761	. . . . . 6 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
22		m2pmfzmapfsupp.e	. . . . . 6 ⊢ ↑ = (._g‘(mulGrp‘𝑃))
23		eqid 2761	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
24	3, 20, 21, 22, 23	ply1moncl 22411	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ ℕ₀) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
25	18, 19, 24	syl2an 607	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
26	2	anim2i 628	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
27	26	3adant3 1148	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
28		simpl 487	. . . . . . 7 ⊢ ((𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑠 ∈ ℕ₀)
29	27, 28	anim12i 624	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ ℕ₀))
30		df-3an 1103	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ₀) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ ℕ₀))
31	29, 30	sylibr 237	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ₀))
32		simprr 784	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))
33	32	anim1i 626	. . . . 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 22883	. . . . 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 697	. . . 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 22567	. . . 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 851	. . 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 3155	. 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 20036	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 400 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ↦ cmpt 5191 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8823 Fincfn 8942 0cc0 11099 ℕ₀cn0 12503 ...cfz 13534 Basecbs 17268 ·_𝑠 cvsca 17313 Σ_g cgsu 17492 ._gcmg 19132 CMndccmn 19849 mulGrpcmgp 20215 Ringcrg 20314 CRingccrg 20315 var₁cv1 22315 Poly₁cpl1 22316 Mat cmat 22543 matToPolyMat cmat2pmat 22840

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-ot 4597 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-ofr 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-sup 9401 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-fz 13535 df-fzo 13682 df-seq 14037 df-hash 14366 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-hom 17333 df-cco 17334 df-0g 17493 df-gsum 17494 df-prds 17499 df-pws 17501 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-submnd 18841 df-grp 19002 df-minusg 19003 df-sbg 19004 df-mulg 19133 df-subg 19188 df-ghm 19283 df-cntz 19386 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-cring 20317 df-subrng 20630 df-subrg 20654 df-lmod 20962 df-lss 21032 df-sra 21273 df-rgmod 21274 df-dsmm 21861 df-frlm 21876 df-ascl 21984 df-psr 22038 df-mvr 22039 df-mpl 22040 df-opsr 22042 df-psr1 22319 df-vr1 22320 df-ply1 22321 df-mamu 22527 df-mat 22544 df-mat2pmat 22843

This theorem is referenced by: (None)

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