m2pmfzgsumcl - Metamath Proof Explorer

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Theorem m2pmfzgsumcl 22974

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 2762	. 2 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		crngring 20385	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3		m2pmfzmap.p	. . . . . . . 8 ⊢ 𝑃 = (Poly₁‘𝑅)
4	3	ply1ring 22473	. . . . . . 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 22666	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ Ring)
8	5, 7	sylan2 605	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
9		ringcmn 20424	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
10	8, 9	syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ CMnd)
11	10	3adant3 1150	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ CMnd)
12	11	adantr 486	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ CMnd)
13		fzfid 14039	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0...𝑠) ∈ Fin)
14		simpll1 1231	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑁 ∈ Fin)
15	5	3ad2ant2 1152	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
16	15	ad2antrr 739	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑃 ∈ Ring)
17	2	3ad2ant2 1152	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
18	17	adantr 486	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑅 ∈ Ring)
19		elfznn0 13677	. . . . 5 ⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ₀)
20		m2pmfzmapfsupp.x	. . . . . 6 ⊢ 𝑋 = (var₁‘𝑅)
21		eqid 2762	. . . . . 6 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
22		m2pmfzmapfsupp.e	. . . . . 6 ⊢ ↑ = (._g‘(mulGrp‘𝑃))
23		eqid 2762	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
24	3, 20, 21, 22, 23	ply1moncl 22498	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ ℕ₀) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
25	18, 19, 24	syl2an 608	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
26	2	anim2i 629	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
27	26	3adant3 1150	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
28		simpl 488	. . . . . . 7 ⊢ ((𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑠 ∈ ℕ₀)
29	27, 28	anim12i 625	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ ℕ₀))
30		df-3an 1105	. . . . . 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 785	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))
33	32	anim1i 627	. . . . 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 22973	. . . . 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 698	. . . 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 22654	. . . 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 852	. . 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 3156	. 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 20095	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ↦ cmpt 5190 ‘cfv 6537 (class class class)co 7416 ↑_m cmap 8829 Fincfn 8955 0cc0 11127 ℕ₀cn0 12531 ...cfz 13563 Basecbs 17305 ·_𝑠 cvsca 17350 Σ_g cgsu 17529 ._gcmg 19191 CMndccmn 19908 mulGrpcmgp 20274 Ringcrg 20373 CRingccrg 20374 var₁cv1 22402 Poly₁cpl1 22403 Mat cmat 22630 matToPolyMat cmat2pmat 22930

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-ofr 7682 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-fz 13564 df-fzo 13712 df-seq 14068 df-hash 14397 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-hom 17370 df-cco 17371 df-0g 17530 df-gsum 17531 df-prds 17536 df-pws 17538 df-mre 17674 df-mrc 17675 df-acs 17677 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-mhm 18892 df-submnd 18893 df-grp 19061 df-minusg 19062 df-sbg 19063 df-mulg 19192 df-subg 19247 df-ghm 19342 df-cntz 19445 df-cmn 19910 df-abl 19911 df-mgp 20275 df-rng 20289 df-ur 20322 df-ring 20375 df-cring 20376 df-subrng 20709 df-subrg 20733 df-lmod 21047 df-lss 21117 df-sra 21358 df-rgmod 21359 df-dsmm 21946 df-frlm 21961 df-ascl 22071 df-psr 22125 df-mvr 22126 df-mpl 22127 df-opsr 22129 df-psr1 22406 df-vr1 22407 df-ply1 22408 df-mamu 22614 df-mat 22631 df-mat2pmat 22933

This theorem is used by: (None)

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