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Theorem mat2pmatbas 20753

Description: The result of a matrix transformation is a polynomial matrix. (Contributed by AV, 1-Aug-2019.)

**Hypotheses**
Ref	Expression
mat2pmatbas.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
mat2pmatbas.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mat2pmatbas.b	⊢ 𝐵 = (Base‘𝐴)
mat2pmatbas.p	⊢ 𝑃 = (Poly₁‘𝑅)
mat2pmatbas.c	⊢ 𝐶 = (𝑁 Mat 𝑃)

**Assertion**
Ref	Expression
mat2pmatbas	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝐶))

Proof of Theorem mat2pmatbas Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mat2pmatbas.t	. . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
2		mat2pmatbas.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		mat2pmatbas.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
4		mat2pmatbas.p	. . 3 ⊢ 𝑃 = (Poly₁‘𝑅)
5		eqid 2760	. . 3 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
6	1, 2, 3, 4, 5	mat2pmatval 20751	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑥𝑀𝑦))))
7		mat2pmatbas.c	. . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃)
8		eqid 2760	. . 3 ⊢ (Base‘𝑃) = (Base‘𝑃)
9		eqid 2760	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
10		simp1 1131	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
11		fvex 6363	. . . . 5 ⊢ (Poly₁‘𝑅) ∈ V
12	4, 11	eqeltri 2835	. . . 4 ⊢ 𝑃 ∈ V
13	12	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ V)
14		eqid 2760	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
15	4	ply1ring 19840	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
16	15	3ad2ant2 1129	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
17	16	3ad2ant1 1128	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑃 ∈ Ring)
18	4	ply1lmod 19844	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
19	18	3ad2ant2 1129	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ LMod)
20	19	3ad2ant1 1128	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑃 ∈ LMod)
21		eqid 2760	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
22	5, 14, 17, 20, 21, 8	asclf 19559	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (algSc‘𝑃):(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
23	4	ply1sca 19845	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
24	23	fveq2d 6357	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
25	24	3ad2ant2 1129	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
26	25	3ad2ant1 1128	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
27	26	feq2d 6192	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃) ↔ (algSc‘𝑃):(Base‘(Scalar‘𝑃))⟶(Base‘𝑃)))
28	22, 27	mpbird 247	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
29		simp2 1132	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
30		simp3 1133	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
31	3	eleq2i 2831	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐴))
32	31	biimpi 206	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐴))
33	32	3ad2ant3 1130	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ (Base‘𝐴))
34	33	3ad2ant1 1128	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴))
35		eqid 2760	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
36	2, 35	matecl 20453	. . . . 5 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝑥𝑀𝑦) ∈ (Base‘𝑅))
37	29, 30, 34, 36	syl3anc 1477	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘𝑅))
38	28, 37	ffvelrnd 6524	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑥𝑀𝑦)) ∈ (Base‘𝑃))
39	7, 8, 9, 10, 13, 38	matbas2d 20451	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑥𝑀𝑦))) ∈ (Base‘𝐶))
40	6, 39	eqeltrd 2839	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 ↦ cmpt2 6816 Fincfn 8123 Basecbs 16079 Scalarcsca 16166 Ringcrg 18767 LModclmod 19085 algSccascl 19533 Poly₁cpl1 19769 Mat cmat 20435 matToPolyMat cmat2pmat 20731

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-ot 4330 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-ofr 7064 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-2o 7731 df-oadd 7734 df-er 7913 df-map 8027 df-pm 8028 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-sup 8515 df-oi 8582 df-card 8975 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-fz 12540 df-fzo 12680 df-seq 13016 df-hash 13332 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-hom 16188 df-cco 16189 df-0g 16324 df-gsum 16325 df-prds 16330 df-pws 16332 df-mre 16468 df-mrc 16469 df-acs 16471 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-mhm 17556 df-submnd 17557 df-grp 17646 df-minusg 17647 df-sbg 17648 df-mulg 17762 df-subg 17812 df-ghm 17879 df-cntz 17970 df-cmn 18415 df-abl 18416 df-mgp 18710 df-ur 18722 df-ring 18769 df-subrg 19000 df-lmod 19087 df-lss 19155 df-sra 19394 df-rgmod 19395 df-ascl 19536 df-psr 19578 df-mpl 19580 df-opsr 19582 df-psr1 19772 df-ply1 19774 df-dsmm 20298 df-frlm 20313 df-mat 20436 df-mat2pmat 20734

This theorem is referenced by: mat2pmatbas0 20754 m2cpm 20768 m2pmfzmap 20774 monmatcollpw 20806 pmatcollpw 20808 chmatcl 20855 chmatval 20856 chpmat1dlem 20862 chpmat1d 20863 chpdmatlem1 20865 chpdmatlem2 20866 chpdmatlem3 20867 chfacfisf 20881 chfacfscmulgsum 20887 chfacfpmmulcl 20888 chfacfpmmul0 20889 chfacfpmmulgsum 20891 chfacfpmmulgsum2 20892 cayhamlem1 20893 cpmadugsumlemC 20902 cpmadugsumlemF 20903 cpmadugsumfi 20904 cpmidgsum2 20906

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