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Theorem mat2pmatbas 21336

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 2823	. . 3 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
6	1, 2, 3, 4, 5	mat2pmatval 21334	. 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 2823	. . 3 ⊢ (Base‘𝑃) = (Base‘𝑃)
9		eqid 2823	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
10		simp1 1132	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
11	4	fvexi 6686	. . . 4 ⊢ 𝑃 ∈ V
12	11	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ V)
13		eqid 2823	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
14	4	ply1ring 20418	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
15	14	3ad2ant2 1130	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
16	15	3ad2ant1 1129	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑃 ∈ Ring)
17	4	ply1lmod 20422	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
18	17	3ad2ant2 1130	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ LMod)
19	18	3ad2ant1 1129	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑃 ∈ LMod)
20		eqid 2823	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
21	5, 13, 16, 19, 20, 8	asclf 20113	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (algSc‘𝑃):(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
22	4	ply1sca 20423	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
23	22	fveq2d 6676	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
24	23	3ad2ant2 1130	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
25	24	3ad2ant1 1129	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
26	25	feq2d 6502	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃) ↔ (algSc‘𝑃):(Base‘(Scalar‘𝑃))⟶(Base‘𝑃)))
27	21, 26	mpbird 259	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
28		simp2 1133	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
29		simp3 1134	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
30	3	eleq2i 2906	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐴))
31	30	biimpi 218	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐴))
32	31	3ad2ant3 1131	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ (Base‘𝐴))
33	32	3ad2ant1 1129	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴))
34		eqid 2823	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
35	2, 34	matecl 21036	. . . . 5 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝑥𝑀𝑦) ∈ (Base‘𝑅))
36	28, 29, 33, 35	syl3anc 1367	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘𝑅))
37	27, 36	ffvelrnd 6854	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑥𝑀𝑦)) ∈ (Base‘𝑃))
38	7, 8, 9, 10, 12, 37	matbas2d 21034	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑥𝑀𝑦))) ∈ (Base‘𝐶))
39	6, 38	eqeltrd 2915	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 Fincfn 8511 Basecbs 16485 Scalarcsca 16570 Ringcrg 19299 LModclmod 19636 algSccascl 20086 Poly₁cpl1 20347 Mat cmat 21018 matToPolyMat cmat2pmat 21314

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-0g 16717 df-gsum 16718 df-prds 16723 df-pws 16725 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-subrg 19535 df-lmod 19638 df-lss 19706 df-sra 19946 df-rgmod 19947 df-ascl 20089 df-psr 20138 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-ply1 20352 df-dsmm 20878 df-frlm 20893 df-mat 21019 df-mat2pmat 21317

This theorem is referenced by: mat2pmatbas0 21337 m2cpm 21351 m2pmfzmap 21357 monmatcollpw 21389 pmatcollpw 21391 chmatcl 21438 chmatval 21439 chpmat1dlem 21445 chpmat1d 21446 chpdmatlem1 21448 chpdmatlem2 21449 chpdmatlem3 21450 chfacfisf 21464 chfacfscmulgsum 21470 chfacfpmmulcl 21471 chfacfpmmul0 21472 chfacfpmmulgsum 21474 chfacfpmmulgsum2 21475 cayhamlem1 21476 cpmadugsumlemC 21485 cpmadugsumlemF 21486 cpmadugsumfi 21487 cpmidgsum2 21489

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