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Theorem mat2pmatbas 22774

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 2761	. . 3 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
6	1, 2, 3, 4, 5	mat2pmatval 22772	. 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 2761	. . 3 ⊢ (Base‘𝑃) = (Base‘𝑃)
9		eqid 2761	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
10		simp1 1148	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
11	4	fvexi 6876	. . . 4 ⊢ 𝑃 ∈ V
12	11	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ V)
13		eqid 2761	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
14	4	ply1ring 22297	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
15	14	3ad2ant2 1146	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
16	15	3ad2ant1 1145	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑃 ∈ Ring)
17	4	ply1lmod 22301	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
18	17	3ad2ant2 1146	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ LMod)
19	18	3ad2ant1 1145	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑃 ∈ LMod)
20		eqid 2761	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
21	5, 13, 16, 19, 20, 8	asclf 21921	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (algSc‘𝑃):(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
22	4	ply1sca 22302	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
23	22	fveq2d 6866	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
24	23	3ad2ant2 1146	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
25	24	3ad2ant1 1145	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
26	25	feq2d 6670	. . . . 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 1149	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
29		simp3 1150	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
30	3	eleq2i 2853	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐴))
31	30	biimpi 218	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐴))
32	31	3ad2ant3 1147	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ (Base‘𝐴))
33	32	3ad2ant1 1145	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴))
34		eqid 2761	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
35	2, 34	matecl 22473	. . . . 5 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝑥𝑀𝑦) ∈ (Base‘𝑅))
36	28, 29, 33, 35	syl3anc 1389	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘𝑅))
37	27, 36	ffvelcdmd 7061	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑥𝑀𝑦)) ∈ (Base‘𝑃))
38	7, 8, 9, 10, 12, 37	matbas2d 22471	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑥𝑀𝑦))) ∈ (Base‘𝐶))
39	6, 38	eqeltrd 2861	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 Vcvv 3453 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 ∈ cmpo 7393 Fincfn 8921 Basecbs 17236 Scalarcsca 17280 Ringcrg 20270 LModclmod 20915 algSccascl 21892 Poly₁cpl1 22227 Mat cmat 22455 matToPolyMat cmat2pmat 22752

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 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 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-ot 4588 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-ofr 7656 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-sup 9382 df-oi 9452 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-z 12563 df-dec 12683 df-uz 12834 df-fz 13507 df-fzo 13654 df-seq 14009 df-hash 14338 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-hom 17301 df-cco 17302 df-0g 17461 df-gsum 17462 df-prds 17467 df-pws 17469 df-mre 17605 df-mrc 17606 df-acs 17608 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18969 df-minusg 18970 df-sbg 18971 df-mulg 19101 df-subg 19156 df-ghm 19245 df-cntz 19348 df-cmn 19813 df-abl 19814 df-mgp 20178 df-rng 20190 df-ur 20219 df-ring 20272 df-subrng 20583 df-subrg 20607 df-lmod 20917 df-lss 20987 df-sra 21228 df-rgmod 21229 df-dsmm 21772 df-frlm 21787 df-ascl 21895 df-psr 21949 df-mpl 21951 df-opsr 21953 df-psr1 22230 df-ply1 22232 df-mat 22456 df-mat2pmat 22755

This theorem is referenced by: mat2pmatbas0 22775 m2cpm 22789 m2pmfzmap 22795 monmatcollpw 22827 pmatcollpw 22829 chmatcl 22876 chmatval 22877 chpmat1dlem 22883 chpmat1d 22884 chpdmatlem1 22886 chpdmatlem2 22887 chpdmatlem3 22888 chfacfisf 22902 chfacfscmulgsum 22908 chfacfpmmulcl 22909 chfacfpmmul0 22910 chfacfpmmulgsum 22912 chfacfpmmulgsum2 22913 cayhamlem1 22914 cpmadugsumlemC 22923 cpmadugsumlemF 22924 cpmadugsumfi 22925 cpmidgsum2 22927

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