mat2pmatbas - Metamath Proof Explorer

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Theorem mat2pmatbas 22732

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 2737	. . 3 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
6	1, 2, 3, 4, 5	mat2pmatval 22730	. 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 2737	. . 3 ⊢ (Base‘𝑃) = (Base‘𝑃)
9		eqid 2737	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
10		simp1 1137	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
11	4	fvexi 6920	. . . 4 ⊢ 𝑃 ∈ V
12	11	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ V)
13		eqid 2737	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
14	4	ply1ring 22249	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
15	14	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
16	15	3ad2ant1 1134	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑃 ∈ Ring)
17	4	ply1lmod 22253	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
18	17	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ LMod)
19	18	3ad2ant1 1134	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑃 ∈ LMod)
20		eqid 2737	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
21	5, 13, 16, 19, 20, 8	asclf 21902	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (algSc‘𝑃):(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
22	4	ply1sca 22254	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
23	22	fveq2d 6910	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
24	23	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
25	24	3ad2ant1 1134	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
26	25	feq2d 6722	. . . . 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 257	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
28		simp2 1138	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
29		simp3 1139	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
30	3	eleq2i 2833	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐴))
31	30	biimpi 216	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐴))
32	31	3ad2ant3 1136	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ (Base‘𝐴))
33	32	3ad2ant1 1134	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴))
34		eqid 2737	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
35	2, 34	matecl 22431	. . . . 5 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝑥𝑀𝑦) ∈ (Base‘𝑅))
36	28, 29, 33, 35	syl3anc 1373	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘𝑅))
37	27, 36	ffvelcdmd 7105	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑥𝑀𝑦)) ∈ (Base‘𝑃))
38	7, 8, 9, 10, 12, 37	matbas2d 22429	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑥𝑀𝑦))) ∈ (Base‘𝐶))
39	6, 38	eqeltrd 2841	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Fincfn 8985 Basecbs 17247 Scalarcsca 17300 Ringcrg 20230 LModclmod 20858 algSccascl 21872 Poly₁cpl1 22178 Mat cmat 22411 matToPolyMat cmat2pmat 22710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 df-ascl 21875 df-psr 21929 df-mpl 21931 df-opsr 21933 df-psr1 22181 df-ply1 22183 df-mat 22412 df-mat2pmat 22713

This theorem is referenced by: mat2pmatbas0 22733 m2cpm 22747 m2pmfzmap 22753 monmatcollpw 22785 pmatcollpw 22787 chmatcl 22834 chmatval 22835 chpmat1dlem 22841 chpmat1d 22842 chpdmatlem1 22844 chpdmatlem2 22845 chpdmatlem3 22846 chfacfisf 22860 chfacfscmulgsum 22866 chfacfpmmulcl 22867 chfacfpmmul0 22868 chfacfpmmulgsum 22870 chfacfpmmulgsum2 22871 cayhamlem1 22872 cpmadugsumlemC 22881 cpmadugsumlemF 22882 cpmadugsumfi 22883 cpmidgsum2 22885

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