m2cpmf - Metamath Proof Explorer

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Theorem m2cpmf 21891

Description: The matrix transformation is a function from the matrices to the constant polynomial matrices. (Contributed by AV, 18-Nov-2019.)

**Hypotheses**
Ref	Expression
m2cpm.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
m2cpm.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
m2cpm.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
m2cpm.b	⊢ 𝐵 = (Base‘𝐴)

**Assertion**
Ref	Expression
m2cpmf	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝑆)

Proof of Theorem m2cpmf Dummy variables 𝑖 𝑗 𝑏 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 483	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin)
2	1, 1	jca 512	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
3	2	adantr 481	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
4		mpoexga 7918	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly₁‘𝑅))‘(𝑖𝑚𝑗))) ∈ V)
5	3, 4	syl 17	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly₁‘𝑅))‘(𝑖𝑚𝑗))) ∈ V)
6		m2cpm.t	. . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
7		m2cpm.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
8		m2cpm.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
9		eqid 2738	. . 3 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
10		eqid 2738	. . 3 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
11	6, 7, 8, 9, 10	mat2pmatfval 21872	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly₁‘𝑅))‘(𝑖𝑚𝑗)))))
12		m2cpm.s	. . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
13	12, 6, 7, 8	m2cpm 21890	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑏 ∈ 𝐵) → (𝑇‘𝑏) ∈ 𝑆)
14	13	3expa 1117	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑏 ∈ 𝐵) → (𝑇‘𝑏) ∈ 𝑆)
15	5, 11, 14	fmpt2d 6997	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 Fincfn 8733 Basecbs 16912 Ringcrg 19783 algSccascl 21059 Poly₁cpl1 21348 Mat cmat 21554 ConstPolyMat ccpmat 21852 matToPolyMat cmat2pmat 21853

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-subrg 20022 df-lmod 20125 df-lss 20194 df-sra 20434 df-rgmod 20435 df-dsmm 20939 df-frlm 20954 df-ascl 21062 df-psr 21112 df-mvr 21113 df-mpl 21114 df-opsr 21116 df-psr1 21351 df-vr1 21352 df-ply1 21353 df-coe1 21354 df-mat 21555 df-cpmat 21855 df-mat2pmat 21856

This theorem is referenced by: m2cpmf1 21892 m2cpmghm 21893 m2cpmmhm 21894 m2cpmfo 21905 m2cpminv 21909

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