pm2mpf - Metamath Proof Explorer

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Theorem pm2mpf 22713

Description: The transformation of polynomial matrices into polynomials over matrices is a function mapping polynomial matrices to polynomials over matrices. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 5-Dec-2019.)

**Hypotheses**
Ref	Expression
pm2mpval.p	⊢ 𝑃 = (Poly₁‘𝑅)
pm2mpval.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pm2mpval.b	⊢ 𝐵 = (Base‘𝐶)
pm2mpval.m	⊢ ∗ = ( ·_𝑠 ‘𝑄)
pm2mpval.e	⊢ ↑ = (._g‘(mulGrp‘𝑄))
pm2mpval.x	⊢ 𝑋 = (var₁‘𝐴)
pm2mpval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
pm2mpval.q	⊢ 𝑄 = (Poly₁‘𝐴)
pm2mpval.t	⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
pm2mpcl.l	⊢ 𝐿 = (Base‘𝑄)

**Assertion**
Ref	Expression
pm2mpf	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝐿)

Proof of Theorem pm2mpf Dummy variables 𝑚 𝑘 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 7455	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) → (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ V)
2		pm2mpval.p	. . 3 ⊢ 𝑃 = (Poly₁‘𝑅)
3		pm2mpval.c	. . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃)
4		pm2mpval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
5		pm2mpval.m	. . 3 ⊢ ∗ = ( ·_𝑠 ‘𝑄)
6		pm2mpval.e	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑄))
7		pm2mpval.x	. . 3 ⊢ 𝑋 = (var₁‘𝐴)
8		pm2mpval.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
9		pm2mpval.q	. . 3 ⊢ 𝑄 = (Poly₁‘𝐴)
10		pm2mpval.t	. . 3 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
11	2, 3, 4, 5, 6, 7, 8, 9, 10	pm2mpval 22710	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))))
12		pm2mpcl.l	. . . 4 ⊢ 𝐿 = (Base‘𝑄)
13	2, 3, 4, 5, 6, 7, 8, 9, 10, 12	pm2mpcl 22712	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑏 ∈ 𝐵) → (𝑇‘𝑏) ∈ 𝐿)
14	13	3expa 1116	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑏 ∈ 𝐵) → (𝑇‘𝑏) ∈ 𝐿)
15	1, 11, 14	fmpt2d 7134	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝐿)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ↦ cmpt 5231 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 Fincfn 8964 ℕ₀cn0 12503 Basecbs 17180 ·_𝑠 cvsca 17237 Σ_g cgsu 17422 ._gcmg 19023 mulGrpcmgp 20074 Ringcrg 20173 var₁cv1 22095 Poly₁cpl1 22096 Mat cmat 22320 decompPMat cdecpmat 22677 pMatToMatPoly cpm2mp 22707

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-ofr 7686 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-subrng 20483 df-subrg 20508 df-lmod 20745 df-lss 20816 df-sra 21058 df-rgmod 21059 df-dsmm 21666 df-frlm 21681 df-psr 21842 df-mvr 21843 df-mpl 21844 df-opsr 21846 df-psr1 22099 df-vr1 22100 df-ply1 22101 df-coe1 22102 df-mamu 22299 df-mat 22321 df-decpmat 22678 df-pm2mp 22708

This theorem is referenced by: pm2mpf1 22714 pm2mpfo 22729 pm2mpghm 22731 pm2mpmhm 22735

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