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Theorem m2cpminvid 22668

Description: The inverse transformation applied to the transformation of a matrix over a ring R results in the matrix itself. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 13-Dec-2019.)

**Hypotheses**
Ref	Expression
m2cpminvid.i	⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅)
m2cpminvid.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
m2cpminvid.k	⊢ 𝐾 = (Base‘𝐴)
m2cpminvid.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)

**Assertion**
Ref	Expression
m2cpminvid	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝐼‘(𝑇‘𝑀)) = 𝑀)

Proof of Theorem m2cpminvid Dummy variables 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . . 4 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
2		m2cpminvid.t	. . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
3		m2cpminvid.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4		m2cpminvid.k	. . . 4 ⊢ 𝐾 = (Base‘𝐴)
5	1, 2, 3, 4	m2cpm 22656	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑇‘𝑀) ∈ (𝑁 ConstPolyMat 𝑅))
6		m2cpminvid.i	. . . 4 ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅)
7	6, 1	cpm2mval 22665	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑇‘𝑀) ∈ (𝑁 ConstPolyMat 𝑅)) → (𝐼‘(𝑇‘𝑀)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0)))
8	5, 7	syld3an3 1411	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝐼‘(𝑇‘𝑀)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0)))
9		eqid 2731	. . . . . . . 8 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
10		eqid 2731	. . . . . . . 8 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
11	2, 3, 4, 9, 10	mat2pmatvalel 22640	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥(𝑇‘𝑀)𝑦) = ((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))
12	11	3impb 1114	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥(𝑇‘𝑀)𝑦) = ((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))
13	12	fveq2d 6826	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (coe₁‘(𝑥(𝑇‘𝑀)𝑦)) = (coe₁‘((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦))))
14	13	fveq1d 6824	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0) = ((coe₁‘((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))‘0))
15		simp12 1205	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ Ring)
16		eqid 2731	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		simp2 1137	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
18		simp3 1138	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
19		simp13 1206	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑀 ∈ 𝐾)
20	3, 16, 4, 17, 18, 19	matecld 22341	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘𝑅))
21	9, 10, 16	ply1sclid 22202	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥𝑀𝑦) ∈ (Base‘𝑅)) → (𝑥𝑀𝑦) = ((coe₁‘((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))‘0))
22	15, 20, 21	syl2anc 584	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) = ((coe₁‘((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))‘0))
23	14, 22	eqtr4d 2769	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0) = (𝑥𝑀𝑦))
24	23	mpoeq3dva 7423	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)))
25		eqidd 2732	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)))
26		oveq12 7355	. . . . . 6 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
27	26	adantl 481	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑥 = 𝑖 ∧ 𝑦 = 𝑗)) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
28		simprl 770	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
29		simprr 772	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
30		ovexd 7381	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ V)
31	25, 27, 28, 29, 30	ovmpod 7498	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦))𝑗) = (𝑖𝑀𝑗))
32	31	ralrimivva 3175	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦))𝑗) = (𝑖𝑀𝑗))
33		simp1 1136	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → 𝑁 ∈ Fin)
34		simp2 1137	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → 𝑅 ∈ Ring)
35	3, 16, 4, 33, 34, 20	matbas2d 22338	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) ∈ 𝐾)
36		simp3 1138	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → 𝑀 ∈ 𝐾)
37	3, 4	eqmat 22339	. . . 4 ⊢ (((𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) ∈ 𝐾 ∧ 𝑀 ∈ 𝐾) → ((𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) = 𝑀 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦))𝑗) = (𝑖𝑀𝑗)))
38	35, 36, 37	syl2anc 584	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → ((𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) = 𝑀 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦))𝑗) = (𝑖𝑀𝑗)))
39	32, 38	mpbird 257	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) = 𝑀)
40	8, 24, 39	3eqtrd 2770	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝐼‘(𝑇‘𝑀)) = 𝑀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 Vcvv 3436 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 Fincfn 8869 0cc0 11006 Basecbs 17120 Ringcrg 20151 algSccascl 21789 Poly₁cpl1 22089 coe₁cco1 22090 Mat cmat 22322 ConstPolyMat ccpmat 22618 matToPolyMat cmat2pmat 22619 cPolyMatToMat ccpmat2mat 22620

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-ot 4582 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-fzo 13555 df-seq 13909 df-hash 14238 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19125 df-cntz 19229 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-subrng 20461 df-subrg 20485 df-lmod 20795 df-lss 20865 df-sra 21107 df-rgmod 21108 df-dsmm 21669 df-frlm 21684 df-ascl 21792 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-coe1 22095 df-mat 22323 df-cpmat 22621 df-mat2pmat 22622 df-cpmat2mat 22623

This theorem is referenced by: m2cpminv 22675 m2cpminv0 22676 cayhamlem4 22803

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