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Theorem m2cpminvid 22673

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 2729	. . . 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 22661	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑇‘𝑀) ∈ (𝑁 ConstPolyMat 𝑅))
6		m2cpminvid.i	. . . 4 ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅)
7	6, 1	cpm2mval 22670	. . 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 2729	. . . . . . . 8 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
10		eqid 2729	. . . . . . . 8 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
11	2, 3, 4, 9, 10	mat2pmatvalel 22645	. . . . . . 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 6844	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (coe₁‘(𝑥(𝑇‘𝑀)𝑦)) = (coe₁‘((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦))))
14	13	fveq1d 6842	. . . 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 2729	. . . . . 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 22346	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘𝑅))
21	9, 10, 16	ply1sclid 22207	. . . . 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 2767	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0) = (𝑥𝑀𝑦))
24	23	mpoeq3dva 7446	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)))
25		eqidd 2730	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)))
26		oveq12 7378	. . . . . 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 7404	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ V)
31	25, 27, 28, 29, 30	ovmpod 7521	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦))𝑗) = (𝑖𝑀𝑗))
32	31	ralrimivva 3178	. . 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 22343	. . . 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 22344	. . . 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 2768	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 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3444 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 Fincfn 8895 0cc0 11044 Basecbs 17155 Ringcrg 20153 algSccascl 21794 Poly₁cpl1 22094 coe₁cco1 22095 Mat cmat 22327 ConstPolyMat ccpmat 22623 matToPolyMat cmat2pmat 22624 cPolyMatToMat ccpmat2mat 22625

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-ofr 7634 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-sup 9369 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-fz 13445 df-fzo 13592 df-seq 13943 df-hash 14272 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17380 df-gsum 17381 df-prds 17386 df-pws 17388 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-mulg 18982 df-subg 19037 df-ghm 19127 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-subrng 20466 df-subrg 20490 df-lmod 20800 df-lss 20870 df-sra 21112 df-rgmod 21113 df-dsmm 21674 df-frlm 21689 df-ascl 21797 df-psr 21851 df-mvr 21852 df-mpl 21853 df-opsr 21855 df-psr1 22097 df-vr1 22098 df-ply1 22099 df-coe1 22100 df-mat 22328 df-cpmat 22626 df-mat2pmat 22627 df-cpmat2mat 22628

This theorem is referenced by: m2cpminv 22680 m2cpminv0 22681 cayhamlem4 22808

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