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Theorem m2cpminvid 20558

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 2622	. . . 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 20546	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑇‘𝑀) ∈ (𝑁 ConstPolyMat 𝑅))
6		m2cpminvid.i	. . . 4 ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅)
7	6, 1	cpm2mval 20555	. . 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 1371	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝐼‘(𝑇‘𝑀)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0)))
9		eqid 2622	. . . . . . . 8 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
10		eqid 2622	. . . . . . . 8 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
11	2, 3, 4, 9, 10	mat2pmatvalel 20530	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥(𝑇‘𝑀)𝑦) = ((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))
12	11	3impb 1260	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥(𝑇‘𝑀)𝑦) = ((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))
13	12	fveq2d 6195	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (coe₁‘(𝑥(𝑇‘𝑀)𝑦)) = (coe₁‘((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦))))
14	13	fveq1d 6193	. . . 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 1092	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ Ring)
16		eqid 2622	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		simp2 1062	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
18		simp3 1063	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
19		simp13 1093	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑀 ∈ 𝐾)
20	3, 16, 4, 17, 18, 19	matecld 20232	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘𝑅))
21	9, 10, 16	ply1sclid 19658	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥𝑀𝑦) ∈ (Base‘𝑅)) → (𝑥𝑀𝑦) = ((coe₁‘((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))‘0))
22	15, 20, 21	syl2anc 693	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) = ((coe₁‘((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))‘0))
23	14, 22	eqtr4d 2659	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0) = (𝑥𝑀𝑦))
24	23	mpt2eq3dva 6719	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)))
25		eqidd 2623	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)))
26		oveq12 6659	. . . . . 6 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
27	26	adantl 482	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑥 = 𝑖 ∧ 𝑦 = 𝑗)) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
28		simprl 794	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
29		simprr 796	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
30		ovexd 6680	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ V)
31	25, 27, 28, 29, 30	ovmpt2d 6788	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦))𝑗) = (𝑖𝑀𝑗))
32	31	ralrimivva 2971	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦))𝑗) = (𝑖𝑀𝑗))
33		simp1 1061	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → 𝑁 ∈ Fin)
34		simp2 1062	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → 𝑅 ∈ Ring)
35	3, 16, 4, 33, 34, 20	matbas2d 20229	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) ∈ 𝐾)
36		simp3 1063	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → 𝑀 ∈ 𝐾)
37	3, 4	eqmat 20230	. . . 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 693	. . 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 247	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) = 𝑀)
40	8, 24, 39	3eqtrd 2660	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝐼‘(𝑇‘𝑀)) = 𝑀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 Fincfn 7955 0cc0 9936 Basecbs 15857 Ringcrg 18547 algSccascl 19311 Poly₁cpl1 19547 coe₁cco1 19548 Mat cmat 20213 ConstPolyMat ccpmat 20508 matToPolyMat cmat2pmat 20509 cPolyMatToMat ccpmat2mat 20510

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-coe1 19553 df-dsmm 20076 df-frlm 20091 df-mat 20214 df-cpmat 20511 df-mat2pmat 20512 df-cpmat2mat 20513

This theorem is referenced by: m2cpminv 20565 m2cpminv0 20566 cayhamlem4 20693

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