m2cpminvid - Metamath Proof Explorer

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Theorem m2cpminvid 23018

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 2760	. . . 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 23006	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑇‘𝑀) ∈ (𝑁 ConstPolyMat 𝑅))
6		m2cpminvid.i	. . . 4 ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅)
7	6, 1	cpm2mval 23015	. . 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 1436	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝐼‘(𝑇‘𝑀)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0)))
9		eqid 2760	. . . . . . . 8 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
10		eqid 2760	. . . . . . . 8 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
11	2, 3, 4, 9, 10	mat2pmatvalel 22990	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥(𝑇‘𝑀)𝑦) = ((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))
12	11	3impb 1132	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥(𝑇‘𝑀)𝑦) = ((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))
13	12	fveq2d 6878	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (coe₁‘(𝑥(𝑇‘𝑀)𝑦)) = (coe₁‘((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦))))
14	13	fveq1d 6876	. . . 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 1223	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ Ring)
16		eqid 2760	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		simp2 1155	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
18		simp3 1156	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
19		simp13 1224	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑀 ∈ 𝐾)
20	3, 16, 4, 17, 18, 19	matecld 22688	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘𝑅))
21	9, 10, 16	ply1sclid 22554	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥𝑀𝑦) ∈ (Base‘𝑅)) → (𝑥𝑀𝑦) = ((coe₁‘((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))‘0))
22	15, 20, 21	syl2anc 596	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) = ((coe₁‘((algSc‘(Poly₁‘𝑅))‘(𝑥𝑀𝑦)))‘0))
23	14, 22	eqtr4d 2798	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0) = (𝑥𝑀𝑦))
24	23	mpoeq3dva 7486	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥(𝑇‘𝑀)𝑦))‘0)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)))
25		eqidd 2761	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)))
26		oveq12 7418	. . . . . 6 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
27	26	adantl 487	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑥 = 𝑖 ∧ 𝑦 = 𝑗)) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
28		simprl 783	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
29		simprr 785	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
30		ovexd 7444	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ V)
31	25, 27, 28, 29, 30	ovmpod 7561	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦))𝑗) = (𝑖𝑀𝑗))
32	31	ralrimivva 3205	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦))𝑗) = (𝑖𝑀𝑗))
33		simp1 1154	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → 𝑁 ∈ Fin)
34		simp2 1155	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → 𝑅 ∈ Ring)
35	3, 16, 4, 33, 34, 20	matbas2d 22685	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) ∈ 𝐾)
36		simp3 1156	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → 𝑀 ∈ 𝐾)
37	3, 4	eqmat 22686	. . . 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 596	. . 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 260	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑥𝑀𝑦)) = 𝑀)
40	8, 24, 39	3eqtrd 2799	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝐼‘(𝑇‘𝑀)) = 𝑀)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 Vcvv 3450 ‘cfv 6528 (class class class)co 7409 ∈ cmpo 7411 Fincfn 8952 0cc0 11157 Basecbs 17334 Ringcrg 20406 algSccascl 22107 Poly₁cpl1 22442 coe₁cco1 22443 Mat cmat 22669 ConstPolyMat ccpmat 22968 matToPolyMat cmat2pmat 22969 cPolyMatToMat ccpmat2mat 22970

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-cnex 11213 ax-resscn 11214 ax-1cn 11215 ax-icn 11216 ax-addcl 11217 ax-addrcl 11218 ax-mulcl 11219 ax-mulrcl 11220 ax-mulcom 11221 ax-addass 11222 ax-mulass 11223 ax-distr 11224 ax-i2m1 11225 ax-1ne0 11226 ax-1rid 11227 ax-rnegex 11228 ax-rrecex 11229 ax-cnre 11230 ax-pre-lttri 11231 ax-pre-lttrn 11232 ax-pre-ltadd 11233 ax-pre-mulgt0 11234

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6294 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-isom 6537 df-riota 7366 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7677 df-ofr 7678 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-sup 9412 df-oi 9482 df-card 9977 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11500 df-neg 11501 df-nn 12291 df-2 12360 df-3 12361 df-4 12362 df-5 12363 df-6 12364 df-7 12365 df-8 12366 df-9 12367 df-n0 12562 df-z 12649 df-dec 12770 df-uz 12921 df-fz 13595 df-fzo 13743 df-seq 14099 df-hash 14428 df-struct 17272 df-sets 17289 df-slot 17307 df-ndx 17319 df-base 17335 df-ress 17356 df-plusg 17388 df-mulr 17389 df-sca 17391 df-vsca 17392 df-ip 17393 df-tset 17394 df-ple 17395 df-ds 17397 df-hom 17399 df-cco 17400 df-0g 17559 df-gsum 17560 df-prds 17565 df-pws 17567 df-mre 17703 df-mrc 17704 df-acs 17706 df-mgm 18763 df-sgrp 18855 df-mnd 18871 df-mhm 18925 df-submnd 18926 df-grp 19094 df-minusg 19095 df-sbg 19096 df-mulg 19225 df-subg 19280 df-ghm 19375 df-cntz 19478 df-cmn 19943 df-abl 19944 df-mgp 20308 df-rng 20322 df-ur 20355 df-ring 20408 df-subrng 20745 df-subrg 20769 df-lmod 21084 df-lss 21154 df-sra 21395 df-rgmod 21396 df-dsmm 21985 df-frlm 22000 df-ascl 22110 df-psr 22164 df-mvr 22165 df-mpl 22166 df-opsr 22168 df-psr1 22445 df-vr1 22446 df-ply1 22447 df-coe1 22448 df-mat 22670 df-cpmat 22971 df-mat2pmat 22972 df-cpmat2mat 22973

This theorem is used by: m2cpminv 23025 m2cpminv0 23026 cayhamlem4 23153

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