m2cpminv - Metamath Proof Explorer

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Theorem m2cpminv 22781

Description: The inverse matrix transformation is a 1-1 function from the constant polynomial matrices onto the matrices over the base ring of the polynomials. (Contributed by AV, 27-Nov-2019.) (Revised by AV, 15-Dec-2019.)

**Hypotheses**
Ref	Expression
m2cpminv.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
m2cpminv.k	⊢ 𝐾 = (Base‘𝐴)
m2cpminv.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
m2cpminv.i	⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅)
m2cpminv.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)

**Assertion**
Ref	Expression
m2cpminv	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼:𝑆–1-1-onto→𝐾 ∧ ^◡𝐼 = 𝑇))

Proof of Theorem m2cpminv Dummy variables 𝑘 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		m2cpminv.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		m2cpminv.k	. . . 4 ⊢ 𝐾 = (Base‘𝐴)
3		m2cpminv.s	. . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
4		m2cpminv.i	. . . 4 ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅)
5	1, 2, 3, 4	cpm2mf 22773	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼:𝑆⟶𝐾)
6		m2cpminv.t	. . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
7	3, 6, 1, 2	m2cpmf 22763	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾⟶𝑆)
8	3, 4, 6	m2cpminvid2 22776	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ 𝑆) → (𝑇‘(𝐼‘𝑠)) = 𝑠)
9	8	3expa 1117	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ 𝑆) → (𝑇‘(𝐼‘𝑠)) = 𝑠)
10	9	ralrimiva 3143	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑠 ∈ 𝑆 (𝑇‘(𝐼‘𝑠)) = 𝑠)
11	4, 1, 2, 6	m2cpminvid 22774	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑘 ∈ 𝐾) → (𝐼‘(𝑇‘𝑘)) = 𝑘)
12	11	3expa 1117	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ 𝐾) → (𝐼‘(𝑇‘𝑘)) = 𝑘)
13	12	ralrimiva 3143	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑘 ∈ 𝐾 (𝐼‘(𝑇‘𝑘)) = 𝑘)
14	5, 7, 10, 13	2fvidf1od 7317	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼:𝑆–1-1-onto→𝐾)
15	5, 7, 10, 13	2fvidinvd 7318	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ^◡𝐼 = 𝑇)
16	14, 15	jca 511	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼:𝑆–1-1-onto→𝐾 ∧ ^◡𝐼 = 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ^◡ccnv 5687 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 Basecbs 17244 Ringcrg 20250 Mat cmat 22426 ConstPolyMat ccpmat 22724 matToPolyMat cmat2pmat 22725 cPolyMatToMat ccpmat2mat 22726

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-ofr 7697 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-srg 20204 df-ring 20252 df-subrng 20562 df-subrg 20586 df-lmod 20876 df-lss 20947 df-sra 21189 df-rgmod 21190 df-dsmm 21769 df-frlm 21784 df-ascl 21892 df-psr 21946 df-mvr 21947 df-mpl 21948 df-opsr 21950 df-psr1 22196 df-vr1 22197 df-ply1 22198 df-coe1 22199 df-mat 22427 df-cpmat 22727 df-mat2pmat 22728 df-cpmat2mat 22729

This theorem is referenced by: (None)

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