m2cpmrngiso - Metamath Proof Explorer

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Theorem m2cpmrngiso 22791

Description: The transformation of matrices into constant polynomial matrices is a ring isomorphism. (Contributed by AV, 19-Nov-2019.)

**Hypotheses**
Ref	Expression
m2cpmfo.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
m2cpmfo.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
m2cpmfo.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
m2cpmfo.k	⊢ 𝐾 = (Base‘𝐴)
m2cpmrngiso.p	⊢ 𝑃 = (Poly₁‘𝑅)
m2cpmrngiso.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
m2cpmrngiso.u	⊢ 𝑈 = (𝐶 ↾_s 𝑆)

**Assertion**
Ref	Expression
m2cpmrngiso	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingIso 𝑈))

Proof of Theorem m2cpmrngiso Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		m2cpmfo.s	. . 3 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
2		m2cpmfo.t	. . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
3		m2cpmfo.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4		m2cpmfo.k	. . 3 ⊢ 𝐾 = (Base‘𝐴)
5		m2cpmrngiso.p	. . 3 ⊢ 𝑃 = (Poly₁‘𝑅)
6		m2cpmrngiso.c	. . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃)
7		m2cpmrngiso.u	. . 3 ⊢ 𝑈 = (𝐶 ↾_s 𝑆)
8	1, 2, 3, 4, 5, 6, 7	m2cpmrhm 22779	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈))
9		crngring 20267	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
10	1, 2, 3, 4	m2cpmf1o 22790	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→𝑆)
11		eqid 2756	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
12	1, 5, 6, 11	cpmatpmat 22743	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ (Base‘𝐶))
13	12	3expia 1130	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ 𝑆 → 𝑚 ∈ (Base‘𝐶)))
14	13	ssrdv 3937	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶))
15	7, 11	ressbas2 17250	. . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝐶) → 𝑆 = (Base‘𝑈))
16	14, 15	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = (Base‘𝑈))
17	16	eqcomd 2762	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑈) = 𝑆)
18	17	f1oeq3d 6792	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇:𝐾–1-1-onto→(Base‘𝑈) ↔ 𝑇:𝐾–1-1-onto→𝑆))
19	10, 18	mpbird 259	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→(Base‘𝑈))
20	9, 19	sylan2 601	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇:𝐾–1-1-onto→(Base‘𝑈))
21		eqid 2756	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
22	4, 21	isrim 20513	. 2 ⊢ (𝑇 ∈ (𝐴 RingIso 𝑈) ↔ (𝑇 ∈ (𝐴 RingHom 𝑈) ∧ 𝑇:𝐾–1-1-onto→(Base‘𝑈)))
23	8, 20, 22	sylanbrc 591	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingIso 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1554 ∈ wcel 2136 ⊆ wss 3899 –1-1-onto→wf1o 6509 ‘cfv 6510 (class class class)co 7385 Fincfn 8916 Basecbs 17221 ↾_s cress 17242 Ringcrg 20255 CRingccrg 20256 RingHom crh 20490 RingIso crs 20491 Poly₁cpl1 22212 Mat cmat 22440 ConstPolyMat ccpmat 22736 matToPolyMat cmat2pmat 22737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-ot 4585 df-uni 4860 df-int 4900 df-iun 4945 df-iin 4946 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-isom 6519 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-of 7649 df-ofr 7650 df-om 7836 df-1st 7959 df-2nd 7960 df-supp 8129 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-2o 8426 df-er 8666 df-map 8798 df-pm 8799 df-ixp 8869 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-fsupp 9298 df-sup 9378 df-oi 9448 df-card 9887 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-nn 12201 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12472 df-z 12559 df-dec 12679 df-uz 12830 df-fz 13503 df-fzo 13650 df-seq 14005 df-hash 14334 df-struct 17159 df-sets 17176 df-slot 17194 df-ndx 17206 df-base 17222 df-ress 17243 df-plusg 17275 df-mulr 17276 df-sca 17278 df-vsca 17279 df-ip 17280 df-tset 17281 df-ple 17282 df-ds 17284 df-hom 17286 df-cco 17287 df-0g 17446 df-gsum 17447 df-prds 17452 df-pws 17454 df-mre 17590 df-mrc 17591 df-acs 17593 df-mgm 18650 df-sgrp 18729 df-mnd 18745 df-mhm 18793 df-submnd 18794 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19230 df-cntz 19333 df-cmn 19798 df-abl 19799 df-mgp 20163 df-rng 20175 df-ur 20204 df-srg 20209 df-ring 20257 df-cring 20258 df-rhm 20493 df-rim 20494 df-subrng 20568 df-subrg 20592 df-lmod 20902 df-lss 20972 df-sra 21213 df-rgmod 21214 df-dsmm 21757 df-frlm 21772 df-assa 21878 df-ascl 21880 df-psr 21934 df-mvr 21935 df-mpl 21936 df-opsr 21938 df-psr1 22215 df-vr1 22216 df-ply1 22217 df-coe1 22218 df-mamu 22424 df-mat 22441 df-cpmat 22739 df-mat2pmat 22740 df-cpmat2mat 22741

This theorem is referenced by: matcpmric 22792

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