m2cpmrngiso - Metamath Proof Explorer

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Theorem m2cpmrngiso 22876

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 22864	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈))
9		crngring 20318	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
10	1, 2, 3, 4	m2cpmf1o 22875	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→𝑆)
11		eqid 2765	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
12	1, 5, 6, 11	cpmatpmat 22828	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ (Base‘𝐶))
13	12	3expia 1137	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ 𝑆 → 𝑚 ∈ (Base‘𝐶)))
14	13	ssrdv 3945	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶))
15	7, 11	ressbas2 17288	. . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝐶) → 𝑆 = (Base‘𝑈))
16	14, 15	syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = (Base‘𝑈))
17	16	eqcomd 2771	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑈) = 𝑆)
18	17	f1oeq3d 6807	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇:𝐾–1-1-onto→(Base‘𝑈) ↔ 𝑇:𝐾–1-1-onto→𝑆))
19	10, 18	mpbird 260	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→(Base‘𝑈))
20	9, 19	sylan2 604	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇:𝐾–1-1-onto→(Base‘𝑈))
21		eqid 2765	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
22	4, 21	isrim 20565	. 2 ⊢ (𝑇 ∈ (𝐴 RingIso 𝑈) ↔ (𝑇 ∈ (𝐴 RingHom 𝑈) ∧ 𝑇:𝐾–1-1-onto→(Base‘𝑈)))
23	8, 20, 22	sylanbrc 594	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingIso 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ⊆ wss 3907 –1-1-onto→wf1o 6524 ‘cfv 6525 (class class class)co 7400 Fincfn 8931 Basecbs 17259 ↾_s cress 17280 Ringcrg 20306 CRingccrg 20307 RingHom crh 20542 RingIso crs 20543 Poly₁cpl1 22297 Mat cmat 22525 ConstPolyMat ccpmat 22821 matToPolyMat cmat2pmat 22822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-ofr 7665 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-sup 9390 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-fz 13527 df-fzo 13674 df-seq 14029 df-hash 14358 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-hom 17324 df-cco 17325 df-0g 17484 df-gsum 17485 df-prds 17490 df-pws 17492 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-mhm 18831 df-submnd 18832 df-grp 18993 df-minusg 18994 df-sbg 18995 df-mulg 19125 df-subg 19180 df-ghm 19275 df-cntz 19378 df-cmn 19843 df-abl 19844 df-mgp 20208 df-rng 20222 df-ur 20255 df-srg 20260 df-ring 20308 df-cring 20309 df-rhm 20545 df-rim 20546 df-subrng 20622 df-subrg 20646 df-lmod 20952 df-lss 21022 df-sra 21263 df-rgmod 21264 df-dsmm 21842 df-frlm 21857 df-assa 21963 df-ascl 21965 df-psr 22019 df-mvr 22020 df-mpl 22021 df-opsr 22023 df-psr1 22300 df-vr1 22301 df-ply1 22302 df-coe1 22303 df-mamu 22509 df-mat 22526 df-cpmat 22824 df-mat2pmat 22825 df-cpmat2mat 22826

This theorem is referenced by: matcpmric 22877

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