m2cpmrngiso - Metamath Proof Explorer

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Theorem m2cpmrngiso 22666

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 22654	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈))
9		crngring 20156	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
10	1, 2, 3, 4	m2cpmf1o 22665	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→𝑆)
11		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
12	1, 5, 6, 11	cpmatpmat 22618	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ (Base‘𝐶))
13	12	3expia 1121	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ 𝑆 → 𝑚 ∈ (Base‘𝐶)))
14	13	ssrdv 3938	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶))
15	7, 11	ressbas2 17141	. . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝐶) → 𝑆 = (Base‘𝑈))
16	14, 15	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = (Base‘𝑈))
17	16	eqcomd 2736	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑈) = 𝑆)
18	17	f1oeq3d 6756	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇:𝐾–1-1-onto→(Base‘𝑈) ↔ 𝑇:𝐾–1-1-onto→𝑆))
19	10, 18	mpbird 257	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→(Base‘𝑈))
20	9, 19	sylan2 593	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇:𝐾–1-1-onto→(Base‘𝑈))
21		eqid 2730	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
22	4, 21	isrim 20402	. 2 ⊢ (𝑇 ∈ (𝐴 RingIso 𝑈) ↔ (𝑇 ∈ (𝐴 RingHom 𝑈) ∧ 𝑇:𝐾–1-1-onto→(Base‘𝑈)))
23	8, 20, 22	sylanbrc 583	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingIso 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2110 ⊆ wss 3900 –1-1-onto→wf1o 6476 ‘cfv 6477 (class class class)co 7341 Fincfn 8864 Basecbs 17112 ↾_s cress 17133 Ringcrg 20144 CRingccrg 20145 RingHom crh 20380 RingIso crs 20381 Poly₁cpl1 22082 Mat cmat 22315 ConstPolyMat ccpmat 22611 matToPolyMat cmat2pmat 22612

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-ot 4583 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-ofr 7606 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-sup 9321 df-oi 9391 df-card 9824 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-fz 13400 df-fzo 13547 df-seq 13901 df-hash 14230 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-hom 17177 df-cco 17178 df-0g 17337 df-gsum 17338 df-prds 17343 df-pws 17345 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-mhm 18683 df-submnd 18684 df-grp 18841 df-minusg 18842 df-sbg 18843 df-mulg 18973 df-subg 19028 df-ghm 19118 df-cntz 19222 df-cmn 19687 df-abl 19688 df-mgp 20052 df-rng 20064 df-ur 20093 df-srg 20098 df-ring 20146 df-cring 20147 df-rhm 20383 df-rim 20384 df-subrng 20454 df-subrg 20478 df-lmod 20788 df-lss 20858 df-sra 21100 df-rgmod 21101 df-dsmm 21662 df-frlm 21677 df-assa 21783 df-ascl 21785 df-psr 21839 df-mvr 21840 df-mpl 21841 df-opsr 21843 df-psr1 22085 df-vr1 22086 df-ply1 22087 df-coe1 22088 df-mamu 22299 df-mat 22316 df-cpmat 22614 df-mat2pmat 22615 df-cpmat2mat 22616

This theorem is referenced by: matcpmric 22667

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