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Mirrors > Home > MPE Home > Th. List > matcpmric | Structured version Visualization version GIF version |
Description: The ring of matrices over a commutative ring is isomorphic to the ring of scalar matrices over the same ring. (Contributed by AV, 30-Dec-2019.) |
Ref | Expression |
---|---|
matcpmric.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
matcpmric.p | ⊢ 𝑃 = (Poly1‘𝑅) |
matcpmric.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
matcpmric.s | ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) |
matcpmric.u | ⊢ 𝑈 = (𝐶 ↾s 𝑆) |
Ref | Expression |
---|---|
matcpmric | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ≃𝑟 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | matcpmric.s | . . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
2 | eqid 2728 | . . . 4 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅) | |
3 | matcpmric.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
4 | eqid 2728 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
5 | matcpmric.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
6 | matcpmric.c | . . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
7 | matcpmric.u | . . . 4 ⊢ 𝑈 = (𝐶 ↾s 𝑆) | |
8 | 1, 2, 3, 4, 5, 6, 7 | m2cpmrngiso 22688 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 matToPolyMat 𝑅) ∈ (𝐴 RingIso 𝑈)) |
9 | 8 | ne0d 4339 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝐴 RingIso 𝑈) ≠ ∅) |
10 | brric 20457 | . 2 ⊢ (𝐴 ≃𝑟 𝑈 ↔ (𝐴 RingIso 𝑈) ≠ ∅) | |
11 | 9, 10 | sylibr 233 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ≃𝑟 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∅c0 4326 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 Fincfn 8972 Basecbs 17189 ↾s cress 17218 CRingccrg 20188 RingIso crs 20423 ≃𝑟 cric 20424 Poly1cpl1 22114 Mat cmat 22335 ConstPolyMat ccpmat 22633 matToPolyMat cmat2pmat 22634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-ofr 7693 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-sup 9475 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-fzo 13670 df-seq 14009 df-hash 14332 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-hom 17266 df-cco 17267 df-0g 17432 df-gsum 17433 df-prds 17438 df-pws 17440 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19038 df-subg 19092 df-ghm 19182 df-cntz 19282 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-srg 20141 df-ring 20189 df-cring 20190 df-rhm 20425 df-rim 20426 df-ric 20428 df-subrng 20497 df-subrg 20522 df-lmod 20759 df-lss 20830 df-sra 21072 df-rgmod 21073 df-dsmm 21680 df-frlm 21695 df-assa 21801 df-ascl 21803 df-psr 21856 df-mvr 21857 df-mpl 21858 df-opsr 21860 df-psr1 22117 df-vr1 22118 df-ply1 22119 df-coe1 22120 df-mamu 22319 df-mat 22336 df-cpmat 22636 df-mat2pmat 22637 df-cpmat2mat 22638 |
This theorem is referenced by: (None) |
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