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Mirrors > Home > MPE Home > Th. List > mat1ric | Structured version Visualization version GIF version |
Description: A ring is isomorphic to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 30-Dec-2019.) |
Ref | Expression |
---|---|
mat1ric.a | ⊢ 𝐴 = ({𝐸} Mat 𝑅) |
Ref | Expression |
---|---|
mat1ric | ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝑅 ≃𝑟 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | mat1ric.a | . . . 4 ⊢ 𝐴 = ({𝐸} Mat 𝑅) | |
3 | eqid 2731 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
4 | eqid 2731 | . . . 4 ⊢ ⟨𝐸, 𝐸⟩ = ⟨𝐸, 𝐸⟩ | |
5 | opeq2 4867 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ⟨⟨𝐸, 𝐸⟩, 𝑥⟩ = ⟨⟨𝐸, 𝐸⟩, 𝑦⟩) | |
6 | 5 | sneqd 4634 | . . . . 5 ⊢ (𝑥 = 𝑦 → {⟨⟨𝐸, 𝐸⟩, 𝑥⟩} = {⟨⟨𝐸, 𝐸⟩, 𝑦⟩}) |
7 | 6 | cbvmptv 5254 | . . . 4 ⊢ (𝑥 ∈ (Base‘𝑅) ↦ {⟨⟨𝐸, 𝐸⟩, 𝑥⟩}) = (𝑦 ∈ (Base‘𝑅) ↦ {⟨⟨𝐸, 𝐸⟩, 𝑦⟩}) |
8 | 1, 2, 3, 4, 7 | mat1rngiso 21917 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝑥 ∈ (Base‘𝑅) ↦ {⟨⟨𝐸, 𝐸⟩, 𝑥⟩}) ∈ (𝑅 RingIso 𝐴)) |
9 | 8 | ne0d 4331 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝑅 RingIso 𝐴) ≠ ∅) |
10 | brric 20233 | . 2 ⊢ (𝑅 ≃𝑟 𝐴 ↔ (𝑅 RingIso 𝐴) ≠ ∅) | |
11 | 9, 10 | sylibr 233 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝑅 ≃𝑟 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∅c0 4318 {csn 4622 ⟨cop 4628 class class class wbr 5141 ↦ cmpt 5224 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 Ringcrg 20014 RingIso crs 20199 ≃𝑟 cric 20200 Mat cmat 21836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-ot 4631 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-sup 9419 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-fzo 13610 df-seq 13949 df-hash 14273 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17369 df-gsum 17370 df-prds 17375 df-pws 17377 df-mre 17512 df-mrc 17513 df-acs 17515 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-mhm 18647 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-mulg 18923 df-subg 18975 df-ghm 19056 df-cntz 19147 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-rnghom 20201 df-rngiso 20202 df-ric 20204 df-subrg 20310 df-lmod 20422 df-lss 20492 df-sra 20734 df-rgmod 20735 df-dsmm 21220 df-frlm 21235 df-mamu 21815 df-mat 21837 |
This theorem is referenced by: (None) |
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