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Mirrors > Home > MPE Home > Th. List > mat1rngiso | Structured version Visualization version GIF version |
Description: There is a ring isomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
Ref | Expression |
---|---|
mat1rhmval.k | ⊢ 𝐾 = (Base‘𝑅) |
mat1rhmval.a | ⊢ 𝐴 = ({𝐸} Mat 𝑅) |
mat1rhmval.b | ⊢ 𝐵 = (Base‘𝐴) |
mat1rhmval.o | ⊢ 𝑂 = 〈𝐸, 𝐸〉 |
mat1rhmval.f | ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) |
Ref | Expression |
---|---|
mat1rngiso | ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹 ∈ (𝑅 RingIso 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mat1rhmval.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
2 | mat1rhmval.a | . . 3 ⊢ 𝐴 = ({𝐸} Mat 𝑅) | |
3 | mat1rhmval.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
4 | mat1rhmval.o | . . 3 ⊢ 𝑂 = 〈𝐸, 𝐸〉 | |
5 | mat1rhmval.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) | |
6 | 1, 2, 3, 4, 5 | mat1rhm 21956 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹 ∈ (𝑅 RingHom 𝐴)) |
7 | 1, 2, 3, 4, 5 | mat1f1o 21949 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹:𝐾–1-1-onto→𝐵) |
8 | 1, 3 | isrim 20248 | . 2 ⊢ (𝐹 ∈ (𝑅 RingIso 𝐴) ↔ (𝐹 ∈ (𝑅 RingHom 𝐴) ∧ 𝐹:𝐾–1-1-onto→𝐵)) |
9 | 6, 7, 8 | sylanbrc 584 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹 ∈ (𝑅 RingIso 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4624 〈cop 4630 ↦ cmpt 5227 –1-1-onto→wf1o 6534 ‘cfv 6535 (class class class)co 7396 Basecbs 17131 Ringcrg 20038 RingHom crh 20226 RingIso crs 20227 Mat cmat 21876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7657 df-om 7843 df-1st 7962 df-2nd 7963 df-supp 8134 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9350 df-sup 9424 df-oi 9492 df-card 9921 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-fz 13472 df-fzo 13615 df-seq 13954 df-hash 14278 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-ds 17206 df-hom 17208 df-cco 17209 df-0g 17374 df-gsum 17375 df-prds 17380 df-pws 17382 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-mhm 18658 df-submnd 18659 df-grp 18809 df-minusg 18810 df-sbg 18811 df-mulg 18936 df-subg 18988 df-ghm 19075 df-cntz 19166 df-cmn 19634 df-abl 19635 df-mgp 19971 df-ur 19988 df-ring 20040 df-rnghom 20229 df-rngiso 20230 df-subrg 20338 df-lmod 20450 df-lss 20520 df-sra 20762 df-rgmod 20763 df-dsmm 21260 df-frlm 21275 df-mamu 21855 df-mat 21877 |
This theorem is referenced by: mat1ric 21958 |
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