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Mirrors > Home > MPE Home > Th. List > scmatrngiso | Structured version Visualization version GIF version |
Description: There is a ring isomorphism from a ring to the ring of scalar matrices over this ring with positive dimension. (Contributed by AV, 29-Dec-2019.) |
Ref | Expression |
---|---|
scmatrhmval.k | ⊢ 𝐾 = (Base‘𝑅) |
scmatrhmval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
scmatrhmval.o | ⊢ 1 = (1r‘𝐴) |
scmatrhmval.t | ⊢ ∗ = ( ·𝑠 ‘𝐴) |
scmatrhmval.f | ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) |
scmatrhmval.c | ⊢ 𝐶 = (𝑁 ScMat 𝑅) |
scmatghm.s | ⊢ 𝑆 = (𝐴 ↾s 𝐶) |
Ref | Expression |
---|---|
scmatrngiso | ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 RingIso 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scmatrhmval.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
2 | scmatrhmval.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | scmatrhmval.o | . . . 4 ⊢ 1 = (1r‘𝐴) | |
4 | scmatrhmval.t | . . . 4 ⊢ ∗ = ( ·𝑠 ‘𝐴) | |
5 | scmatrhmval.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) | |
6 | scmatrhmval.c | . . . 4 ⊢ 𝐶 = (𝑁 ScMat 𝑅) | |
7 | scmatghm.s | . . . 4 ⊢ 𝑆 = (𝐴 ↾s 𝐶) | |
8 | 1, 2, 3, 4, 5, 6, 7 | scmatrhm 21143 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
9 | 8 | 3adant2 1127 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
10 | 1, 2, 3, 4, 5, 6 | scmatf1o 21140 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾–1-1-onto→𝐶) |
11 | 2, 6, 7 | scmatstrbas 21134 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑆) = 𝐶) |
12 | 11 | 3adant2 1127 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → (Base‘𝑆) = 𝐶) |
13 | 12 | f1oeq3d 6611 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → (𝐹:𝐾–1-1-onto→(Base‘𝑆) ↔ 𝐹:𝐾–1-1-onto→𝐶)) |
14 | 10, 13 | mpbird 259 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾–1-1-onto→(Base‘𝑆)) |
15 | simp3 1134 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring) | |
16 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
17 | eqid 2821 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
18 | 2, 16, 1, 17, 6 | scmatsrng 21128 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ (SubRing‘𝐴)) |
19 | 7 | subrgring 19537 | . . . 4 ⊢ (𝐶 ∈ (SubRing‘𝐴) → 𝑆 ∈ Ring) |
20 | 18, 19 | syl 17 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ Ring) |
21 | eqid 2821 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
22 | 1, 21 | isrim 19484 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐾–1-1-onto→(Base‘𝑆)))) |
23 | 15, 20, 22 | 3imp3i2an 1341 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐾–1-1-onto→(Base‘𝑆)))) |
24 | 9, 14, 23 | mpbir2and 711 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 RingIso 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∅c0 4290 ↦ cmpt 5145 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 Fincfn 8508 Basecbs 16482 ↾s cress 16483 ·𝑠 cvsca 16568 0gc0g 16712 1rcur 19250 Ringcrg 19296 RingHom crh 19463 RingIso crs 19464 SubRingcsubrg 19530 Mat cmat 21015 ScMat cscmat 21097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-hom 16588 df-cco 16589 df-0g 16714 df-gsum 16715 df-prds 16720 df-pws 16722 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-ghm 18355 df-cntz 18446 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-rnghom 19466 df-rngiso 19467 df-subrg 19532 df-lmod 19635 df-lss 19703 df-sra 19943 df-rgmod 19944 df-dsmm 20875 df-frlm 20890 df-mamu 20994 df-mat 21016 df-dmat 21098 df-scmat 21099 |
This theorem is referenced by: scmatric 21145 |
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