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Mirrors > Home > MPE Home > Th. List > mat0dimcrng | Structured version Visualization version GIF version |
Description: The algebra of matrices with dimension 0 (over an arbitrary ring!) is a commutative ring. (Contributed by AV, 10-Aug-2019.) |
Ref | Expression |
---|---|
mat0dim.a | ⊢ 𝐴 = (∅ Mat 𝑅) |
Ref | Expression |
---|---|
mat0dimcrng | ⊢ (𝑅 ∈ Ring → 𝐴 ∈ CRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0fin 9111 | . . 3 ⊢ ∅ ∈ Fin | |
2 | mat0dim.a | . . . 4 ⊢ 𝐴 = (∅ Mat 𝑅) | |
3 | 2 | matring 21776 | . . 3 ⊢ ((∅ ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
4 | 1, 3 | mpan 688 | . 2 ⊢ (𝑅 ∈ Ring → 𝐴 ∈ Ring) |
5 | mat0dimbas0 21799 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(∅ Mat 𝑅)) = {∅}) | |
6 | 2 | eqcomi 2745 | . . . . . 6 ⊢ (∅ Mat 𝑅) = 𝐴 |
7 | 6 | fveq2i 6842 | . . . . 5 ⊢ (Base‘(∅ Mat 𝑅)) = (Base‘𝐴) |
8 | 7 | eqeq1i 2741 | . . . 4 ⊢ ((Base‘(∅ Mat 𝑅)) = {∅} ↔ (Base‘𝐴) = {∅}) |
9 | eqidd 2737 | . . . . . . 7 ⊢ (((Base‘𝐴) = {∅} ∧ 𝑅 ∈ Ring) → (∅(.r‘𝐴)∅) = (∅(.r‘𝐴)∅)) | |
10 | 0ex 5262 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
11 | oveq1 7360 | . . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥(.r‘𝐴)𝑦) = (∅(.r‘𝐴)𝑦)) | |
12 | oveq2 7361 | . . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑦(.r‘𝐴)𝑥) = (𝑦(.r‘𝐴)∅)) | |
13 | 11, 12 | eqeq12d 2752 | . . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥) ↔ (∅(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)∅))) |
14 | 13 | ralbidv 3172 | . . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ {∅} (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥) ↔ ∀𝑦 ∈ {∅} (∅(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)∅))) |
15 | 10, 14 | ralsn 4640 | . . . . . . . 8 ⊢ (∀𝑥 ∈ {∅}∀𝑦 ∈ {∅} (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥) ↔ ∀𝑦 ∈ {∅} (∅(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)∅)) |
16 | oveq2 7361 | . . . . . . . . . 10 ⊢ (𝑦 = ∅ → (∅(.r‘𝐴)𝑦) = (∅(.r‘𝐴)∅)) | |
17 | oveq1 7360 | . . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑦(.r‘𝐴)∅) = (∅(.r‘𝐴)∅)) | |
18 | 16, 17 | eqeq12d 2752 | . . . . . . . . 9 ⊢ (𝑦 = ∅ → ((∅(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)∅) ↔ (∅(.r‘𝐴)∅) = (∅(.r‘𝐴)∅))) |
19 | 10, 18 | ralsn 4640 | . . . . . . . 8 ⊢ (∀𝑦 ∈ {∅} (∅(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)∅) ↔ (∅(.r‘𝐴)∅) = (∅(.r‘𝐴)∅)) |
20 | 15, 19 | bitri 274 | . . . . . . 7 ⊢ (∀𝑥 ∈ {∅}∀𝑦 ∈ {∅} (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥) ↔ (∅(.r‘𝐴)∅) = (∅(.r‘𝐴)∅)) |
21 | 9, 20 | sylibr 233 | . . . . . 6 ⊢ (((Base‘𝐴) = {∅} ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ {∅}∀𝑦 ∈ {∅} (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥)) |
22 | raleq 3307 | . . . . . . . 8 ⊢ ((Base‘𝐴) = {∅} → (∀𝑦 ∈ (Base‘𝐴)(𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥) ↔ ∀𝑦 ∈ {∅} (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥))) | |
23 | 22 | raleqbi1dv 3305 | . . . . . . 7 ⊢ ((Base‘𝐴) = {∅} → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥) ↔ ∀𝑥 ∈ {∅}∀𝑦 ∈ {∅} (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥))) |
24 | 23 | adantr 481 | . . . . . 6 ⊢ (((Base‘𝐴) = {∅} ∧ 𝑅 ∈ Ring) → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥) ↔ ∀𝑥 ∈ {∅}∀𝑦 ∈ {∅} (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥))) |
25 | 21, 24 | mpbird 256 | . . . . 5 ⊢ (((Base‘𝐴) = {∅} ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥)) |
26 | 25 | ex 413 | . . . 4 ⊢ ((Base‘𝐴) = {∅} → (𝑅 ∈ Ring → ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥))) |
27 | 8, 26 | sylbi 216 | . . 3 ⊢ ((Base‘(∅ Mat 𝑅)) = {∅} → (𝑅 ∈ Ring → ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥))) |
28 | 5, 27 | mpcom 38 | . 2 ⊢ (𝑅 ∈ Ring → ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥)) |
29 | eqid 2736 | . . 3 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
30 | eqid 2736 | . . 3 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
31 | 29, 30 | iscrng2 19969 | . 2 ⊢ (𝐴 ∈ CRing ↔ (𝐴 ∈ Ring ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥))) |
32 | 4, 28, 31 | sylanbrc 583 | 1 ⊢ (𝑅 ∈ Ring → 𝐴 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ∅c0 4280 {csn 4584 ‘cfv 6493 (class class class)co 7353 Fincfn 8879 Basecbs 17075 .rcmulr 17126 Ringcrg 19950 CRingccrg 19951 Mat cmat 21738 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-om 7799 df-1st 7917 df-2nd 7918 df-supp 8089 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-map 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9302 df-sup 9374 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12410 df-z 12496 df-dec 12615 df-uz 12760 df-fz 13417 df-fzo 13560 df-seq 13899 df-hash 14223 df-struct 17011 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-mulr 17139 df-sca 17141 df-vsca 17142 df-ip 17143 df-tset 17144 df-ple 17145 df-ds 17147 df-hom 17149 df-cco 17150 df-0g 17315 df-gsum 17316 df-prds 17321 df-pws 17323 df-mre 17458 df-mrc 17459 df-acs 17461 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-mhm 18593 df-submnd 18594 df-grp 18743 df-minusg 18744 df-sbg 18745 df-mulg 18864 df-subg 18916 df-ghm 18997 df-cntz 19088 df-cmn 19555 df-abl 19556 df-mgp 19888 df-ur 19905 df-ring 19952 df-cring 19953 df-subrg 20205 df-lmod 20309 df-lss 20378 df-sra 20618 df-rgmod 20619 df-dsmm 21123 df-frlm 21138 df-mamu 21717 df-mat 21739 |
This theorem is referenced by: (None) |
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