![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mat1 | Structured version Visualization version GIF version |
Description: Value of an identity matrix, see also the statement in [Lang] p. 504: "The unit element of the ring of n x n matrices is the matrix In ... whose components are equal to 0 except on the diagonal, in which case they are equal to 1.". (Contributed by Stefan O'Rear, 7-Sep-2015.) |
Ref | Expression |
---|---|
mat1.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mat1.o | ⊢ 1 = (1r‘𝑅) |
mat1.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
mat1 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | simpr 483 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring) | |
3 | mat1.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
4 | mat1.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
5 | eqid 2726 | . . . 4 ⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )) | |
6 | simpl 481 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin) | |
7 | 1, 2, 3, 4, 5, 6 | mamumat1cl 22435 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
8 | mat1.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
9 | 8, 1 | matbas2 22417 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((Base‘𝑅) ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) |
10 | 7, 9 | eleqtrd 2828 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )) ∈ (Base‘𝐴)) |
11 | eqid 2726 | . . . . . . . 8 ⊢ (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) | |
12 | 8, 11 | matmulr 22434 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) |
13 | 12 | adantr 479 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) |
14 | 13 | oveqd 7443 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)𝑥) = ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))(.r‘𝐴)𝑥)) |
15 | simplr 767 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑅 ∈ Ring) | |
16 | simpll 765 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑁 ∈ Fin) | |
17 | 9 | eleq2d 2812 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) ↔ 𝑥 ∈ (Base‘𝐴))) |
18 | 17 | biimpar 476 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑥 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
19 | 1, 15, 3, 4, 5, 16, 16, 11, 18 | mamulid 22437 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)𝑥) = 𝑥) |
20 | 14, 19 | eqtr3d 2768 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))(.r‘𝐴)𝑥) = 𝑥) |
21 | 13 | oveqd 7443 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑥(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) = (𝑥(.r‘𝐴)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )))) |
22 | 1, 15, 3, 4, 5, 16, 16, 11, 18 | mamurid 22438 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑥(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) = 𝑥) |
23 | 21, 22 | eqtr3d 2768 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑥(.r‘𝐴)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) = 𝑥) |
24 | 20, 23 | jca 510 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → (((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))(.r‘𝐴)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐴)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) = 𝑥)) |
25 | 24 | ralrimiva 3136 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ (Base‘𝐴)(((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))(.r‘𝐴)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐴)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) = 𝑥)) |
26 | 8 | matring 22439 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
27 | eqid 2726 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
28 | eqid 2726 | . . . 4 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
29 | eqid 2726 | . . . 4 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
30 | 27, 28, 29 | isringid 20252 | . . 3 ⊢ (𝐴 ∈ Ring → (((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )) ∈ (Base‘𝐴) ∧ ∀𝑥 ∈ (Base‘𝐴)(((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))(.r‘𝐴)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐴)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) = 𝑥)) ↔ (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )))) |
31 | 26, 30 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )) ∈ (Base‘𝐴) ∧ ∀𝑥 ∈ (Base‘𝐴)(((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))(.r‘𝐴)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐴)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) = 𝑥)) ↔ (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )))) |
32 | 10, 25, 31 | mpbi2and 710 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ifcif 4533 〈cotp 4641 × cxp 5682 ‘cfv 6556 (class class class)co 7426 ∈ cmpo 7428 ↑m cmap 8857 Fincfn 8976 Basecbs 17215 .rcmulr 17269 0gc0g 17456 1rcur 20166 Ringcrg 20218 maMul cmmul 22384 Mat cmat 22401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-ot 4642 df-uni 4916 df-int 4957 df-iun 5005 df-iin 5006 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-se 5640 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-isom 6565 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8005 df-2nd 8006 df-supp 8177 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-2o 8499 df-er 8736 df-map 8859 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9408 df-sup 9487 df-oi 9555 df-card 9984 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12613 df-dec 12732 df-uz 12877 df-fz 13541 df-fzo 13684 df-seq 14024 df-hash 14350 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-sca 17284 df-vsca 17285 df-ip 17286 df-tset 17287 df-ple 17288 df-ds 17290 df-hom 17292 df-cco 17293 df-0g 17458 df-gsum 17459 df-prds 17464 df-pws 17466 df-mre 17601 df-mrc 17602 df-acs 17604 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-mhm 18775 df-submnd 18776 df-grp 18933 df-minusg 18934 df-sbg 18935 df-mulg 19064 df-subg 19119 df-ghm 19209 df-cntz 19313 df-cmn 19782 df-abl 19783 df-mgp 20120 df-rng 20138 df-ur 20167 df-ring 20220 df-subrg 20555 df-lmod 20840 df-lss 20911 df-sra 21153 df-rgmod 21154 df-dsmm 21732 df-frlm 21747 df-mamu 22385 df-mat 22402 |
This theorem is referenced by: mat1ov 22444 matsc 22446 mattpos1 22452 mat1dimid 22470 1mavmul 22544 1marepvsma1 22579 pmat1op 22692 decpmatid 22766 |
Copyright terms: Public domain | W3C validator |