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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 2818 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | simpr 485 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring) | |
3 | mat1.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
4 | mat1.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
5 | eqid 2818 | . . . 4 ⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )) | |
6 | simpl 483 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin) | |
7 | 1, 2, 3, 4, 5, 6 | mamumat1cl 20976 | . . 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 20958 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((Base‘𝑅) ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) |
10 | 7, 9 | eleqtrd 2912 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )) ∈ (Base‘𝐴)) |
11 | eqid 2818 | . . . . . . . 8 ⊢ (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) | |
12 | 8, 11 | matmulr 20975 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) |
13 | 12 | adantr 481 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) |
14 | 13 | oveqd 7162 | . . . . 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 765 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑅 ∈ Ring) | |
16 | simpll 763 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑁 ∈ Fin) | |
17 | 9 | eleq2d 2895 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) ↔ 𝑥 ∈ (Base‘𝐴))) |
18 | 17 | biimpar 478 | . . . . . 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 20978 | . . . . 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 2855 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐴)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))(.r‘𝐴)𝑥) = 𝑥) |
21 | 13 | oveqd 7162 | . . . . 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 20979 | . . . . 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 2855 | . . . 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 512 | . . 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 3179 | . 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 20980 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
27 | eqid 2818 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
28 | eqid 2818 | . . . 4 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
29 | eqid 2818 | . . . 4 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
30 | 27, 28, 29 | isringid 19252 | . . 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 708 | 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 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ifcif 4463 〈cotp 4565 × cxp 5546 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ↑m cmap 8395 Fincfn 8497 Basecbs 16471 .rcmulr 16554 0gc0g 16701 1rcur 19180 Ringcrg 19226 maMul cmmul 20922 Mat cmat 20944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-0g 16703 df-gsum 16704 df-prds 16709 df-pws 16711 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-subrg 19462 df-lmod 19565 df-lss 19633 df-sra 19873 df-rgmod 19874 df-dsmm 20804 df-frlm 20819 df-mamu 20923 df-mat 20945 |
This theorem is referenced by: mat1ov 20985 matsc 20987 mattpos1 20993 mat1dimid 21011 1mavmul 21085 1marepvsma1 21120 pmat1op 21232 decpmatid 21306 |
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