Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mdetcl | Structured version Visualization version GIF version |
Description: The determinant evaluates to an element of the base ring. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by AV, 7-Feb-2019.) |
Ref | Expression |
---|---|
mdetf.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
mdetf.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mdetf.b | ⊢ 𝐵 = (Base‘𝐴) |
mdetf.k | ⊢ 𝐾 = (Base‘𝑅) |
Ref | Expression |
---|---|
mdetcl | ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdetf.d | . . 3 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
2 | mdetf.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | mdetf.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
4 | mdetf.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
5 | 1, 2, 3, 4 | mdetf 21134 | . 2 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶𝐾) |
6 | 5 | ffvelrnda 6844 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 CRingccrg 19229 Mat cmat 20946 maDet cmdat 21123 |
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 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-xor 1496 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 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 3497 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 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-ot 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-supp 7822 df-tpos 7883 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-pm 8399 df-ixp 8451 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-fsupp 8823 df-sup 8895 df-oi 8963 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-xnn0 11957 df-z 11971 df-dec 12088 df-uz 12233 df-rp 12380 df-fz 12883 df-fzo 13024 df-seq 13360 df-exp 13420 df-hash 13681 df-word 13852 df-lsw 13905 df-concat 13913 df-s1 13940 df-substr 13993 df-pfx 14023 df-splice 14102 df-reverse 14111 df-s2 14200 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-starv 16570 df-sca 16571 df-vsca 16572 df-ip 16573 df-tset 16574 df-ple 16575 df-ds 16577 df-unif 16578 df-hom 16579 df-cco 16580 df-0g 16705 df-gsum 16706 df-prds 16711 df-pws 16713 df-mre 16847 df-mrc 16848 df-acs 16850 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-mhm 17946 df-submnd 17947 df-grp 18046 df-minusg 18047 df-mulg 18165 df-subg 18216 df-ghm 18296 df-gim 18339 df-cntz 18387 df-oppg 18414 df-symg 18436 df-pmtr 18501 df-psgn 18550 df-cmn 18839 df-abl 18840 df-mgp 19171 df-ur 19183 df-ring 19230 df-cring 19231 df-oppr 19304 df-dvdsr 19322 df-unit 19323 df-invr 19353 df-dvr 19364 df-rnghom 19398 df-drng 19435 df-subrg 19464 df-sra 19875 df-rgmod 19876 df-cnfld 20476 df-zring 20548 df-zrh 20581 df-dsmm 20806 df-frlm 20821 df-mat 20947 df-mdet 21124 |
This theorem is referenced by: mdetuni 21161 matunit 21217 chpmatply1 21370 mdetpmtr12 30990 madjusmdetlem1 30992 mdetlap 30997 matunitlindflem2 34771 matunitlindf 34772 |
Copyright terms: Public domain | W3C validator |