Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > maduf | Structured version Visualization version GIF version |
Description: Creating the adjunct of matrices is a function from the set of matrices into the set of matrices. (Contributed by Stefan O'Rear, 11-Jul-2018.) |
Ref | Expression |
---|---|
maduf.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
maduf.j | ⊢ 𝐽 = (𝑁 maAdju 𝑅) |
maduf.b | ⊢ 𝐵 = (Base‘𝐴) |
Ref | Expression |
---|---|
maduf | ⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | maduf.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | eqid 2738 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | maduf.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
4 | 1, 3 | matrcl 21333 | . . . . 5 ⊢ (𝑚 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
5 | 4 | adantl 485 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
6 | 5 | simpld 498 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → 𝑁 ∈ Fin) |
7 | simpl 486 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → 𝑅 ∈ CRing) | |
8 | eqid 2738 | . . . . . . 7 ⊢ (𝑁 maDet 𝑅) = (𝑁 maDet 𝑅) | |
9 | 8, 1, 3, 2 | mdetf 21516 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (𝑁 maDet 𝑅):𝐵⟶(Base‘𝑅)) |
10 | 9 | adantr 484 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → (𝑁 maDet 𝑅):𝐵⟶(Base‘𝑅)) |
11 | 10 | 3ad2ant1 1135 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑁 maDet 𝑅):𝐵⟶(Base‘𝑅)) |
12 | 6 | 3ad2ant1 1135 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑁 ∈ Fin) |
13 | simp1l 1199 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ CRing) | |
14 | simp11l 1286 | . . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑅 ∈ CRing) | |
15 | crngring 19598 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
16 | eqid 2738 | . . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
17 | 2, 16 | ringidcl 19610 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
18 | eqid 2738 | . . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
19 | 2, 18 | ring0cl 19611 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅)) |
20 | 17, 19 | ifcld 4499 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅)) |
21 | 14, 15, 20 | 3syl 18 | . . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅)) |
22 | simp2 1139 | . . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑘 ∈ 𝑁) | |
23 | simp3 1140 | . . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑙 ∈ 𝑁) | |
24 | simp11r 1287 | . . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑚 ∈ 𝐵) | |
25 | 1, 2, 3, 22, 23, 24 | matecld 21347 | . . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘𝑚𝑙) ∈ (Base‘𝑅)) |
26 | 21, 25 | ifcld 4499 | . . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙)) ∈ (Base‘𝑅)) |
27 | 1, 2, 3, 12, 13, 26 | matbas2d 21344 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙))) ∈ 𝐵) |
28 | 11, 27 | ffvelrnd 6923 | . . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝑁 maDet 𝑅)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙)))) ∈ (Base‘𝑅)) |
29 | 1, 2, 3, 6, 7, 28 | matbas2d 21344 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙))))) ∈ 𝐵) |
30 | maduf.j | . . 3 ⊢ 𝐽 = (𝑁 maAdju 𝑅) | |
31 | 1, 8, 30, 3, 16, 18 | madufval 21558 | . 2 ⊢ 𝐽 = (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑅), (0g‘𝑅)), (𝑘𝑚𝑙)))))) |
32 | 29, 31 | fmptd 6949 | 1 ⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 Vcvv 3420 ifcif 4453 ⟶wf 6393 ‘cfv 6397 (class class class)co 7231 ∈ cmpo 7233 Fincfn 8646 Basecbs 16784 0gc0g 16968 1rcur 19540 Ringcrg 19586 CRingccrg 19587 Mat cmat 21328 maDet cmdat 21505 maAdju cmadu 21553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 ax-addf 10832 ax-mulf 10833 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-xor 1508 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-ot 4564 df-uni 4834 df-int 4874 df-iun 4920 df-iin 4921 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-se 5524 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-isom 6406 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-supp 7924 df-tpos 7988 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-2o 8223 df-er 8411 df-map 8530 df-pm 8531 df-ixp 8599 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-fsupp 9010 df-sup 9082 df-oi 9150 df-card 9579 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-div 11514 df-nn 11855 df-2 11917 df-3 11918 df-4 11919 df-5 11920 df-6 11921 df-7 11922 df-8 11923 df-9 11924 df-n0 12115 df-xnn0 12187 df-z 12201 df-dec 12318 df-uz 12463 df-rp 12611 df-fz 13120 df-fzo 13263 df-seq 13599 df-exp 13660 df-hash 13921 df-word 14094 df-lsw 14142 df-concat 14150 df-s1 14177 df-substr 14230 df-pfx 14260 df-splice 14339 df-reverse 14348 df-s2 14437 df-struct 16724 df-sets 16741 df-slot 16759 df-ndx 16769 df-base 16785 df-ress 16809 df-plusg 16839 df-mulr 16840 df-starv 16841 df-sca 16842 df-vsca 16843 df-ip 16844 df-tset 16845 df-ple 16846 df-ds 16848 df-unif 16849 df-hom 16850 df-cco 16851 df-0g 16970 df-gsum 16971 df-prds 16976 df-pws 16978 df-mre 17113 df-mrc 17114 df-acs 17116 df-mgm 18138 df-sgrp 18187 df-mnd 18198 df-mhm 18242 df-submnd 18243 df-efmnd 18320 df-grp 18392 df-minusg 18393 df-mulg 18513 df-subg 18564 df-ghm 18644 df-gim 18687 df-cntz 18735 df-oppg 18762 df-symg 18784 df-pmtr 18858 df-psgn 18907 df-cmn 19196 df-abl 19197 df-mgp 19529 df-ur 19541 df-ring 19588 df-cring 19589 df-oppr 19665 df-dvdsr 19683 df-unit 19684 df-invr 19714 df-dvr 19725 df-rnghom 19759 df-drng 19793 df-subrg 19822 df-sra 20233 df-rgmod 20234 df-cnfld 20388 df-zring 20460 df-zrh 20494 df-dsmm 20718 df-frlm 20733 df-mat 21329 df-mdet 21506 df-madu 21555 |
This theorem is referenced by: madutpos 21563 madugsum 21564 madurid 21565 madulid 21566 matinv 21598 cpmadugsumfi 21798 |
Copyright terms: Public domain | W3C validator |