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Mirrors > Home > MPE Home > Th. List > matsc | Structured version Visualization version GIF version |
Description: The identity matrix multiplied with a scalar. (Contributed by Stefan O'Rear, 16-Jul-2018.) |
Ref | Expression |
---|---|
matsc.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
matsc.k | ⊢ 𝐾 = (Base‘𝑅) |
matsc.m | ⊢ · = ( ·𝑠 ‘𝐴) |
matsc.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
matsc | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝐿 · (1r‘𝐴)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 𝐿, 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1063 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → 𝐿 ∈ 𝐾) | |
2 | 3simpa 1058 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) | |
3 | matsc.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
4 | 3 | matring 20249 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
5 | eqid 2622 | . . . . 5 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
6 | eqid 2622 | . . . . 5 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
7 | 5, 6 | ringidcl 18568 | . . . 4 ⊢ (𝐴 ∈ Ring → (1r‘𝐴) ∈ (Base‘𝐴)) |
8 | 2, 4, 7 | 3syl 18 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (1r‘𝐴) ∈ (Base‘𝐴)) |
9 | matsc.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
10 | matsc.m | . . . 4 ⊢ · = ( ·𝑠 ‘𝐴) | |
11 | eqid 2622 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
12 | eqid 2622 | . . . 4 ⊢ (𝑁 × 𝑁) = (𝑁 × 𝑁) | |
13 | 3, 5, 9, 10, 11, 12 | matvsca2 20234 | . . 3 ⊢ ((𝐿 ∈ 𝐾 ∧ (1r‘𝐴) ∈ (Base‘𝐴)) → (𝐿 · (1r‘𝐴)) = (((𝑁 × 𝑁) × {𝐿}) ∘𝑓 (.r‘𝑅)(1r‘𝐴))) |
14 | 1, 8, 13 | syl2anc 693 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝐿 · (1r‘𝐴)) = (((𝑁 × 𝑁) × {𝐿}) ∘𝑓 (.r‘𝑅)(1r‘𝐴))) |
15 | simp1 1061 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → 𝑁 ∈ Fin) | |
16 | simp13 1093 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐿 ∈ 𝐾) | |
17 | fvex 6201 | . . . . 5 ⊢ (1r‘𝑅) ∈ V | |
18 | matsc.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
19 | fvex 6201 | . . . . . 6 ⊢ (0g‘𝑅) ∈ V | |
20 | 18, 19 | eqeltri 2697 | . . . . 5 ⊢ 0 ∈ V |
21 | 17, 20 | ifex 4156 | . . . 4 ⊢ if(𝑖 = 𝑗, (1r‘𝑅), 0 ) ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝑗, (1r‘𝑅), 0 ) ∈ V) |
23 | fconstmpt2 6755 | . . . 4 ⊢ ((𝑁 × 𝑁) × {𝐿}) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝐿) | |
24 | 23 | a1i 11 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → ((𝑁 × 𝑁) × {𝐿}) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝐿)) |
25 | eqid 2622 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
26 | 3, 25, 18 | mat1 20253 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘𝑅), 0 ))) |
27 | 26 | 3adant3 1081 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘𝑅), 0 ))) |
28 | 15, 15, 16, 22, 24, 27 | offval22 7253 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (((𝑁 × 𝑁) × {𝐿}) ∘𝑓 (.r‘𝑅)(1r‘𝐴)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐿(.r‘𝑅)if(𝑖 = 𝑗, (1r‘𝑅), 0 )))) |
29 | ovif2 6738 | . . . 4 ⊢ (𝐿(.r‘𝑅)if(𝑖 = 𝑗, (1r‘𝑅), 0 )) = if(𝑖 = 𝑗, (𝐿(.r‘𝑅)(1r‘𝑅)), (𝐿(.r‘𝑅) 0 )) | |
30 | 9, 11, 25 | ringridm 18572 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝐿(.r‘𝑅)(1r‘𝑅)) = 𝐿) |
31 | 30 | 3adant1 1079 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝐿(.r‘𝑅)(1r‘𝑅)) = 𝐿) |
32 | 9, 11, 18 | ringrz 18588 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝐿(.r‘𝑅) 0 ) = 0 ) |
33 | 32 | 3adant1 1079 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝐿(.r‘𝑅) 0 ) = 0 ) |
34 | 31, 33 | ifeq12d 4106 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → if(𝑖 = 𝑗, (𝐿(.r‘𝑅)(1r‘𝑅)), (𝐿(.r‘𝑅) 0 )) = if(𝑖 = 𝑗, 𝐿, 0 )) |
35 | 29, 34 | syl5eq 2668 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝐿(.r‘𝑅)if(𝑖 = 𝑗, (1r‘𝑅), 0 )) = if(𝑖 = 𝑗, 𝐿, 0 )) |
36 | 35 | mpt2eq3dv 6721 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐿(.r‘𝑅)if(𝑖 = 𝑗, (1r‘𝑅), 0 ))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 𝐿, 0 ))) |
37 | 14, 28, 36 | 3eqtrd 2660 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝐿 · (1r‘𝐴)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 𝐿, 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ifcif 4086 {csn 4177 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ∘𝑓 cof 6895 Fincfn 7955 Basecbs 15857 .rcmulr 15942 ·𝑠 cvsca 15945 0gc0g 16100 1rcur 18501 Ringcrg 18547 Mat cmat 20213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-dsmm 20076 df-frlm 20091 df-mamu 20190 df-mat 20214 |
This theorem is referenced by: scmatscm 20319 madurid 20450 chmatval 20634 |
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