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Mirrors > Home > MPE Home > Th. List > scmatmats | Structured version Visualization version GIF version |
Description: The set of an 𝑁 x 𝑁 scalar matrices over the ring 𝑅 expressed as a subset of 𝑁 x 𝑁 matrices over the ring 𝑅 with certain properties for their entries. (Contributed by AV, 31-Oct-2019.) (Revised by AV, 19-Dec-2019.) |
Ref | Expression |
---|---|
scmatmat.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
scmatmat.b | ⊢ 𝐵 = (Base‘𝐴) |
scmatmat.s | ⊢ 𝑆 = (𝑁 ScMat 𝑅) |
scmate.k | ⊢ 𝐾 = (Base‘𝑅) |
scmate.0 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
scmatmats | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scmate.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
2 | scmatmat.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | scmatmat.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
4 | eqid 2825 | . . 3 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
5 | eqid 2825 | . . 3 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴) | |
6 | scmatmat.s | . . 3 ⊢ 𝑆 = (𝑁 ScMat 𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | scmatval 20678 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))}) |
8 | simpr 479 | . . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) → 𝑚 ∈ 𝐵) | |
9 | 8 | adantr 474 | . . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) → 𝑚 ∈ 𝐵) |
10 | simpll 785 | . . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) | |
11 | 2 | matring 20616 | . . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
12 | 3, 4 | ringidcl 18922 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ Ring → (1r‘𝐴) ∈ 𝐵) |
13 | 11, 12 | syl 17 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) ∈ 𝐵) |
14 | 13 | adantr 474 | . . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) → (1r‘𝐴) ∈ 𝐵) |
15 | 14 | anim1i 610 | . . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) → ((1r‘𝐴) ∈ 𝐵 ∧ 𝑐 ∈ 𝐾)) |
16 | 15 | ancomd 455 | . . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) → (𝑐 ∈ 𝐾 ∧ (1r‘𝐴) ∈ 𝐵)) |
17 | 1, 2, 3, 5 | matvscl 20604 | . . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑐 ∈ 𝐾 ∧ (1r‘𝐴) ∈ 𝐵)) → (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴)) ∈ 𝐵) |
18 | 10, 16, 17 | syl2anc 581 | . . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) → (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴)) ∈ 𝐵) |
19 | 2, 3 | eqmat 20597 | . . . . . 6 ⊢ ((𝑚 ∈ 𝐵 ∧ (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴)) ∈ 𝐵) → (𝑚 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴)) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = (𝑖(𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))𝑗))) |
20 | 9, 18, 19 | syl2anc 581 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) → (𝑚 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴)) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = (𝑖(𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))𝑗))) |
21 | simplll 793 | . . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) → 𝑁 ∈ Fin) | |
22 | simpllr 795 | . . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) → 𝑅 ∈ Ring) | |
23 | simpr 479 | . . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) → 𝑐 ∈ 𝐾) | |
24 | 21, 22, 23 | 3jca 1164 | . . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑐 ∈ 𝐾)) |
25 | scmate.0 | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
26 | 2, 1, 25, 4, 5 | scmatscmide 20681 | . . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑐 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )) |
27 | 24, 26 | sylan 577 | . . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )) |
28 | 27 | eqeq2d 2835 | . . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑖𝑚𝑗) = (𝑖(𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))𝑗) ↔ (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ))) |
29 | 28 | 2ralbidva 3197 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = (𝑖(𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))𝑗) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ))) |
30 | 20, 29 | bitrd 271 | . . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) ∧ 𝑐 ∈ 𝐾) → (𝑚 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴)) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ))) |
31 | 30 | rexbidva 3259 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝐵) → (∃𝑐 ∈ 𝐾 𝑚 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴)) ↔ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ))) |
32 | 31 | rabbidva 3401 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))} = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )}) |
33 | 7, 32 | eqtrd 2861 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3117 ∃wrex 3118 {crab 3121 ifcif 4306 ‘cfv 6123 (class class class)co 6905 Fincfn 8222 Basecbs 16222 ·𝑠 cvsca 16309 0gc0g 16453 1rcur 18855 Ringcrg 18901 Mat cmat 20580 ScMat cscmat 20663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-ot 4406 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-ixp 8176 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-sup 8617 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-fz 12620 df-fzo 12761 df-seq 13096 df-hash 13411 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-hom 16329 df-cco 16330 df-0g 16455 df-gsum 16456 df-prds 16461 df-pws 16463 df-mre 16599 df-mrc 16600 df-acs 16602 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-mhm 17688 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-mulg 17895 df-subg 17942 df-ghm 18009 df-cntz 18100 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-subrg 19134 df-lmod 19221 df-lss 19289 df-sra 19533 df-rgmod 19534 df-dsmm 20439 df-frlm 20454 df-mamu 20557 df-mat 20581 df-scmat 20665 |
This theorem is referenced by: scmateALT 20686 scmatdmat 20689 |
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