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Mirrors > Home > MPE Home > Th. List > scmate | Structured version Visualization version GIF version |
Description: An entry of an 𝑁 x 𝑁 scalar matrix over the ring 𝑅. (Contributed by AV, 18-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 |
---|---|
scmate | ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scmate.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
2 | scmatmat.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | scmatmat.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
4 | eqid 2821 | . . . 4 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
5 | eqid 2821 | . . . 4 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴) | |
6 | scmatmat.s | . . . 4 ⊢ 𝑆 = (𝑁 ScMat 𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | scmatscmid 21109 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝑀 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))) |
8 | oveq 7156 | . . . . . . 7 ⊢ (𝑀 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴)) → (𝐼𝑀𝐽) = (𝐼(𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))𝐽)) | |
9 | simpll1 1208 | . . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 𝑐 ∈ 𝐾) → 𝑁 ∈ Fin) | |
10 | simpll2 1209 | . . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 𝑐 ∈ 𝐾) → 𝑅 ∈ Ring) | |
11 | simpr 487 | . . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 𝑐 ∈ 𝐾) → 𝑐 ∈ 𝐾) | |
12 | simplr 767 | . . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 𝑐 ∈ 𝐾) → (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) | |
13 | scmate.0 | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
14 | 2, 1, 13, 4, 5 | scmatscmide 21110 | . . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑐 ∈ 𝐾) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )) |
15 | 9, 10, 11, 12, 14 | syl31anc 1369 | . . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 𝑐 ∈ 𝐾) → (𝐼(𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )) |
16 | 8, 15 | sylan9eqr 2878 | . . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 𝑐 ∈ 𝐾) ∧ 𝑀 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))) → (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )) |
17 | 16 | ex 415 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 𝑐 ∈ 𝐾) → (𝑀 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴)) → (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 ))) |
18 | 17 | reximdva 3274 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (∃𝑐 ∈ 𝐾 𝑀 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴)) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 ))) |
19 | 18 | ex 415 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (∃𝑐 ∈ 𝐾 𝑀 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴)) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )))) |
20 | 7, 19 | mpid 44 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 ))) |
21 | 20 | imp 409 | 1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ifcif 4466 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 Basecbs 16477 ·𝑠 cvsca 16563 0gc0g 16707 1rcur 19245 Ringcrg 19291 Mat cmat 21010 ScMat cscmat 21092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 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 3496 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 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-ot 4569 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 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 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-gsum 16710 df-prds 16715 df-pws 16717 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-subrg 19527 df-lmod 19630 df-lss 19698 df-sra 19938 df-rgmod 19939 df-dsmm 20870 df-frlm 20885 df-mamu 20989 df-mat 21011 df-scmat 21094 |
This theorem is referenced by: (None) |
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