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Mirrors > Home > MPE Home > Th. List > matecld | Structured version Visualization version GIF version |
Description: Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring, deduction form. (Contributed by AV, 27-Nov-2019.) |
Ref | Expression |
---|---|
matecl.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
matecl.k | ⊢ 𝐾 = (Base‘𝑅) |
matecld.b | ⊢ 𝐵 = (Base‘𝐴) |
matecld.i | ⊢ (𝜑 → 𝐼 ∈ 𝑁) |
matecld.j | ⊢ (𝜑 → 𝐽 ∈ 𝑁) |
matecld.m | ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
Ref | Expression |
---|---|
matecld | ⊢ (𝜑 → (𝐼𝑀𝐽) ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | matecld.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑁) | |
2 | matecld.j | . 2 ⊢ (𝜑 → 𝐽 ∈ 𝑁) | |
3 | matecld.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵) | |
4 | matecld.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
5 | 3, 4 | eleqtrdi 2848 | . 2 ⊢ (𝜑 → 𝑀 ∈ (Base‘𝐴)) |
6 | matecl.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
7 | matecl.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
8 | 6, 7 | matecl 21790 | . 2 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾) |
9 | 1, 2, 5, 8 | syl3anc 1372 | 1 ⊢ (𝜑 → (𝐼𝑀𝐽) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6501 (class class class)co 7362 Basecbs 17090 Mat cmat 21770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-sup 9385 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-hom 17164 df-cco 17165 df-0g 17330 df-prds 17336 df-pws 17338 df-sra 20649 df-rgmod 20650 df-dsmm 21154 df-frlm 21169 df-mat 21771 |
This theorem is referenced by: mat1mhm 21849 dmatmulcl 21865 dmatcrng 21867 scmatscm 21878 scmatcrng 21886 maduf 22006 pmatcoe1fsupp 22066 cpmatel2 22078 cpmatmcllem 22083 mat2pmatf1 22094 mat2pmatghm 22095 mat2pmatmul 22096 mat2pmatlin 22100 m2cpm 22106 cpm2mf 22117 m2cpminvid 22118 m2cpminvid2lem 22119 m2cpminvid2 22120 m2cpmfo 22121 decpmatcl 22132 decpmatmullem 22136 decpmatmul 22137 pmatcollpw1lem1 22139 pmatcollpw1lem2 22140 pmatcollpw1 22141 pmatcollpw2 22143 monmatcollpw 22144 pmatcollpwlem 22145 pmatcollpw 22146 pmatcollpw3lem 22148 pmatcollpwscmatlem2 22155 pm2mpf1 22164 mptcoe1matfsupp 22167 mply1topmatcl 22170 mp2pm2mplem2 22172 mp2pm2mplem4 22174 mdetpmtr1 32444 mdetpmtr2 32445 mdetpmtr12 32446 madjusmdetlem1 32448 madjusmdetlem3 32450 mdetlap 32453 |
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