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Mirrors > Home > MPE Home > Th. List > marep01ma | Structured version Visualization version GIF version |
Description: Replacing a row of a square matrix by a row with 0's and a 1 results in a square matrix of the same dimension. (Contributed by AV, 30-Dec-2018.) |
Ref | Expression |
---|---|
marep01ma.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
marep01ma.b | ⊢ 𝐵 = (Base‘𝐴) |
marep01ma.r | ⊢ 𝑅 ∈ CRing |
marep01ma.0 | ⊢ 0 = (0g‘𝑅) |
marep01ma.1 | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
marep01ma | ⊢ (𝑀 ∈ 𝐵 → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | marep01ma.a | . 2 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | eqid 2799 | . 2 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | marep01ma.b | . 2 ⊢ 𝐵 = (Base‘𝐴) | |
4 | 1, 3 | matrcl 20543 | . . 3 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
5 | 4 | simpld 489 | . 2 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
6 | marep01ma.r | . . 3 ⊢ 𝑅 ∈ CRing | |
7 | 6 | a1i 11 | . 2 ⊢ (𝑀 ∈ 𝐵 → 𝑅 ∈ CRing) |
8 | crngring 18874 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
9 | marep01ma.1 | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
10 | 2, 9 | ringidcl 18884 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
11 | 6, 8, 10 | mp2b 10 | . . . . 5 ⊢ 1 ∈ (Base‘𝑅) |
12 | marep01ma.0 | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
13 | 2, 12 | ring0cl 18885 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
14 | 6, 8, 13 | mp2b 10 | . . . . 5 ⊢ 0 ∈ (Base‘𝑅) |
15 | 11, 14 | ifcli 4323 | . . . 4 ⊢ if(𝑙 = 𝐼, 1 , 0 ) ∈ (Base‘𝑅) |
16 | 15 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, 1 , 0 ) ∈ (Base‘𝑅)) |
17 | simp2 1168 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑘 ∈ 𝑁) | |
18 | simp3 1169 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑙 ∈ 𝑁) | |
19 | id 22 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ 𝐵) | |
20 | 19, 3 | syl6eleq 2888 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐴)) |
21 | 20 | 3ad2ant1 1164 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴)) |
22 | 1, 2 | matecl 20556 | . . . 4 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝑘𝑀𝑙) ∈ (Base‘𝑅)) |
23 | 17, 18, 21, 22 | syl3anc 1491 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘𝑀𝑙) ∈ (Base‘𝑅)) |
24 | 16, 23 | ifcld 4322 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)) ∈ (Base‘𝑅)) |
25 | 1, 2, 3, 5, 7, 24 | matbas2d 20554 | 1 ⊢ (𝑀 ∈ 𝐵 → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 Vcvv 3385 ifcif 4277 ‘cfv 6101 (class class class)co 6878 ↦ cmpt2 6880 Fincfn 8195 Basecbs 16184 0gc0g 16415 1rcur 18817 Ringcrg 18863 CRingccrg 18864 Mat cmat 20538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-ot 4377 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-sup 8590 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-fz 12581 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-hom 16291 df-cco 16292 df-0g 16417 df-prds 16423 df-pws 16425 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-grp 17741 df-mgp 18806 df-ur 18818 df-ring 18865 df-cring 18866 df-sra 19495 df-rgmod 19496 df-dsmm 20401 df-frlm 20416 df-mat 20539 |
This theorem is referenced by: smadiadetlem0 20794 smadiadetlem1 20795 smadiadet 20803 |
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