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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 2821 | . 2 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | marep01ma.b | . 2 ⊢ 𝐵 = (Base‘𝐴) | |
4 | 1, 3 | matrcl 21015 | . . 3 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
5 | 4 | simpld 497 | . 2 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
6 | marep01ma.r | . . 3 ⊢ 𝑅 ∈ CRing | |
7 | 6 | a1i 11 | . 2 ⊢ (𝑀 ∈ 𝐵 → 𝑅 ∈ CRing) |
8 | crngring 19302 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
9 | marep01ma.1 | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
10 | 2, 9 | ringidcl 19312 | . . . . . 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 19313 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
14 | 6, 8, 13 | mp2b 10 | . . . . 5 ⊢ 0 ∈ (Base‘𝑅) |
15 | 11, 14 | ifcli 4512 | . . . 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 1133 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑘 ∈ 𝑁) | |
18 | simp3 1134 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑙 ∈ 𝑁) | |
19 | id 22 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ 𝐵) | |
20 | 19, 3 | eleqtrdi 2923 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐴)) |
21 | 20 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴)) |
22 | 1, 2 | matecl 21028 | . . . 4 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝑘𝑀𝑙) ∈ (Base‘𝑅)) |
23 | 17, 18, 21, 22 | syl3anc 1367 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘𝑀𝑙) ∈ (Base‘𝑅)) |
24 | 16, 23 | ifcld 4511 | . 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 21026 | 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 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ifcif 4466 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 Fincfn 8503 Basecbs 16477 0gc0g 16707 1rcur 19245 Ringcrg 19291 CRingccrg 19292 Mat cmat 21010 |
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-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-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-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 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-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-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-prds 16715 df-pws 16717 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-sra 19938 df-rgmod 19939 df-dsmm 20870 df-frlm 20885 df-mat 21011 |
This theorem is referenced by: smadiadetlem0 21264 smadiadetlem1 21265 smadiadet 21273 |
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