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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 21021 | . . 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 19308 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
9 | marep01ma.1 | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
10 | 2, 9 | ringidcl 19318 | . . . . . 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 19319 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
14 | 6, 8, 13 | mp2b 10 | . . . . 5 ⊢ 0 ∈ (Base‘𝑅) |
15 | 11, 14 | ifcli 4513 | . . . 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 21034 | . . . 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 4512 | . 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 21032 | 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 1537 ∈ wcel 2114 Vcvv 3494 ifcif 4467 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 Fincfn 8509 Basecbs 16483 0gc0g 16713 1rcur 19251 Ringcrg 19297 CRingccrg 19298 Mat cmat 21016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-0g 16715 df-prds 16721 df-pws 16723 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-sra 19944 df-rgmod 19945 df-dsmm 20876 df-frlm 20891 df-mat 21017 |
This theorem is referenced by: smadiadetlem0 21270 smadiadetlem1 21271 smadiadet 21279 |
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