Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > minmar1marrep | Structured version Visualization version GIF version |
Description: The minor matrix is a special case of a matrix with a replaced row. (Contributed by AV, 12-Feb-2019.) (Revised by AV, 4-Jul-2022.) |
Ref | Expression |
---|---|
minmar1marrep.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
minmar1marrep.b | ⊢ 𝐵 = (Base‘𝐴) |
minmar1marrep.o | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
minmar1marrep | ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅) 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minmar1marrep.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | minmar1marrep.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
3 | eqid 2821 | . . . 4 ⊢ (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅) | |
4 | minmar1marrep.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
5 | eqid 2821 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | 1, 2, 3, 4, 5 | minmar1val0 21250 | . . 3 ⊢ (𝑀 ∈ 𝐵 → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , (0g‘𝑅)), (𝑖𝑀𝑗))))) |
7 | 6 | adantl 484 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , (0g‘𝑅)), (𝑖𝑀𝑗))))) |
8 | simpr 487 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) | |
9 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | 9, 4 | ringidcl 19312 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 1 ∈ (Base‘𝑅)) |
12 | eqid 2821 | . . . 4 ⊢ (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅) | |
13 | 1, 2, 12, 5 | marrepval0 21164 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 1 ∈ (Base‘𝑅)) → (𝑀(𝑁 matRRep 𝑅) 1 ) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , (0g‘𝑅)), (𝑖𝑀𝑗))))) |
14 | 8, 11, 13 | syl2anc 586 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀(𝑁 matRRep 𝑅) 1 ) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , (0g‘𝑅)), (𝑖𝑀𝑗))))) |
15 | 7, 14 | eqtr4d 2859 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅) 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ifcif 4467 ‘cfv 6350 (class class class)co 7150 ∈ cmpo 7152 Basecbs 16477 0gc0g 16707 1rcur 19245 Ringcrg 19291 Mat cmat 21010 matRRep cmarrep 21159 minMatR1 cminmar1 21236 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 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-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mgp 19234 df-ur 19246 df-ring 19293 df-mat 21011 df-marrep 21161 df-minmar1 21238 |
This theorem is referenced by: minmar1cl 21254 smadiadetglem1 21274 submatminr1 31070 |
Copyright terms: Public domain | W3C validator |