![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > minmar1cl | Structured version Visualization version GIF version |
Description: Closure of the row replacement function for square matrices: The matrix for a minor is a matrix. (Contributed by AV, 13-Feb-2019.) |
Ref | Expression |
---|---|
minmar1cl.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
minmar1cl.b | ⊢ 𝐵 = (Base‘𝐴) |
Ref | Expression |
---|---|
minmar1cl | ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐿) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minmar1cl.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | minmar1cl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
3 | eqid 2726 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
4 | 1, 2, 3 | minmar1marrep 22507 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅)(1r‘𝑅))) |
5 | 4 | adantr 480 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅)(1r‘𝑅))) |
6 | 5 | oveqd 7422 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐿) = (𝐾(𝑀(𝑁 matRRep 𝑅)(1r‘𝑅))𝐿)) |
7 | simpl 482 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) | |
8 | simpr 484 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) | |
9 | eqid 2726 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | 9, 3 | ringidcl 20165 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
11 | 10 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (1r‘𝑅) ∈ (Base‘𝑅)) |
12 | 7, 8, 11 | 3jca 1125 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (1r‘𝑅) ∈ (Base‘𝑅))) |
13 | 1, 2 | marrepcl 22421 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (1r‘𝑅) ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)(1r‘𝑅))𝐿) ∈ 𝐵) |
14 | 12, 13 | sylan 579 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)(1r‘𝑅))𝐿) ∈ 𝐵) |
15 | 6, 14 | eqeltrd 2827 | 1 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐿) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 1rcur 20086 Ringcrg 20138 Mat cmat 22262 matRRep cmarrep 22413 minMatR1 cminmar1 22490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-prds 17402 df-pws 17404 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-mgp 20040 df-ur 20087 df-ring 20140 df-sra 21021 df-rgmod 21022 df-dsmm 21627 df-frlm 21642 df-mat 22263 df-marrep 22415 df-minmar1 22492 |
This theorem is referenced by: smadiadetg 22530 madjusmdetlem1 33337 madjusmdetlem4 33340 |
Copyright terms: Public domain | W3C validator |