Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > slesolex | Structured version Visualization version GIF version |
Description: Every system of linear equations represented by a matrix with a unit as determinant has a solution. (Contributed by AV, 11-Feb-2019.) (Revised by AV, 28-Feb-2019.) |
Ref | Expression |
---|---|
slesolex.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
slesolex.b | ⊢ 𝐵 = (Base‘𝐴) |
slesolex.v | ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) |
slesolex.x | ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) |
slesolex.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
Ref | Expression |
---|---|
slesolex | ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∃𝑧 ∈ 𝑉 (𝑋 · 𝑧) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slesolex.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | slesolex.x | . . . . 5 ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) | |
3 | eqid 2823 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | eqid 2823 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | crngring 19310 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
6 | 5 | adantl 484 | . . . . . 6 ⊢ ((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring) |
7 | 6 | 3ad2ant1 1129 | . . . . 5 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring) |
8 | slesolex.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐴) | |
9 | 1, 8 | matrcl 21023 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
10 | 9 | simpld 497 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin) |
11 | 10 | adantr 483 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑁 ∈ Fin) |
12 | 11 | 3ad2ant2 1130 | . . . . 5 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → 𝑁 ∈ Fin) |
13 | 6, 11 | anim12ci 615 | . . . . . . . 8 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
14 | 13 | 3adant3 1128 | . . . . . . 7 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
15 | 1 | matring 21054 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
16 | 14, 15 | syl 17 | . . . . . 6 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → 𝐴 ∈ Ring) |
17 | slesolex.d | . . . . . . . . . 10 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
18 | eqid 2823 | . . . . . . . . . 10 ⊢ (Unit‘𝐴) = (Unit‘𝐴) | |
19 | eqid 2823 | . . . . . . . . . 10 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
20 | 1, 17, 8, 18, 19 | matunit 21289 | . . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ (Unit‘𝐴) ↔ (𝐷‘𝑋) ∈ (Unit‘𝑅))) |
21 | 20 | bicomd 225 | . . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((𝐷‘𝑋) ∈ (Unit‘𝑅) ↔ 𝑋 ∈ (Unit‘𝐴))) |
22 | 21 | ad2ant2lr 746 | . . . . . . 7 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝐷‘𝑋) ∈ (Unit‘𝑅) ↔ 𝑋 ∈ (Unit‘𝐴))) |
23 | 22 | biimp3a 1465 | . . . . . 6 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → 𝑋 ∈ (Unit‘𝐴)) |
24 | eqid 2823 | . . . . . . 7 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
25 | eqid 2823 | . . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
26 | 18, 24, 25 | ringinvcl 19428 | . . . . . 6 ⊢ ((𝐴 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝐴)) → ((invr‘𝐴)‘𝑋) ∈ (Base‘𝐴)) |
27 | 16, 23, 26 | syl2anc 586 | . . . . 5 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ((invr‘𝐴)‘𝑋) ∈ (Base‘𝐴)) |
28 | slesolex.v | . . . . . . . . 9 ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) | |
29 | 28 | eleq2i 2906 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 ↔ 𝑌 ∈ ((Base‘𝑅) ↑m 𝑁)) |
30 | 29 | biimpi 218 | . . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ ((Base‘𝑅) ↑m 𝑁)) |
31 | 30 | adantl 484 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ ((Base‘𝑅) ↑m 𝑁)) |
32 | 31 | 3ad2ant2 1130 | . . . . 5 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → 𝑌 ∈ ((Base‘𝑅) ↑m 𝑁)) |
33 | 1, 2, 3, 4, 7, 12, 27, 32 | mavmulcl 21158 | . . . 4 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (((invr‘𝐴)‘𝑋) · 𝑌) ∈ ((Base‘𝑅) ↑m 𝑁)) |
34 | 33, 28 | eleqtrrdi 2926 | . . 3 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (((invr‘𝐴)‘𝑋) · 𝑌) ∈ 𝑉) |
35 | 1, 8, 28, 2, 17, 24 | slesolinvbi 21292 | . . . . . 6 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ((𝑋 · 𝑧) = 𝑌 ↔ 𝑧 = (((invr‘𝐴)‘𝑋) · 𝑌))) |
36 | 35 | adantr 483 | . . . . 5 ⊢ ((((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ ((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → ((𝑋 · 𝑧) = 𝑌 ↔ 𝑧 = (((invr‘𝐴)‘𝑋) · 𝑌))) |
37 | 36 | biimprd 250 | . . . 4 ⊢ ((((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ ((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑧 = (((invr‘𝐴)‘𝑋) · 𝑌) → (𝑋 · 𝑧) = 𝑌)) |
38 | 37 | impancom 454 | . . 3 ⊢ ((((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 = (((invr‘𝐴)‘𝑋) · 𝑌)) → (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑋 · 𝑧) = 𝑌)) |
39 | 34, 38 | rspcimedv 3616 | . 2 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∃𝑧 ∈ 𝑉 (𝑋 · 𝑧) = 𝑌)) |
40 | 39 | pm2.43i 52 | 1 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∃𝑧 ∈ 𝑉 (𝑋 · 𝑧) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 Vcvv 3496 ∅c0 4293 〈cop 4575 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 Fincfn 8511 Basecbs 16485 .rcmulr 16568 Ringcrg 19299 CRingccrg 19300 Unitcui 19391 invrcinvr 19423 Mat cmat 21018 maVecMul cmvmul 21151 maDet cmdat 21195 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-word 13865 df-lsw 13917 df-concat 13925 df-s1 13952 df-substr 14005 df-pfx 14035 df-splice 14114 df-reverse 14123 df-s2 14212 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-0g 16717 df-gsum 16718 df-prds 16723 df-pws 16725 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-efmnd 18036 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-gim 18401 df-cntz 18449 df-oppg 18476 df-symg 18498 df-pmtr 18572 df-psgn 18621 df-evpm 18622 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-srg 19258 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-rnghom 19469 df-drng 19506 df-subrg 19535 df-lmod 19638 df-lss 19706 df-sra 19946 df-rgmod 19947 df-assa 20087 df-cnfld 20548 df-zring 20620 df-zrh 20653 df-dsmm 20878 df-frlm 20893 df-mamu 20997 df-mat 21019 df-mvmul 21152 df-mdet 21196 df-madu 21245 |
This theorem is referenced by: cramerlem3 21300 |
Copyright terms: Public domain | W3C validator |