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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 2771 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | eqid 2771 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | crngring 18766 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
6 | 5 | adantl 467 | . . . . . 6 ⊢ ((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring) |
7 | 6 | 3ad2ant1 1127 | . . . . 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 20435 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
10 | 9 | simpld 482 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin) |
11 | 10 | adantr 466 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑁 ∈ Fin) |
12 | 11 | 3ad2ant2 1128 | . . . . 5 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → 𝑁 ∈ Fin) |
13 | 6, 11 | anim12ci 601 | . . . . . . . 8 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
14 | 13 | 3adant3 1126 | . . . . . . 7 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
15 | 1 | matring 20466 | . . . . . . 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 2771 | . . . . . . . . . 10 ⊢ (Unit‘𝐴) = (Unit‘𝐴) | |
19 | eqid 2771 | . . . . . . . . . 10 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
20 | 1, 17, 8, 18, 19 | matunit 20703 | . . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ (Unit‘𝐴) ↔ (𝐷‘𝑋) ∈ (Unit‘𝑅))) |
21 | 20 | bicomd 213 | . . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((𝐷‘𝑋) ∈ (Unit‘𝑅) ↔ 𝑋 ∈ (Unit‘𝐴))) |
22 | 21 | ad2ant2lr 742 | . . . . . . 7 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝐷‘𝑋) ∈ (Unit‘𝑅) ↔ 𝑋 ∈ (Unit‘𝐴))) |
23 | 22 | biimp3a 1580 | . . . . . 6 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → 𝑋 ∈ (Unit‘𝐴)) |
24 | eqid 2771 | . . . . . . 7 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
25 | eqid 2771 | . . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
26 | 18, 24, 25 | ringinvcl 18884 | . . . . . 6 ⊢ ((𝐴 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝐴)) → ((invr‘𝐴)‘𝑋) ∈ (Base‘𝐴)) |
27 | 16, 23, 26 | syl2anc 573 | . . . . 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 2842 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 ↔ 𝑌 ∈ ((Base‘𝑅) ↑𝑚 𝑁)) |
30 | 29 | biimpi 206 | . . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ ((Base‘𝑅) ↑𝑚 𝑁)) |
31 | 30 | adantl 467 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ ((Base‘𝑅) ↑𝑚 𝑁)) |
32 | 31 | 3ad2ant2 1128 | . . . . 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 20571 | . . . 4 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (((invr‘𝐴)‘𝑋) · 𝑌) ∈ ((Base‘𝑅) ↑𝑚 𝑁)) |
34 | 33, 28 | syl6eleqr 2861 | . . 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 20706 | . . . . . 6 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ((𝑋 · 𝑧) = 𝑌 ↔ 𝑧 = (((invr‘𝐴)‘𝑋) · 𝑌))) |
36 | 35 | adantr 466 | . . . . 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 238 | . . . 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 439 | . . 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 3462 | . 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 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∃wrex 3062 Vcvv 3351 ∅c0 4063 〈cop 4323 ‘cfv 6030 (class class class)co 6796 ↑𝑚 cmap 8013 Fincfn 8113 Basecbs 16064 .rcmulr 16150 Ringcrg 18755 CRingccrg 18756 Unitcui 18847 invrcinvr 18879 Mat cmat 20430 maVecMul cmvmul 20564 maDet cmdat 20608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-addf 10221 ax-mulf 10222 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-xor 1613 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-ot 4326 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-of 7048 df-om 7217 df-1st 7319 df-2nd 7320 df-supp 7451 df-tpos 7508 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-er 7900 df-map 8015 df-pm 8016 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8436 df-sup 8508 df-oi 8575 df-card 8969 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-7 11290 df-8 11291 df-9 11292 df-n0 11500 df-xnn0 11571 df-z 11585 df-dec 11701 df-uz 11894 df-rp 12036 df-fz 12534 df-fzo 12674 df-seq 13009 df-exp 13068 df-hash 13322 df-word 13495 df-lsw 13496 df-concat 13497 df-s1 13498 df-substr 13499 df-splice 13500 df-reverse 13501 df-s2 13802 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-0g 16310 df-gsum 16311 df-prds 16316 df-pws 16318 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-mhm 17543 df-submnd 17544 df-grp 17633 df-minusg 17634 df-sbg 17635 df-mulg 17749 df-subg 17799 df-ghm 17866 df-gim 17909 df-cntz 17957 df-oppg 17983 df-symg 18005 df-pmtr 18069 df-psgn 18118 df-evpm 18119 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-srg 18714 df-ring 18757 df-cring 18758 df-oppr 18831 df-dvdsr 18849 df-unit 18850 df-invr 18880 df-dvr 18891 df-rnghom 18925 df-drng 18959 df-subrg 18988 df-lmod 19075 df-lss 19143 df-sra 19387 df-rgmod 19388 df-assa 19527 df-cnfld 19962 df-zring 20034 df-zrh 20067 df-dsmm 20293 df-frlm 20308 df-mamu 20407 df-mat 20431 df-mvmul 20565 df-mdet 20609 df-madu 20658 |
This theorem is referenced by: cramerlem3 20715 |
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