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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 2772 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | eqid 2772 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | crngring 19025 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
6 | 5 | adantl 474 | . . . . . 6 ⊢ ((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring) |
7 | 6 | 3ad2ant1 1113 | . . . . 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 20719 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
10 | 9 | simpld 487 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin) |
11 | 10 | adantr 473 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑁 ∈ Fin) |
12 | 11 | 3ad2ant2 1114 | . . . . 5 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → 𝑁 ∈ Fin) |
13 | 6, 11 | anim12ci 604 | . . . . . . . 8 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
14 | 13 | 3adant3 1112 | . . . . . . 7 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
15 | 1 | matring 20750 | . . . . . . 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 2772 | . . . . . . . . . 10 ⊢ (Unit‘𝐴) = (Unit‘𝐴) | |
19 | eqid 2772 | . . . . . . . . . 10 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
20 | 1, 17, 8, 18, 19 | matunit 20985 | . . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ (Unit‘𝐴) ↔ (𝐷‘𝑋) ∈ (Unit‘𝑅))) |
21 | 20 | bicomd 215 | . . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((𝐷‘𝑋) ∈ (Unit‘𝑅) ↔ 𝑋 ∈ (Unit‘𝐴))) |
22 | 21 | ad2ant2lr 735 | . . . . . . 7 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝐷‘𝑋) ∈ (Unit‘𝑅) ↔ 𝑋 ∈ (Unit‘𝐴))) |
23 | 22 | biimp3a 1448 | . . . . . 6 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → 𝑋 ∈ (Unit‘𝐴)) |
24 | eqid 2772 | . . . . . . 7 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
25 | eqid 2772 | . . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
26 | 18, 24, 25 | ringinvcl 19143 | . . . . . 6 ⊢ ((𝐴 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝐴)) → ((invr‘𝐴)‘𝑋) ∈ (Base‘𝐴)) |
27 | 16, 23, 26 | syl2anc 576 | . . . . 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 2851 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 ↔ 𝑌 ∈ ((Base‘𝑅) ↑𝑚 𝑁)) |
30 | 29 | biimpi 208 | . . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ ((Base‘𝑅) ↑𝑚 𝑁)) |
31 | 30 | adantl 474 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ ((Base‘𝑅) ↑𝑚 𝑁)) |
32 | 31 | 3ad2ant2 1114 | . . . . 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 20854 | . . . 4 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (((invr‘𝐴)‘𝑋) · 𝑌) ∈ ((Base‘𝑅) ↑𝑚 𝑁)) |
34 | 33, 28 | syl6eleqr 2871 | . . 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 20988 | . . . . . 6 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ((𝑋 · 𝑧) = 𝑌 ↔ 𝑧 = (((invr‘𝐴)‘𝑋) · 𝑌))) |
36 | 35 | adantr 473 | . . . . 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 240 | . . . 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 444 | . . 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 3531 | . 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 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 ∃wrex 3083 Vcvv 3409 ∅c0 4172 〈cop 4441 ‘cfv 6182 (class class class)co 6970 ↑𝑚 cmap 8200 Fincfn 8300 Basecbs 16333 .rcmulr 16416 Ringcrg 19014 CRingccrg 19015 Unitcui 19106 invrcinvr 19138 Mat cmat 20714 maVecMul cmvmul 20847 maDet cmdat 20891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-addf 10408 ax-mulf 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-xor 1489 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-ot 4444 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7495 df-2nd 7496 df-supp 7628 df-tpos 7689 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-2o 7900 df-oadd 7903 df-er 8083 df-map 8202 df-pm 8203 df-ixp 8254 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-fsupp 8623 df-sup 8695 df-oi 8763 df-card 9156 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-xnn0 11774 df-z 11788 df-dec 11906 df-uz 12053 df-rp 12199 df-fz 12703 df-fzo 12844 df-seq 13179 df-exp 13239 df-hash 13500 df-word 13667 df-lsw 13720 df-concat 13728 df-s1 13753 df-substr 13798 df-pfx 13847 df-splice 13954 df-reverse 13972 df-s2 14066 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-starv 16430 df-sca 16431 df-vsca 16432 df-ip 16433 df-tset 16434 df-ple 16435 df-ds 16437 df-unif 16438 df-hom 16439 df-cco 16440 df-0g 16565 df-gsum 16566 df-prds 16571 df-pws 16573 df-mre 16709 df-mrc 16710 df-acs 16712 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-mhm 17797 df-submnd 17798 df-grp 17888 df-minusg 17889 df-sbg 17890 df-mulg 18006 df-subg 18054 df-ghm 18121 df-gim 18164 df-cntz 18212 df-oppg 18239 df-symg 18261 df-pmtr 18325 df-psgn 18374 df-evpm 18375 df-cmn 18662 df-abl 18663 df-mgp 18957 df-ur 18969 df-srg 18973 df-ring 19016 df-cring 19017 df-oppr 19090 df-dvdsr 19108 df-unit 19109 df-invr 19139 df-dvr 19150 df-rnghom 19184 df-drng 19221 df-subrg 19250 df-lmod 19352 df-lss 19420 df-sra 19660 df-rgmod 19661 df-assa 19800 df-cnfld 20242 df-zring 20314 df-zrh 20347 df-dsmm 20572 df-frlm 20587 df-mamu 20691 df-mat 20715 df-mvmul 20848 df-mdet 20892 df-madu 20941 |
This theorem is referenced by: cramerlem3 20996 |
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