slesolex - Metamath Proof Explorer

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Theorem slesolex 21739

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.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
slesolex	⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∃𝑧 ∈ 𝑉 (𝑋 · 𝑧) = 𝑌)

Distinct variable groups: 𝑧,𝐴 𝑧,𝐵 𝑧,𝐷 𝑧,𝑁 𝑧,𝑅 𝑧,𝑉 𝑧,𝑋 𝑧,𝑌 𝑧, ·

**Proof of Theorem slesolex**
Step	Hyp	Ref	Expression
1		slesolex.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		slesolex.x	. . . . 5 ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
3		eqid 2738	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
4		eqid 2738	. . . . 5 ⊢ (._r‘𝑅) = (._r‘𝑅)
5		crngring 19710	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
6	5	adantl 481	. . . . . 6 ⊢ ((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
7	6	3ad2ant1 1131	. . . . 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 21469	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
10	9	simpld 494	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin)
11	10	adantr 480	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑁 ∈ Fin)
12	11	3ad2ant2 1132	. . . . 5 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → 𝑁 ∈ Fin)
13	6, 11	anim12ci 613	. . . . . . . 8 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
14	13	3adant3 1130	. . . . . . 7 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
15	1	matring 21500	. . . . . . 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 2738	. . . . . . . . . 10 ⊢ (Unit‘𝐴) = (Unit‘𝐴)
19		eqid 2738	. . . . . . . . . 10 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
20	1, 17, 8, 18, 19	matunit 21735	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ (Unit‘𝐴) ↔ (𝐷‘𝑋) ∈ (Unit‘𝑅)))
21	20	bicomd 222	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((𝐷‘𝑋) ∈ (Unit‘𝑅) ↔ 𝑋 ∈ (Unit‘𝐴)))
22	21	ad2ant2lr 744	. . . . . . 7 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝐷‘𝑋) ∈ (Unit‘𝑅) ↔ 𝑋 ∈ (Unit‘𝐴)))
23	22	biimp3a 1467	. . . . . 6 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → 𝑋 ∈ (Unit‘𝐴))
24		eqid 2738	. . . . . . 7 ⊢ (inv_r‘𝐴) = (inv_r‘𝐴)
25		eqid 2738	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
26	18, 24, 25	ringinvcl 19833	. . . . . 6 ⊢ ((𝐴 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝐴)) → ((inv_r‘𝐴)‘𝑋) ∈ (Base‘𝐴))
27	16, 23, 26	syl2anc 583	. . . . 5 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ((inv_r‘𝐴)‘𝑋) ∈ (Base‘𝐴))
28		slesolex.v	. . . . . . . . 9 ⊢ 𝑉 = ((Base‘𝑅) ↑_m 𝑁)
29	28	eleq2i 2830	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 ↔ 𝑌 ∈ ((Base‘𝑅) ↑_m 𝑁))
30	29	biimpi 215	. . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ ((Base‘𝑅) ↑_m 𝑁))
31	30	adantl 481	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ ((Base‘𝑅) ↑_m 𝑁))
32	31	3ad2ant2 1132	. . . . 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 21604	. . . 4 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (((inv_r‘𝐴)‘𝑋) · 𝑌) ∈ ((Base‘𝑅) ↑_m 𝑁))
34	33, 28	eleqtrrdi 2850	. . 3 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (((inv_r‘𝐴)‘𝑋) · 𝑌) ∈ 𝑉)
35	1, 8, 28, 2, 17, 24	slesolinvbi 21738	. . . . . 6 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ((𝑋 · 𝑧) = 𝑌 ↔ 𝑧 = (((inv_r‘𝐴)‘𝑋) · 𝑌)))
36	35	adantr 480	. . . . 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 = (((inv_r‘𝐴)‘𝑋) · 𝑌)))
37	36	biimprd 247	. . . 4 ⊢ ((((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ ((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑧 = (((inv_r‘𝐴)‘𝑋) · 𝑌) → (𝑋 · 𝑧) = 𝑌))
38	37	impancom 451	. . 3 ⊢ ((((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 = (((inv_r‘𝐴)‘𝑋) · 𝑌)) → (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑋 · 𝑧) = 𝑌))
39	34, 38	rspcimedv 3542	. 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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 Vcvv 3422 ∅c0 4253 ⟨cop 4564 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 Fincfn 8691 Basecbs 16840 ._rcmulr 16889 Ringcrg 19698 CRingccrg 19699 Unitcui 19796 inv_rcinvr 19828 Mat cmat 21464 maVecMul cmvmul 21597 maDet cmdat 21641

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-xor 1504 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-word 14146 df-lsw 14194 df-concat 14202 df-s1 14229 df-substr 14282 df-pfx 14312 df-splice 14391 df-reverse 14400 df-s2 14489 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-efmnd 18423 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-gim 18790 df-cntz 18838 df-oppg 18865 df-symg 18890 df-pmtr 18965 df-psgn 19014 df-evpm 19015 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-srg 19657 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-lmod 20040 df-lss 20109 df-sra 20349 df-rgmod 20350 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-dsmm 20849 df-frlm 20864 df-assa 20970 df-mamu 21443 df-mat 21465 df-mvmul 21598 df-mdet 21642 df-madu 21691

This theorem is referenced by: cramerlem3 21746

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