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Theorem cramerimp 20406
Description: One direction of Cramer's rule (according to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule: "[Cramer's rule] ... expresses the solution [of a system of linear equations] in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations."): The ith component of the solution vector of a system of linear equations equals the determinant of the matrix of the system of linear equations with the ith column replaced by the righthand side vector of the system of linear equations divided by the determinant of the matrix of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.)
Hypotheses
Ref Expression
cramerimp.a 𝐴 = (𝑁 Mat 𝑅)
cramerimp.b 𝐵 = (Base‘𝐴)
cramerimp.v 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)
cramerimp.e 𝐸 = (((1r𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)
cramerimp.h 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)
cramerimp.x · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
cramerimp.d 𝐷 = (𝑁 maDet 𝑅)
cramerimp.q / = (/r𝑅)
Assertion
Ref Expression
cramerimp (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (𝑍𝐼) = ((𝐷𝐻) / (𝐷𝑋)))

Proof of Theorem cramerimp
StepHypRef Expression
1 crngring 18474 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21adantr 481 . . . . 5 ((𝑅 ∈ CRing ∧ 𝐼𝑁) → 𝑅 ∈ Ring)
323ad2ant1 1080 . . . 4 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → 𝑅 ∈ Ring)
4 cramerimp.d . . . . . . . 8 𝐷 = (𝑁 maDet 𝑅)
5 cramerimp.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
6 cramerimp.b . . . . . . . 8 𝐵 = (Base‘𝐴)
7 eqid 2626 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
84, 5, 6, 7mdetf 20315 . . . . . . 7 (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅))
98adantr 481 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝐼𝑁) → 𝐷:𝐵⟶(Base‘𝑅))
1093ad2ant1 1080 . . . . 5 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → 𝐷:𝐵⟶(Base‘𝑅))
11 cramerimp.e . . . . . 6 𝐸 = (((1r𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)
125, 6matrcl 20132 . . . . . . . . . . 11 (𝑋𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
1312simpld 475 . . . . . . . . . 10 (𝑋𝐵𝑁 ∈ Fin)
1413adantr 481 . . . . . . . . 9 ((𝑋𝐵𝑌𝑉) → 𝑁 ∈ Fin)
152, 14anim12i 589 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
16153adant3 1079 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
17 ne0i 3902 . . . . . . . . . . 11 (𝐼𝑁𝑁 ≠ ∅)
181, 17anim12ci 590 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼𝑁) → (𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring))
1918anim1i 591 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉)) → ((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐵𝑌𝑉)))
20193adant3 1079 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → ((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐵𝑌𝑉)))
21 simpl 473 . . . . . . . . 9 (((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅)) → (𝑋 · 𝑍) = 𝑌)
22213ad2ant3 1082 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (𝑋 · 𝑍) = 𝑌)
23 cramerimp.v . . . . . . . . 9 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)
24 cramerimp.x . . . . . . . . 9 · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
255, 6, 23, 24slesolvec 20399 . . . . . . . 8 (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐵𝑌𝑉)) → ((𝑋 · 𝑍) = 𝑌𝑍𝑉))
2620, 22, 25sylc 65 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → 𝑍𝑉)
27 simpr 477 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝐼𝑁) → 𝐼𝑁)
28273ad2ant1 1080 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → 𝐼𝑁)
29 eqid 2626 . . . . . . . 8 (1r𝐴) = (1r𝐴)
305, 6, 23, 29ma1repvcl 20290 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍𝑉𝐼𝑁)) → (((1r𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ 𝐵)
3116, 26, 28, 30syl12anc 1321 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (((1r𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ 𝐵)
3211, 31syl5eqel 2708 . . . . 5 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → 𝐸𝐵)
3310, 32ffvelrnd 6317 . . . 4 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (𝐷𝐸) ∈ (Base‘𝑅))
34 simpr 477 . . . . 5 (((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅)) → (𝐷𝑋) ∈ (Unit‘𝑅))
35343ad2ant3 1082 . . . 4 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (𝐷𝑋) ∈ (Unit‘𝑅))
36 eqid 2626 . . . . 5 (Unit‘𝑅) = (Unit‘𝑅)
37 cramerimp.q . . . . 5 / = (/r𝑅)
38 eqid 2626 . . . . 5 (.r𝑅) = (.r𝑅)
397, 36, 37, 38dvrcan3 18608 . . . 4 ((𝑅 ∈ Ring ∧ (𝐷𝐸) ∈ (Base‘𝑅) ∧ (𝐷𝑋) ∈ (Unit‘𝑅)) → (((𝐷𝐸)(.r𝑅)(𝐷𝑋)) / (𝐷𝑋)) = (𝐷𝐸))
403, 33, 35, 39syl3anc 1323 . . 3 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (((𝐷𝐸)(.r𝑅)(𝐷𝑋)) / (𝐷𝑋)) = (𝐷𝐸))
41 simpl 473 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝐼𝑁) → 𝑅 ∈ CRing)
42413ad2ant1 1080 . . . . 5 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → 𝑅 ∈ CRing)
437, 36unitcl 18575 . . . . . . 7 ((𝐷𝑋) ∈ (Unit‘𝑅) → (𝐷𝑋) ∈ (Base‘𝑅))
4443adantl 482 . . . . . 6 (((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅)) → (𝐷𝑋) ∈ (Base‘𝑅))
45443ad2ant3 1082 . . . . 5 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (𝐷𝑋) ∈ (Base‘𝑅))
467, 38crngcom 18478 . . . . 5 ((𝑅 ∈ CRing ∧ (𝐷𝐸) ∈ (Base‘𝑅) ∧ (𝐷𝑋) ∈ (Base‘𝑅)) → ((𝐷𝐸)(.r𝑅)(𝐷𝑋)) = ((𝐷𝑋)(.r𝑅)(𝐷𝐸)))
4742, 33, 45, 46syl3anc 1323 . . . 4 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → ((𝐷𝐸)(.r𝑅)(𝐷𝑋)) = ((𝐷𝑋)(.r𝑅)(𝐷𝐸)))
4847oveq1d 6620 . . 3 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (((𝐷𝐸)(.r𝑅)(𝐷𝑋)) / (𝐷𝑋)) = (((𝐷𝑋)(.r𝑅)(𝐷𝐸)) / (𝐷𝑋)))
4914adantl 482 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉)) → 𝑁 ∈ Fin)
5041adantr 481 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉)) → 𝑅 ∈ CRing)
5127adantr 481 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉)) → 𝐼𝑁)
5249, 50, 513jca 1240 . . . . 5 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝐼𝑁))
53523adant3 1079 . . . 4 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝐼𝑁))
545, 6, 23, 11, 4cramerimplem1 20403 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ 𝑍𝑉) → (𝐷𝐸) = (𝑍𝐼))
5553, 26, 54syl2anc 692 . . 3 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (𝐷𝐸) = (𝑍𝐼))
5640, 48, 553eqtr3rd 2669 . 2 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (𝑍𝐼) = (((𝐷𝑋)(.r𝑅)(𝐷𝐸)) / (𝐷𝑋)))
57 cramerimp.h . . . . 5 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)
585, 6, 23, 11, 57, 24, 4, 38cramerimplem3 20405 . . . 4 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → ((𝐷𝑋)(.r𝑅)(𝐷𝐸)) = (𝐷𝐻))
59583adant3r 1320 . . 3 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → ((𝐷𝑋)(.r𝑅)(𝐷𝐸)) = (𝐷𝐻))
6059oveq1d 6620 . 2 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (((𝐷𝑋)(.r𝑅)(𝐷𝐸)) / (𝐷𝑋)) = ((𝐷𝐻) / (𝐷𝑋)))
6156, 60eqtrd 2660 1 (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (𝑍𝐼) = ((𝐷𝐻) / (𝐷𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  Vcvv 3191  c0 3896  cop 4159  wf 5846  cfv 5850  (class class class)co 6605  𝑚 cmap 7803  Fincfn 7900  Basecbs 15776  .rcmulr 15858  1rcur 18417  Ringcrg 18463  CRingccrg 18464  Unitcui 18555  /rcdvr 18598   Mat cmat 20127   maVecMul cmvmul 20260   matRepV cmatrepV 20277   maDet cmdat 20304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-addf 9960  ax-mulf 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-xor 1462  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-om 7014  df-1st 7116  df-2nd 7117  df-supp 7242  df-tpos 7298  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-fsupp 8221  df-sup 8293  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-xnn0 11309  df-z 11323  df-dec 11438  df-uz 11632  df-rp 11777  df-fz 12266  df-fzo 12404  df-seq 12739  df-exp 12798  df-hash 13055  df-word 13233  df-lsw 13234  df-concat 13235  df-s1 13236  df-substr 13237  df-splice 13238  df-reverse 13239  df-s2 13525  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-mulr 15871  df-starv 15872  df-sca 15873  df-vsca 15874  df-ip 15875  df-tset 15876  df-ple 15877  df-ds 15880  df-unif 15881  df-hom 15882  df-cco 15883  df-0g 16018  df-gsum 16019  df-prds 16024  df-pws 16026  df-mre 16162  df-mrc 16163  df-acs 16165  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-submnd 17252  df-grp 17341  df-minusg 17342  df-sbg 17343  df-mulg 17457  df-subg 17507  df-ghm 17574  df-gim 17617  df-cntz 17666  df-oppg 17692  df-symg 17714  df-pmtr 17778  df-psgn 17827  df-evpm 17828  df-cmn 18111  df-abl 18112  df-mgp 18406  df-ur 18418  df-srg 18422  df-ring 18465  df-cring 18466  df-oppr 18539  df-dvdsr 18557  df-unit 18558  df-invr 18588  df-dvr 18599  df-rnghom 18631  df-drng 18665  df-subrg 18694  df-lmod 18781  df-lss 18847  df-sra 19086  df-rgmod 19087  df-cnfld 19661  df-zring 19733  df-zrh 19766  df-dsmm 19990  df-frlm 20005  df-mamu 20104  df-mat 20128  df-mvmul 20261  df-marrep 20278  df-marepv 20279  df-subma 20297  df-mdet 20305  df-minmar1 20355
This theorem is referenced by:  cramerlem1  20407
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