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Theorem cramer0 20415
Description: Special case of Cramer's rule for 0-dimensional matrices/vectors. (Contributed by AV, 28-Feb-2019.)
Hypotheses
Ref Expression
cramer.a 𝐴 = (𝑁 Mat 𝑅)
cramer.b 𝐵 = (Base‘𝐴)
cramer.v 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)
cramer.d 𝐷 = (𝑁 maDet 𝑅)
cramer.x · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
cramer.q / = (/r𝑅)
Assertion
Ref Expression
cramer0 (((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐵𝑌𝑉) ∧ (𝐷𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) → (𝑋 · 𝑍) = 𝑌))
Distinct variable groups:   𝐵,𝑖   𝐷,𝑖   𝑖,𝑁   𝑅,𝑖   𝑖,𝑉   𝑖,𝑋   𝑖,𝑌   𝑖,𝑍   · ,𝑖   / ,𝑖
Allowed substitution hint:   𝐴(𝑖)

Proof of Theorem cramer0
StepHypRef Expression
1 cramer.b . . . . . . . . 9 𝐵 = (Base‘𝐴)
2 cramer.a . . . . . . . . . 10 𝐴 = (𝑁 Mat 𝑅)
32fveq2i 6151 . . . . . . . . 9 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
41, 3eqtri 2643 . . . . . . . 8 𝐵 = (Base‘(𝑁 Mat 𝑅))
5 oveq1 6611 . . . . . . . . 9 (𝑁 = ∅ → (𝑁 Mat 𝑅) = (∅ Mat 𝑅))
65fveq2d 6152 . . . . . . . 8 (𝑁 = ∅ → (Base‘(𝑁 Mat 𝑅)) = (Base‘(∅ Mat 𝑅)))
74, 6syl5eq 2667 . . . . . . 7 (𝑁 = ∅ → 𝐵 = (Base‘(∅ Mat 𝑅)))
87adantr 481 . . . . . 6 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → 𝐵 = (Base‘(∅ Mat 𝑅)))
98eleq2d 2684 . . . . 5 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → (𝑋𝐵𝑋 ∈ (Base‘(∅ Mat 𝑅))))
10 mat0dimbas0 20191 . . . . . . 7 (𝑅 ∈ CRing → (Base‘(∅ Mat 𝑅)) = {∅})
1110eleq2d 2684 . . . . . 6 (𝑅 ∈ CRing → (𝑋 ∈ (Base‘(∅ Mat 𝑅)) ↔ 𝑋 ∈ {∅}))
1211adantl 482 . . . . 5 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → (𝑋 ∈ (Base‘(∅ Mat 𝑅)) ↔ 𝑋 ∈ {∅}))
139, 12bitrd 268 . . . 4 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → (𝑋𝐵𝑋 ∈ {∅}))
14 cramer.v . . . . . . . 8 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)
1514a1i 11 . . . . . . 7 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁))
16 oveq2 6612 . . . . . . . 8 (𝑁 = ∅ → ((Base‘𝑅) ↑𝑚 𝑁) = ((Base‘𝑅) ↑𝑚 ∅))
1716adantr 481 . . . . . . 7 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → ((Base‘𝑅) ↑𝑚 𝑁) = ((Base‘𝑅) ↑𝑚 ∅))
18 fvex 6158 . . . . . . . 8 (Base‘𝑅) ∈ V
19 map0e 7839 . . . . . . . 8 ((Base‘𝑅) ∈ V → ((Base‘𝑅) ↑𝑚 ∅) = 1𝑜)
2018, 19mp1i 13 . . . . . . 7 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → ((Base‘𝑅) ↑𝑚 ∅) = 1𝑜)
2115, 17, 203eqtrd 2659 . . . . . 6 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → 𝑉 = 1𝑜)
2221eleq2d 2684 . . . . 5 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → (𝑌𝑉𝑌 ∈ 1𝑜))
23 el1o 7524 . . . . 5 (𝑌 ∈ 1𝑜𝑌 = ∅)
2422, 23syl6bb 276 . . . 4 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → (𝑌𝑉𝑌 = ∅))
2513, 24anbi12d 746 . . 3 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → ((𝑋𝐵𝑌𝑉) ↔ (𝑋 ∈ {∅} ∧ 𝑌 = ∅)))
26 elsni 4165 . . . 4 (𝑋 ∈ {∅} → 𝑋 = ∅)
27 mpteq1 4697 . . . . . . . . . 10 (𝑁 = ∅ → (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) = (𝑖 ∈ ∅ ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))))
28 mpt0 5978 . . . . . . . . . 10 (𝑖 ∈ ∅ ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) = ∅
2927, 28syl6eq 2671 . . . . . . . . 9 (𝑁 = ∅ → (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) = ∅)
3029eqeq2d 2631 . . . . . . . 8 (𝑁 = ∅ → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) ↔ 𝑍 = ∅))
3130ad2antrr 761 . . . . . . 7 (((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) ↔ 𝑍 = ∅))
32 simplrl 799 . . . . . . . . . 10 ((((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) ∧ 𝑍 = ∅) → 𝑋 = ∅)
33 simpr 477 . . . . . . . . . 10 ((((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) ∧ 𝑍 = ∅) → 𝑍 = ∅)
3432, 33oveq12d 6622 . . . . . . . . 9 ((((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) ∧ 𝑍 = ∅) → (𝑋 · 𝑍) = (∅ · ∅))
35 cramer.x . . . . . . . . . . 11 · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
3635mavmul0 20277 . . . . . . . . . 10 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → (∅ · ∅) = ∅)
3736ad2antrr 761 . . . . . . . . 9 ((((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) ∧ 𝑍 = ∅) → (∅ · ∅) = ∅)
38 simpr 477 . . . . . . . . . . 11 ((𝑋 = ∅ ∧ 𝑌 = ∅) → 𝑌 = ∅)
3938eqcomd 2627 . . . . . . . . . 10 ((𝑋 = ∅ ∧ 𝑌 = ∅) → ∅ = 𝑌)
4039ad2antlr 762 . . . . . . . . 9 ((((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) ∧ 𝑍 = ∅) → ∅ = 𝑌)
4134, 37, 403eqtrd 2659 . . . . . . . 8 ((((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) ∧ 𝑍 = ∅) → (𝑋 · 𝑍) = 𝑌)
4241ex 450 . . . . . . 7 (((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) → (𝑍 = ∅ → (𝑋 · 𝑍) = 𝑌))
4331, 42sylbid 230 . . . . . 6 (((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) → (𝑋 · 𝑍) = 𝑌))
4443a1d 25 . . . . 5 (((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) → ((𝐷𝑋) ∈ (Unit‘𝑅) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) → (𝑋 · 𝑍) = 𝑌)))
4544ex 450 . . . 4 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → ((𝑋 = ∅ ∧ 𝑌 = ∅) → ((𝐷𝑋) ∈ (Unit‘𝑅) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) → (𝑋 · 𝑍) = 𝑌))))
4626, 45sylani 685 . . 3 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → ((𝑋 ∈ {∅} ∧ 𝑌 = ∅) → ((𝐷𝑋) ∈ (Unit‘𝑅) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) → (𝑋 · 𝑍) = 𝑌))))
4725, 46sylbid 230 . 2 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → ((𝑋𝐵𝑌𝑉) → ((𝐷𝑋) ∈ (Unit‘𝑅) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) → (𝑋 · 𝑍) = 𝑌))))
48473imp 1254 1 (((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐵𝑌𝑉) ∧ (𝐷𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) → (𝑋 · 𝑍) = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  c0 3891  {csn 4148  cop 4154  cmpt 4673  cfv 5847  (class class class)co 6604  1𝑜c1o 7498  𝑚 cmap 7802  Basecbs 15781  CRingccrg 18469  Unitcui 18560  /rcdvr 18603   Mat cmat 20132   maVecMul cmvmul 20265   matRepV cmatrepV 20282   maDet cmdat 20309
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-hom 15887  df-cco 15888  df-0g 16023  df-prds 16029  df-pws 16031  df-sra 19091  df-rgmod 19092  df-dsmm 19995  df-frlm 20010  df-mat 20133  df-mvmul 20266
This theorem is referenced by: (None)
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