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Theorem cramer0 20315
 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 6106 . . . . . . . . 9 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
41, 3eqtri 2632 . . . . . . . 8 𝐵 = (Base‘(𝑁 Mat 𝑅))
5 oveq1 6556 . . . . . . . . 9 (𝑁 = ∅ → (𝑁 Mat 𝑅) = (∅ Mat 𝑅))
65fveq2d 6107 . . . . . . . 8 (𝑁 = ∅ → (Base‘(𝑁 Mat 𝑅)) = (Base‘(∅ Mat 𝑅)))
74, 6syl5eq 2656 . . . . . . 7 (𝑁 = ∅ → 𝐵 = (Base‘(∅ Mat 𝑅)))
87adantr 480 . . . . . 6 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → 𝐵 = (Base‘(∅ Mat 𝑅)))
98eleq2d 2673 . . . . 5 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → (𝑋𝐵𝑋 ∈ (Base‘(∅ Mat 𝑅))))
10 mat0dimbas0 20091 . . . . . . 7 (𝑅 ∈ CRing → (Base‘(∅ Mat 𝑅)) = {∅})
1110eleq2d 2673 . . . . . 6 (𝑅 ∈ CRing → (𝑋 ∈ (Base‘(∅ Mat 𝑅)) ↔ 𝑋 ∈ {∅}))
1211adantl 481 . . . . 5 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → (𝑋 ∈ (Base‘(∅ Mat 𝑅)) ↔ 𝑋 ∈ {∅}))
139, 12bitrd 267 . . . 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 6557 . . . . . . . 8 (𝑁 = ∅ → ((Base‘𝑅) ↑𝑚 𝑁) = ((Base‘𝑅) ↑𝑚 ∅))
1716adantr 480 . . . . . . 7 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → ((Base‘𝑅) ↑𝑚 𝑁) = ((Base‘𝑅) ↑𝑚 ∅))
18 fvex 6113 . . . . . . . 8 (Base‘𝑅) ∈ V
19 map0e 7781 . . . . . . . 8 ((Base‘𝑅) ∈ V → ((Base‘𝑅) ↑𝑚 ∅) = 1𝑜)
2018, 19mp1i 13 . . . . . . 7 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → ((Base‘𝑅) ↑𝑚 ∅) = 1𝑜)
2115, 17, 203eqtrd 2648 . . . . . 6 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → 𝑉 = 1𝑜)
2221eleq2d 2673 . . . . 5 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → (𝑌𝑉𝑌 ∈ 1𝑜))
23 el1o 7466 . . . . 5 (𝑌 ∈ 1𝑜𝑌 = ∅)
2422, 23syl6bb 275 . . . 4 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → (𝑌𝑉𝑌 = ∅))
2513, 24anbi12d 743 . . 3 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → ((𝑋𝐵𝑌𝑉) ↔ (𝑋 ∈ {∅} ∧ 𝑌 = ∅)))
26 elsni 4142 . . . 4 (𝑋 ∈ {∅} → 𝑋 = ∅)
27 mpteq1 4665 . . . . . . . . . 10 (𝑁 = ∅ → (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) = (𝑖 ∈ ∅ ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))))
28 mpt0 5934 . . . . . . . . . 10 (𝑖 ∈ ∅ ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) = ∅
2927, 28syl6eq 2660 . . . . . . . . 9 (𝑁 = ∅ → (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) = ∅)
3029eqeq2d 2620 . . . . . . . 8 (𝑁 = ∅ → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) ↔ 𝑍 = ∅))
3130ad2antrr 758 . . . . . . 7 (((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) ↔ 𝑍 = ∅))
32 simplrl 796 . . . . . . . . . 10 ((((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) ∧ 𝑍 = ∅) → 𝑋 = ∅)
33 simpr 476 . . . . . . . . . 10 ((((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) ∧ 𝑍 = ∅) → 𝑍 = ∅)
3432, 33oveq12d 6567 . . . . . . . . 9 ((((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) ∧ 𝑍 = ∅) → (𝑋 · 𝑍) = (∅ · ∅))
35 cramer.x . . . . . . . . . . 11 · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
3635mavmul0 20177 . . . . . . . . . 10 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → (∅ · ∅) = ∅)
3736ad2antrr 758 . . . . . . . . 9 ((((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) ∧ 𝑍 = ∅) → (∅ · ∅) = ∅)
38 simpr 476 . . . . . . . . . . 11 ((𝑋 = ∅ ∧ 𝑌 = ∅) → 𝑌 = ∅)
3938eqcomd 2616 . . . . . . . . . 10 ((𝑋 = ∅ ∧ 𝑌 = ∅) → ∅ = 𝑌)
4039ad2antlr 759 . . . . . . . . 9 ((((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) ∧ 𝑍 = ∅) → ∅ = 𝑌)
4134, 37, 403eqtrd 2648 . . . . . . . 8 ((((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) ∧ 𝑍 = ∅) → (𝑋 · 𝑍) = 𝑌)
4241ex 449 . . . . . . 7 (((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 = ∅ ∧ 𝑌 = ∅)) → (𝑍 = ∅ → (𝑋 · 𝑍) = 𝑌))
4331, 42sylbid 229 . . . . . 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 449 . . . 4 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → ((𝑋 = ∅ ∧ 𝑌 = ∅) → ((𝐷𝑋) ∈ (Unit‘𝑅) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) → (𝑋 · 𝑍) = 𝑌))))
4626, 45sylani 684 . . 3 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → ((𝑋 ∈ {∅} ∧ 𝑌 = ∅) → ((𝐷𝑋) ∈ (Unit‘𝑅) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) → (𝑋 · 𝑍) = 𝑌))))
4725, 46sylbid 229 . 2 ((𝑁 = ∅ ∧ 𝑅 ∈ CRing) → ((𝑋𝐵𝑌𝑉) → ((𝐷𝑋) ∈ (Unit‘𝑅) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) → (𝑋 · 𝑍) = 𝑌))))
48473imp 1249 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125  ⟨cop 4131   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  1𝑜c1o 7440   ↑𝑚 cmap 7744  Basecbs 15695  CRingccrg 18371  Unitcui 18462  /rcdvr 18505   Mat cmat 20032   maVecMul cmvmul 20165   matRepV cmatrepV 20182   maDet cmdat 20209 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-prds 15931  df-pws 15933  df-sra 18993  df-rgmod 18994  df-dsmm 19895  df-frlm 19910  df-mat 20033  df-mvmul 20166 This theorem is referenced by: (None)
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