MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  marepvval Structured version   Visualization version   GIF version

Theorem marepvval 22589
Description: Third substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
Hypotheses
Ref Expression
marepvfval.a 𝐴 = (𝑁 Mat 𝑅)
marepvfval.b 𝐵 = (Base‘𝐴)
marepvfval.q 𝑄 = (𝑁 matRepV 𝑅)
marepvfval.v 𝑉 = ((Base‘𝑅) ↑m 𝑁)
Assertion
Ref Expression
marepvval ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))
Distinct variable groups:   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   𝐶,𝑖,𝑗   𝑖,𝑀,𝑗   𝑖,𝐾,𝑗
Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐵(𝑖,𝑗)   𝑄(𝑖,𝑗)   𝑉(𝑖,𝑗)

Proof of Theorem marepvval
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 marepvfval.a . . . . 5 𝐴 = (𝑁 Mat 𝑅)
2 marepvfval.b . . . . 5 𝐵 = (Base‘𝐴)
3 marepvfval.q . . . . 5 𝑄 = (𝑁 matRepV 𝑅)
4 marepvfval.v . . . . 5 𝑉 = ((Base‘𝑅) ↑m 𝑁)
51, 2, 3, 4marepvval0 22588 . . . 4 ((𝑀𝐵𝐶𝑉) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
653adant3 1131 . . 3 ((𝑀𝐵𝐶𝑉𝐾𝑁) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
76fveq1d 6909 . 2 ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑀𝑄𝐶)‘𝐾) = ((𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))‘𝐾))
8 eqid 2735 . . 3 (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))
9 eqeq2 2747 . . . . 5 (𝑘 = 𝐾 → (𝑗 = 𝑘𝑗 = 𝐾))
109ifbid 4554 . . . 4 (𝑘 = 𝐾 → if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)) = if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗)))
1110mpoeq3dv 7512 . . 3 (𝑘 = 𝐾 → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))
12 simp3 1137 . . 3 ((𝑀𝐵𝐶𝑉𝐾𝑁) → 𝐾𝑁)
131, 2matrcl 22432 . . . . . . 7 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
1413simpld 494 . . . . . 6 (𝑀𝐵𝑁 ∈ Fin)
1514, 14jca 511 . . . . 5 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
16153ad2ant1 1132 . . . 4 ((𝑀𝐵𝐶𝑉𝐾𝑁) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
17 mpoexga 8101 . . . 4 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))) ∈ V)
1816, 17syl 17 . . 3 ((𝑀𝐵𝐶𝑉𝐾𝑁) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))) ∈ V)
198, 11, 12, 18fvmptd3 7039 . 2 ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))
207, 19eqtrd 2775 1 ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  ifcif 4531  cmpt 5231  cfv 6563  (class class class)co 7431  cmpo 7433  m cmap 8865  Fincfn 8984  Basecbs 17245   Mat cmat 22427   matRepV cmatrepV 22579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-slot 17216  df-ndx 17228  df-base 17246  df-mat 22428  df-marepv 22581
This theorem is referenced by:  marepveval  22590  marepvcl  22591  1marepvmarrepid  22597  cramerimplem2  22706
  Copyright terms: Public domain W3C validator