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Theorem marepvval 22389
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 22388 . . . 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 6893 . 2 ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑀𝑄𝐶)‘𝐾) = ((𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))‘𝐾))
8 eqid 2731 . . 3 (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))
9 eqeq2 2743 . . . . 5 (𝑘 = 𝐾 → (𝑗 = 𝑘𝑗 = 𝐾))
109ifbid 4551 . . . 4 (𝑘 = 𝐾 → if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)) = if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗)))
1110mpoeq3dv 7491 . . 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 22232 . . . . . . 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 8068 . . . 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 7021 . 2 ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))
207, 19eqtrd 2771 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 1540  wcel 2105  Vcvv 3473  ifcif 4528  cmpt 5231  cfv 6543  (class class class)co 7412  cmpo 7414  m cmap 8826  Fincfn 8945  Basecbs 17151   Mat cmat 22227   matRepV cmatrepV 22379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-1cn 11174  ax-addcl 11176
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-nn 12220  df-slot 17122  df-ndx 17134  df-base 17152  df-mat 22228  df-marepv 22381
This theorem is referenced by:  marepveval  22390  marepvcl  22391  1marepvmarrepid  22397  cramerimplem2  22506
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