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Theorem marepvval 20782
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 20781 . . . 4 ((𝑀𝐵𝐶𝑉) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
653adant3 1123 . . 3 ((𝑀𝐵𝐶𝑉𝐾𝑁) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
76fveq1d 6450 . 2 ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑀𝑄𝐶)‘𝐾) = ((𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))‘𝐾))
8 eqid 2778 . . 3 (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))
9 eqeq2 2789 . . . . 5 (𝑘 = 𝐾 → (𝑗 = 𝑘𝑗 = 𝐾))
109ifbid 4329 . . . 4 (𝑘 = 𝐾 → if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)) = if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗)))
1110mpt2eq3dv 7000 . . 3 (𝑘 = 𝐾 → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))
12 simp3 1129 . . 3 ((𝑀𝐵𝐶𝑉𝐾𝑁) → 𝐾𝑁)
131, 2matrcl 20626 . . . . . . 7 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
1413simpld 490 . . . . . 6 (𝑀𝐵𝑁 ∈ Fin)
1514, 14jca 507 . . . . 5 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
16153ad2ant1 1124 . . . 4 ((𝑀𝐵𝐶𝑉𝐾𝑁) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
17 mpt2exga 7528 . . . 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 6566 . 2 ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))
207, 19eqtrd 2814 1 ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  Vcvv 3398  ifcif 4307  cmpt 4967  cfv 6137  (class class class)co 6924  cmpt2 6926  𝑚 cmap 8142  Fincfn 8243  Basecbs 16259   Mat cmat 20621   matRepV cmatrepV 20772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-slot 16263  df-base 16265  df-mat 20622  df-marepv 20774
This theorem is referenced by:  marepveval  20783  marepvcl  20784  1marepvmarrepid  20790  cramerimplem2  20901
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