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Theorem marepvfval 21866
Description: First 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.) (Proof shortened by AV, 2-Mar-2024.)
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
marepvfval 𝑄 = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))))
Distinct variable groups:   𝐵,𝑚,𝑣   𝑖,𝑁,𝑗,𝑘,𝑚,𝑣   𝑅,𝑖,𝑗,𝑘,𝑚,𝑣   𝑚,𝑉,𝑣
Allowed substitution hints:   𝐴(𝑣,𝑖,𝑗,𝑘,𝑚)   𝐵(𝑖,𝑗,𝑘)   𝑄(𝑣,𝑖,𝑗,𝑘,𝑚)   𝑉(𝑖,𝑗,𝑘)

Proof of Theorem marepvfval
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marepvfval.q . 2 𝑄 = (𝑁 matRepV 𝑅)
2 marepvfval.b . . . . . 6 𝐵 = (Base‘𝐴)
32fvexi 6853 . . . . 5 𝐵 ∈ V
4 marepvfval.v . . . . . . 7 𝑉 = ((Base‘𝑅) ↑m 𝑁)
54ovexi 7385 . . . . . 6 𝑉 ∈ V
65a1i 11 . . . . 5 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝑉 ∈ V)
7 mpoexga 8002 . . . . 5 ((𝐵 ∈ V ∧ 𝑉 ∈ V) → (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))) ∈ V)
83, 6, 7sylancr 587 . . . 4 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))) ∈ V)
9 oveq12 7360 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
10 marepvfval.a . . . . . . . . 9 𝐴 = (𝑁 Mat 𝑅)
119, 10eqtr4di 2795 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
1211fveq2d 6843 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
1312, 2eqtr4di 2795 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
14 fveq2 6839 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
1514adantl 482 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘𝑟) = (Base‘𝑅))
16 simpl 483 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
1715, 16oveq12d 7369 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((Base‘𝑟) ↑m 𝑛) = ((Base‘𝑅) ↑m 𝑁))
1817, 4eqtr4di 2795 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → ((Base‘𝑟) ↑m 𝑛) = 𝑉)
19 eqidd 2738 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)) = if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))
2016, 16, 19mpoeq123dv 7426 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑖𝑛, 𝑗𝑛 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))
2116, 20mpteq12dv 5194 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))))
2213, 18, 21mpoeq123dv 7426 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑘𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))) = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))
23 df-marepv 21860 . . . . 5 matRepV = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑘𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))
2422, 23ovmpoga 7503 . . . 4 ((𝑁 ∈ V ∧ 𝑅 ∈ V ∧ (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))) ∈ V) → (𝑁 matRepV 𝑅) = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))
258, 24mpd3an3 1462 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))
2623mpondm0 7586 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = ∅)
2710fveq2i 6842 . . . . . . . 8 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
282, 27eqtri 2765 . . . . . . 7 𝐵 = (Base‘(𝑁 Mat 𝑅))
29 matbas0pc 21708 . . . . . . 7 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
3028, 29eqtrid 2789 . . . . . 6 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
3130orcd 871 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝐵 = ∅ ∨ 𝑉 = ∅))
32 0mpo0 7434 . . . . 5 ((𝐵 = ∅ ∨ 𝑉 = ∅) → (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))) = ∅)
3331, 32syl 17 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))) = ∅)
3426, 33eqtr4d 2780 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))
3525, 34pm2.61i 182 . 2 (𝑁 matRepV 𝑅) = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))))
361, 35eqtri 2765 1 𝑄 = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wo 845   = wceq 1541  wcel 2106  Vcvv 3443  c0 4280  ifcif 4484  cmpt 5186  cfv 6493  (class class class)co 7351  cmpo 7353  m cmap 8723  Basecbs 17043   Mat cmat 21706   matRepV cmatrepV 21858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-1cn 11067  ax-addcl 11069
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1st 7913  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-nn 12112  df-slot 17014  df-ndx 17026  df-base 17044  df-mat 21707  df-marepv 21860
This theorem is referenced by:  marepvval0  21867
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