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Theorem marepvfval 20602
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.)
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 6431 . . . . 5 𝐵 ∈ V
4 marepvfval.v . . . . . . 7 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)
5 ovex 6915 . . . . . . 7 ((Base‘𝑅) ↑𝑚 𝑁) ∈ V
64, 5eqeltri 2892 . . . . . 6 𝑉 ∈ V
76a1i 11 . . . . 5 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝑉 ∈ V)
8 mpt2exga 7488 . . . . 5 ((𝐵 ∈ V ∧ 𝑉 ∈ V) → (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))) ∈ V)
93, 7, 8sylancr 577 . . . 4 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))) ∈ V)
10 oveq12 6892 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
11 marepvfval.a . . . . . . . . 9 𝐴 = (𝑁 Mat 𝑅)
1210, 11syl6eqr 2869 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
1312fveq2d 6421 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
1413, 2syl6eqr 2869 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
15 fveq2 6417 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
1615adantl 469 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘𝑟) = (Base‘𝑅))
17 simpl 470 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
1816, 17oveq12d 6901 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((Base‘𝑟) ↑𝑚 𝑛) = ((Base‘𝑅) ↑𝑚 𝑁))
1918, 4syl6eqr 2869 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → ((Base‘𝑟) ↑𝑚 𝑛) = 𝑉)
20 eqidd 2818 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)) = if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))
2117, 17, 20mpt2eq123dv 6956 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑖𝑛, 𝑗𝑛 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))
2217, 21mpteq12dv 4938 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))))
2314, 19, 22mpt2eq123dv 6956 . . . . 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))))))
24 df-marepv 20596 . . . . 5 matRepV = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑𝑚 𝑛) ↦ (𝑘𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))
2523, 24ovmpt2ga 7029 . . . 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))))))
269, 25mpd3an3 1579 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))
2724mpt2ndm0 7114 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = ∅)
28 mpt20 6964 . . . . 5 (𝑚 ∈ ∅, 𝑣 ∈ ((Base‘𝑅) ↑𝑚 𝑁) ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))) = ∅
2927, 28syl6eqr 2869 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = (𝑚 ∈ ∅, 𝑣 ∈ ((Base‘𝑅) ↑𝑚 𝑁) ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))
3011fveq2i 6420 . . . . . . 7 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
312, 30eqtri 2839 . . . . . 6 𝐵 = (Base‘(𝑁 Mat 𝑅))
32 matbas0pc 20445 . . . . . 6 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
3331, 32syl5eq 2863 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
34 mpt2eq12 6954 . . . . 5 ((𝐵 = ∅ ∧ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)) → (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))) = (𝑚 ∈ ∅, 𝑣 ∈ ((Base‘𝑅) ↑𝑚 𝑁) ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))
3533, 4, 34sylancl 576 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))) = (𝑚 ∈ ∅, 𝑣 ∈ ((Base‘𝑅) ↑𝑚 𝑁) ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))
3629, 35eqtr4d 2854 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))
3726, 36pm2.61i 176 . 2 (𝑁 matRepV 𝑅) = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))))
381, 37eqtri 2839 1 𝑄 = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1637  wcel 2157  Vcvv 3402  c0 4127  ifcif 4290  cmpt 4934  cfv 6110  (class class class)co 6883  cmpt2 6885  𝑚 cmap 8101  Basecbs 16087   Mat cmat 20443   matRepV cmatrepV 20594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-ov 6886  df-oprab 6887  df-mpt2 6888  df-1st 7407  df-2nd 7408  df-slot 16091  df-base 16093  df-mat 20444  df-marepv 20596
This theorem is referenced by:  marepvval0  20603
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