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Theorem marepvval0 21938
Description: Second 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
marepvval0 ((𝑀𝐵𝐶𝑉) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
Distinct variable groups:   𝑖,𝑁,𝑗,𝑘   𝑅,𝑖,𝑗,𝑘   𝐶,𝑖,𝑗,𝑘   𝑖,𝑀,𝑗,𝑘
Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘)   𝐵(𝑖,𝑗,𝑘)   𝑄(𝑖,𝑗,𝑘)   𝑉(𝑖,𝑗,𝑘)

Proof of Theorem marepvval0
Dummy variables 𝑚 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marepvfval.a . . . . . 6 𝐴 = (𝑁 Mat 𝑅)
2 marepvfval.b . . . . . 6 𝐵 = (Base‘𝐴)
31, 2matrcl 21782 . . . . 5 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
43simpld 496 . . . 4 (𝑀𝐵𝑁 ∈ Fin)
54adantr 482 . . 3 ((𝑀𝐵𝐶𝑉) → 𝑁 ∈ Fin)
65mptexd 7178 . 2 ((𝑀𝐵𝐶𝑉) → (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))) ∈ V)
7 fveq1 6845 . . . . . . 7 (𝑐 = 𝐶 → (𝑐𝑖) = (𝐶𝑖))
87adantl 483 . . . . . 6 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑐𝑖) = (𝐶𝑖))
9 oveq 7367 . . . . . . 7 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
109adantr 482 . . . . . 6 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
118, 10ifeq12d 4511 . . . . 5 ((𝑚 = 𝑀𝑐 = 𝐶) → if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗)) = if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))
1211mpoeq3dv 7440 . . . 4 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))
1312mpteq2dv 5211 . . 3 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗)))) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
14 marepvfval.q . . . 4 𝑄 = (𝑁 matRepV 𝑅)
15 marepvfval.v . . . 4 𝑉 = ((Base‘𝑅) ↑m 𝑁)
161, 2, 14, 15marepvfval 21937 . . 3 𝑄 = (𝑚𝐵, 𝑐𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗)))))
1713, 16ovmpoga 7513 . 2 ((𝑀𝐵𝐶𝑉 ∧ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))) ∈ V) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
186, 17mpd3an3 1463 1 ((𝑀𝐵𝐶𝑉) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3447  ifcif 4490  cmpt 5192  cfv 6500  (class class class)co 7361  cmpo 7363  m cmap 8771  Fincfn 8889  Basecbs 17091   Mat cmat 21777   matRepV cmatrepV 21929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-1cn 11117  ax-addcl 11119
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-nn 12162  df-slot 17062  df-ndx 17074  df-base 17092  df-mat 21778  df-marepv 21931
This theorem is referenced by:  marepvval  21939
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