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Theorem marepvval0 20574
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 20420 . . . . 5 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
43simpld 477 . . . 4 (𝑀𝐵𝑁 ∈ Fin)
54adantr 472 . . 3 ((𝑀𝐵𝐶𝑉) → 𝑁 ∈ Fin)
6 mptexg 6648 . . 3 (𝑁 ∈ Fin → (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))) ∈ V)
75, 6syl 17 . 2 ((𝑀𝐵𝐶𝑉) → (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))) ∈ V)
8 fveq1 6351 . . . . . . 7 (𝑐 = 𝐶 → (𝑐𝑖) = (𝐶𝑖))
98adantl 473 . . . . . 6 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑐𝑖) = (𝐶𝑖))
10 oveq 6819 . . . . . . 7 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
1110adantr 472 . . . . . 6 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
129, 11ifeq12d 4250 . . . . 5 ((𝑚 = 𝑀𝑐 = 𝐶) → if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗)) = if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))
1312mpt2eq3dv 6886 . . . 4 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))
1413mpteq2dv 4897 . . 3 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗)))) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
15 marepvfval.q . . . 4 𝑄 = (𝑁 matRepV 𝑅)
16 marepvfval.v . . . 4 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)
171, 2, 15, 16marepvfval 20573 . . 3 𝑄 = (𝑚𝐵, 𝑐𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗)))))
1814, 17ovmpt2ga 6955 . 2 ((𝑀𝐵𝐶𝑉 ∧ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))) ∈ V) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
197, 18mpd3an3 1574 1 ((𝑀𝐵𝐶𝑉) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  ifcif 4230  cmpt 4881  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023  Fincfn 8121  Basecbs 16059   Mat cmat 20415   matRepV cmatrepV 20565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-slot 16063  df-base 16065  df-mat 20416  df-marepv 20567
This theorem is referenced by:  marepvval  20575
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