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Theorem marepvval0 20291
 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 20137 . . . . 5 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
43simpld 475 . . . 4 (𝑀𝐵𝑁 ∈ Fin)
54adantr 481 . . 3 ((𝑀𝐵𝐶𝑉) → 𝑁 ∈ Fin)
6 mptexg 6438 . . 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 6147 . . . . . . 7 (𝑐 = 𝐶 → (𝑐𝑖) = (𝐶𝑖))
98adantl 482 . . . . . 6 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑐𝑖) = (𝐶𝑖))
10 oveq 6610 . . . . . . 7 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
1110adantr 481 . . . . . 6 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
129, 11ifeq12d 4078 . . . . 5 ((𝑚 = 𝑀𝑐 = 𝐶) → if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗)) = if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))
1312mpt2eq3dv 6674 . . . 4 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))
1413mpteq2dv 4705 . . 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 20290 . . 3 𝑄 = (𝑚𝐵, 𝑐𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗)))))
1814, 17ovmpt2ga 6743 . 2 ((𝑀𝐵𝐶𝑉 ∧ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))) ∈ V) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
197, 18mpd3an3 1422 1 ((𝑀𝐵𝐶𝑉) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ifcif 4058   ↦ cmpt 4673  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606   ↑𝑚 cmap 7802  Fincfn 7899  Basecbs 15781   Mat cmat 20132   matRepV cmatrepV 20282 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-slot 15785  df-base 15786  df-mat 20133  df-marepv 20284 This theorem is referenced by:  marepvval  20292
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