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Theorem marepvval0 21167
 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 21013 . . . . 5 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
43simpld 497 . . . 4 (𝑀𝐵𝑁 ∈ Fin)
54adantr 483 . . 3 ((𝑀𝐵𝐶𝑉) → 𝑁 ∈ Fin)
65mptexd 6979 . 2 ((𝑀𝐵𝐶𝑉) → (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))) ∈ V)
7 fveq1 6662 . . . . . . 7 (𝑐 = 𝐶 → (𝑐𝑖) = (𝐶𝑖))
87adantl 484 . . . . . 6 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑐𝑖) = (𝐶𝑖))
9 oveq 7154 . . . . . . 7 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
109adantr 483 . . . . . 6 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
118, 10ifeq12d 4485 . . . . 5 ((𝑚 = 𝑀𝑐 = 𝐶) → if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗)) = if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))
1211mpoeq3dv 7225 . . . 4 ((𝑚 = 𝑀𝑐 = 𝐶) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗))))
1312mpteq2dv 5153 . . 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 21166 . . 3 𝑄 = (𝑚𝐵, 𝑐𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑐𝑖), (𝑖𝑚𝑗)))))
1713, 16ovmpoga 7296 . 2 ((𝑀𝐵𝐶𝑉 ∧ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))) ∈ V) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
186, 17mpd3an3 1455 1 ((𝑀𝐵𝐶𝑉) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1530   ∈ wcel 2107  Vcvv 3493  ifcif 4465   ↦ cmpt 5137  ‘cfv 6348  (class class class)co 7148   ∈ cmpo 7150   ↑m cmap 8398  Fincfn 8501  Basecbs 16475   Mat cmat 21008   matRepV cmatrepV 21158 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-slot 16479  df-base 16481  df-mat 21009  df-marepv 21160 This theorem is referenced by:  marepvval  21168
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