MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uvcval Structured version   Visualization version   GIF version

Theorem uvcval 21001
Description: Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
Hypotheses
Ref Expression
uvcfval.u 𝑈 = (𝑅 unitVec 𝐼)
uvcfval.o 1 = (1r𝑅)
uvcfval.z 0 = (0g𝑅)
Assertion
Ref Expression
uvcval ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
Distinct variable groups:   1 ,𝑘   𝑅,𝑘   𝑘,𝐼   0 ,𝑘   𝑘,𝐽
Allowed substitution hints:   𝑈(𝑘)   𝑉(𝑘)   𝑊(𝑘)

Proof of Theorem uvcval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 uvcfval.u . . . . 5 𝑈 = (𝑅 unitVec 𝐼)
2 uvcfval.o . . . . 5 1 = (1r𝑅)
3 uvcfval.z . . . . 5 0 = (0g𝑅)
41, 2, 3uvcfval 21000 . . . 4 ((𝑅𝑉𝐼𝑊) → 𝑈 = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
54fveq1d 6785 . . 3 ((𝑅𝑉𝐼𝑊) → (𝑈𝐽) = ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽))
653adant3 1131 . 2 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽))
7 eqid 2739 . . 3 (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))
8 eqeq2 2751 . . . . 5 (𝑗 = 𝐽 → (𝑘 = 𝑗𝑘 = 𝐽))
98ifbid 4483 . . . 4 (𝑗 = 𝐽 → if(𝑘 = 𝑗, 1 , 0 ) = if(𝑘 = 𝐽, 1 , 0 ))
109mpteq2dv 5177 . . 3 (𝑗 = 𝐽 → (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
11 simp3 1137 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → 𝐽𝐼)
12 mptexg 7106 . . . 4 (𝐼𝑊 → (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V)
13123ad2ant2 1133 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V)
147, 10, 11, 13fvmptd3 6907 . 2 ((𝑅𝑉𝐼𝑊𝐽𝐼) → ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
156, 14eqtrd 2779 1 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2107  Vcvv 3433  ifcif 4460  cmpt 5158  cfv 6437  (class class class)co 7284  0gc0g 17159  1rcur 19746   unitVec cuvc 20998
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 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pr 5353
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-ov 7287  df-oprab 7288  df-mpo 7289  df-uvc 20999
This theorem is referenced by:  uvcvval  21002
  Copyright terms: Public domain W3C validator