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

Theorem uvcvval 21727
Description: Value of a unit vector coordinate 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
uvcvval (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))

Proof of Theorem uvcvval
Dummy variable 𝑘 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, 3uvcval 21726 . . . 4 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
54fveq1d 6904 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → ((𝑈𝐽)‘𝐾) = ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾))
65adantr 479 . 2 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾))
7 simpr 483 . . 3 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → 𝐾𝐼)
82fvexi 6916 . . . 4 1 ∈ V
93fvexi 6916 . . . 4 0 ∈ V
108, 9ifex 4582 . . 3 if(𝐾 = 𝐽, 1 , 0 ) ∈ V
11 eqeq1 2732 . . . . 5 (𝑘 = 𝐾 → (𝑘 = 𝐽𝐾 = 𝐽))
1211ifbid 4555 . . . 4 (𝑘 = 𝐾 → if(𝑘 = 𝐽, 1 , 0 ) = if(𝐾 = 𝐽, 1 , 0 ))
13 eqid 2728 . . . 4 (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))
1412, 13fvmptg 7008 . . 3 ((𝐾𝐼 ∧ if(𝐾 = 𝐽, 1 , 0 ) ∈ V) → ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
157, 10, 14sylancl 584 . 2 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
166, 15eqtrd 2768 1 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3473  ifcif 4532  cmpt 5235  cfv 6553  (class class class)co 7426  0gc0g 17428  1rcur 20128   unitVec cuvc 21723
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-uvc 21724
This theorem is referenced by:  uvcvvcl  21728  uvcvvcl2  21729  uvcvv1  21730  uvcvv0  21731  matunitlindflem2  37123
  Copyright terms: Public domain W3C validator