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Theorem uvcvval 21042
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 21041 . . . 4 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
54fveq1d 6806 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → ((𝑈𝐽)‘𝐾) = ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾))
65adantr 482 . 2 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾))
7 simpr 486 . . 3 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → 𝐾𝐼)
82fvexi 6818 . . . 4 1 ∈ V
93fvexi 6818 . . . 4 0 ∈ V
108, 9ifex 4515 . . 3 if(𝐾 = 𝐽, 1 , 0 ) ∈ V
11 eqeq1 2740 . . . . 5 (𝑘 = 𝐾 → (𝑘 = 𝐽𝐾 = 𝐽))
1211ifbid 4488 . . . 4 (𝑘 = 𝐾 → if(𝑘 = 𝐽, 1 , 0 ) = if(𝐾 = 𝐽, 1 , 0 ))
13 eqid 2736 . . . 4 (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))
1412, 13fvmptg 6905 . . 3 ((𝐾𝐼 ∧ if(𝐾 = 𝐽, 1 , 0 ) ∈ V) → ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
157, 10, 14sylancl 587 . 2 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
166, 15eqtrd 2776 1 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1087   = wceq 1539  wcel 2104  Vcvv 3437  ifcif 4465  cmpt 5164  cfv 6458  (class class class)co 7307  0gc0g 17199  1rcur 19786   unitVec cuvc 21038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-uvc 21039
This theorem is referenced by:  uvcvvcl  21043  uvcvvcl2  21044  uvcvv1  21045  uvcvv0  21046  matunitlindflem2  35822
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