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Theorem uvcvval 21676
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 21675 . . . 4 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
54fveq1d 6886 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → ((𝑈𝐽)‘𝐾) = ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾))
65adantr 480 . 2 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾))
7 simpr 484 . . 3 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → 𝐾𝐼)
82fvexi 6898 . . . 4 1 ∈ V
93fvexi 6898 . . . 4 0 ∈ V
108, 9ifex 4573 . . 3 if(𝐾 = 𝐽, 1 , 0 ) ∈ V
11 eqeq1 2730 . . . . 5 (𝑘 = 𝐾 → (𝑘 = 𝐽𝐾 = 𝐽))
1211ifbid 4546 . . . 4 (𝑘 = 𝐾 → if(𝑘 = 𝐽, 1 , 0 ) = if(𝐾 = 𝐽, 1 , 0 ))
13 eqid 2726 . . . 4 (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))
1412, 13fvmptg 6989 . . 3 ((𝐾𝐼 ∧ if(𝐾 = 𝐽, 1 , 0 ) ∈ V) → ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
157, 10, 14sylancl 585 . 2 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
166, 15eqtrd 2766 1 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3468  ifcif 4523  cmpt 5224  cfv 6536  (class class class)co 7404  0gc0g 17391  1rcur 20083   unitVec cuvc 21672
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-uvc 21673
This theorem is referenced by:  uvcvvcl  21677  uvcvvcl2  21678  uvcvv1  21679  uvcvv0  21680  matunitlindflem2  36997
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