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Theorem uvcvval 20323
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 20322 . . . 4 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
54fveq1d 6350 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → ((𝑈𝐽)‘𝐾) = ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾))
65adantr 472 . 2 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾))
7 simpr 479 . . 3 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → 𝐾𝐼)
8 fvex 6358 . . . . 5 (1r𝑅) ∈ V
92, 8eqeltri 2831 . . . 4 1 ∈ V
10 fvex 6358 . . . . 5 (0g𝑅) ∈ V
113, 10eqeltri 2831 . . . 4 0 ∈ V
129, 11ifex 4296 . . 3 if(𝐾 = 𝐽, 1 , 0 ) ∈ V
13 eqeq1 2760 . . . . 5 (𝑘 = 𝐾 → (𝑘 = 𝐽𝐾 = 𝐽))
1413ifbid 4248 . . . 4 (𝑘 = 𝐾 → if(𝑘 = 𝐽, 1 , 0 ) = if(𝐾 = 𝐽, 1 , 0 ))
15 eqid 2756 . . . 4 (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))
1614, 15fvmptg 6438 . . 3 ((𝐾𝐼 ∧ if(𝐾 = 𝐽, 1 , 0 ) ∈ V) → ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
177, 12, 16sylancl 697 . 2 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
186, 17eqtrd 2790 1 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1628  wcel 2135  Vcvv 3336  ifcif 4226  cmpt 4877  cfv 6045  (class class class)co 6809  0gc0g 16298  1rcur 18697   unitVec cuvc 20319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pr 5051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-uvc 20320
This theorem is referenced by:  uvcvvcl  20324  uvcvvcl2  20325  uvcvv1  20326  uvcvv0  20327  matunitlindflem2  33715
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