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Theorem uvcvv1 21809
Description: The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
Hypotheses
Ref Expression
uvcvv.u 𝑈 = (𝑅 unitVec 𝐼)
uvcvv.r (𝜑𝑅𝑉)
uvcvv.i (𝜑𝐼𝑊)
uvcvv.j (𝜑𝐽𝐼)
uvcvv1.o 1 = (1r𝑅)
Assertion
Ref Expression
uvcvv1 (𝜑 → ((𝑈𝐽)‘𝐽) = 1 )

Proof of Theorem uvcvv1
StepHypRef Expression
1 uvcvv.r . . 3 (𝜑𝑅𝑉)
2 uvcvv.i . . 3 (𝜑𝐼𝑊)
3 uvcvv.j . . 3 (𝜑𝐽𝐼)
4 uvcvv.u . . . 4 𝑈 = (𝑅 unitVec 𝐼)
5 uvcvv1.o . . . 4 1 = (1r𝑅)
6 eqid 2737 . . . 4 (0g𝑅) = (0g𝑅)
74, 5, 6uvcvval 21806 . . 3 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐽𝐼) → ((𝑈𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0g𝑅)))
81, 2, 3, 3, 7syl31anc 1375 . 2 (𝜑 → ((𝑈𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0g𝑅)))
9 eqid 2737 . . 3 𝐽 = 𝐽
10 iftrue 4531 . . 3 (𝐽 = 𝐽 → if(𝐽 = 𝐽, 1 , (0g𝑅)) = 1 )
119, 10mp1i 13 . 2 (𝜑 → if(𝐽 = 𝐽, 1 , (0g𝑅)) = 1 )
128, 11eqtrd 2777 1 (𝜑 → ((𝑈𝐽)‘𝐽) = 1 )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  ifcif 4525  cfv 6561  (class class class)co 7431  0gc0g 17484  1rcur 20178   unitVec cuvc 21802
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-uvc 21803
This theorem is referenced by:  uvcf1  21812  uvcresum  21813  frlmssuvc2  21815  frlmup2  21819  uvcn0  42552  0prjspnrel  42637
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