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Theorem uvcvv1 21827
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 2735 . . . 4 (0g𝑅) = (0g𝑅)
74, 5, 6uvcvval 21824 . . 3 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐽𝐼) → ((𝑈𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0g𝑅)))
81, 2, 3, 3, 7syl31anc 1372 . 2 (𝜑 → ((𝑈𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0g𝑅)))
9 eqid 2735 . . 3 𝐽 = 𝐽
10 iftrue 4537 . . 3 (𝐽 = 𝐽 → if(𝐽 = 𝐽, 1 , (0g𝑅)) = 1 )
119, 10mp1i 13 . 2 (𝜑 → if(𝐽 = 𝐽, 1 , (0g𝑅)) = 1 )
128, 11eqtrd 2775 1 (𝜑 → ((𝑈𝐽)‘𝐽) = 1 )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  ifcif 4531  cfv 6563  (class class class)co 7431  0gc0g 17486  1rcur 20199   unitVec cuvc 21820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-uvc 21821
This theorem is referenced by:  uvcf1  21830  uvcresum  21831  frlmssuvc2  21833  frlmup2  21837  uvcn0  42529  0prjspnrel  42614
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