MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uvcvv1 Structured version   Visualization version   GIF version

Theorem uvcvv1 20861
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 2818 . . . 4 (0g𝑅) = (0g𝑅)
74, 5, 6uvcvval 20858 . . 3 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐽𝐼) → ((𝑈𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0g𝑅)))
81, 2, 3, 3, 7syl31anc 1365 . 2 (𝜑 → ((𝑈𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0g𝑅)))
9 eqid 2818 . . 3 𝐽 = 𝐽
10 iftrue 4469 . . 3 (𝐽 = 𝐽 → if(𝐽 = 𝐽, 1 , (0g𝑅)) = 1 )
119, 10mp1i 13 . 2 (𝜑 → if(𝐽 = 𝐽, 1 , (0g𝑅)) = 1 )
128, 11eqtrd 2853 1 (𝜑 → ((𝑈𝐽)‘𝐽) = 1 )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  ifcif 4463  cfv 6348  (class class class)co 7145  0gc0g 16701  1rcur 19180   unitVec cuvc 20854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-uvc 20855
This theorem is referenced by:  uvcf1  20864  uvcresum  20865  frlmssuvc2  20867  frlmup2  20871  uvcn0  39029  0prjspnrel  39147
  Copyright terms: Public domain W3C validator