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Theorem uvcval 20172
Description: Value of a single unit vector 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
uvcval ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
Distinct variable groups:   1 ,𝑘   𝑅,𝑘   𝑘,𝐼   0 ,𝑘   𝑘,𝐽
Allowed substitution hints:   𝑈(𝑘)   𝑉(𝑘)   𝑊(𝑘)

Proof of Theorem uvcval
Dummy variable 𝑗 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, 3uvcfval 20171 . . . 4 ((𝑅𝑉𝐼𝑊) → 𝑈 = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
54fveq1d 6231 . . 3 ((𝑅𝑉𝐼𝑊) → (𝑈𝐽) = ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽))
653adant3 1101 . 2 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽))
7 simp3 1083 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → 𝐽𝐼)
8 mptexg 6525 . . . 4 (𝐼𝑊 → (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V)
983ad2ant2 1103 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V)
10 eqeq2 2662 . . . . . 6 (𝑗 = 𝐽 → (𝑘 = 𝑗𝑘 = 𝐽))
1110ifbid 4141 . . . . 5 (𝑗 = 𝐽 → if(𝑘 = 𝑗, 1 , 0 ) = if(𝑘 = 𝐽, 1 , 0 ))
1211mpteq2dv 4778 . . . 4 (𝑗 = 𝐽 → (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
13 eqid 2651 . . . 4 (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))
1412, 13fvmptg 6319 . . 3 ((𝐽𝐼 ∧ (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V) → ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
157, 9, 14syl2anc 694 . 2 ((𝑅𝑉𝐼𝑊𝐽𝐼) → ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
166, 15eqtrd 2685 1 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  ifcif 4119  cmpt 4762  cfv 5926  (class class class)co 6690  0gc0g 16147  1rcur 18547   unitVec cuvc 20169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-uvc 20170
This theorem is referenced by:  uvcvval  20173
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