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Theorem uvcval 20528
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 20527 . . . 4 ((𝑅𝑉𝐼𝑊) → 𝑈 = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
54fveq1d 6448 . . 3 ((𝑅𝑉𝐼𝑊) → (𝑈𝐽) = ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽))
653adant3 1123 . 2 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽))
7 simp3 1129 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → 𝐽𝐼)
8 mptexg 6756 . . . 4 (𝐼𝑊 → (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V)
983ad2ant2 1125 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V)
10 eqeq2 2788 . . . . . 6 (𝑗 = 𝐽 → (𝑘 = 𝑗𝑘 = 𝐽))
1110ifbid 4328 . . . . 5 (𝑗 = 𝐽 → if(𝑘 = 𝑗, 1 , 0 ) = if(𝑘 = 𝐽, 1 , 0 ))
1211mpteq2dv 4980 . . . 4 (𝑗 = 𝐽 → (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
13 eqid 2777 . . . 4 (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))
1412, 13fvmptg 6540 . . 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 579 . 2 ((𝑅𝑉𝐼𝑊𝐽𝐼) → ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
166, 15eqtrd 2813 1 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2106  Vcvv 3397  ifcif 4306  cmpt 4965  cfv 6135  (class class class)co 6922  0gc0g 16486  1rcur 18888   unitVec cuvc 20525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-uvc 20526
This theorem is referenced by:  uvcvval  20529
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