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Theorem uvcvval 20342
Description: Value of a unit vector coordinate 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
uvcvval (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))

Proof of Theorem uvcvval
Dummy variable 𝑘 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, 3uvcval 20341 . . . 4 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
54fveq1d 6334 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → ((𝑈𝐽)‘𝐾) = ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾))
65adantr 466 . 2 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾))
7 simpr 471 . . 3 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → 𝐾𝐼)
8 fvex 6342 . . . . 5 (1r𝑅) ∈ V
92, 8eqeltri 2846 . . . 4 1 ∈ V
10 fvex 6342 . . . . 5 (0g𝑅) ∈ V
113, 10eqeltri 2846 . . . 4 0 ∈ V
129, 11ifex 4295 . . 3 if(𝐾 = 𝐽, 1 , 0 ) ∈ V
13 eqeq1 2775 . . . . 5 (𝑘 = 𝐾 → (𝑘 = 𝐽𝐾 = 𝐽))
1413ifbid 4247 . . . 4 (𝑘 = 𝐾 → if(𝑘 = 𝐽, 1 , 0 ) = if(𝐾 = 𝐽, 1 , 0 ))
15 eqid 2771 . . . 4 (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))
1614, 15fvmptg 6422 . . 3 ((𝐾𝐼 ∧ if(𝐾 = 𝐽, 1 , 0 ) ∈ V) → ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
177, 12, 16sylancl 574 . 2 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
186, 17eqtrd 2805 1 (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  ifcif 4225  cmpt 4863  cfv 6031  (class class class)co 6793  0gc0g 16308  1rcur 18709   unitVec cuvc 20338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-uvc 20339
This theorem is referenced by:  uvcvvcl  20343  uvcvvcl2  20344  uvcvv1  20345  uvcvv0  20346  matunitlindflem2  33739
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