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Theorem uvcval 20601
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 20600 . . . 4 ((𝑅𝑉𝐼𝑊) → 𝑈 = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
54fveq1d 6676 . . 3 ((𝑅𝑉𝐼𝑊) → (𝑈𝐽) = ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽))
653adant3 1133 . 2 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽))
7 eqid 2738 . . 3 (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))
8 eqeq2 2750 . . . . 5 (𝑗 = 𝐽 → (𝑘 = 𝑗𝑘 = 𝐽))
98ifbid 4437 . . . 4 (𝑗 = 𝐽 → if(𝑘 = 𝑗, 1 , 0 ) = if(𝑘 = 𝐽, 1 , 0 ))
109mpteq2dv 5126 . . 3 (𝑗 = 𝐽 → (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
11 simp3 1139 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → 𝐽𝐼)
12 mptexg 6994 . . . 4 (𝐼𝑊 → (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V)
13123ad2ant2 1135 . . 3 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V)
147, 10, 11, 13fvmptd3 6798 . 2 ((𝑅𝑉𝐼𝑊𝐽𝐼) → ((𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
156, 14eqtrd 2773 1 ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  Vcvv 3398  ifcif 4414  cmpt 5110  cfv 6339  (class class class)co 7170  0gc0g 16816  1rcur 19370   unitVec cuvc 20598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-uvc 20599
This theorem is referenced by:  uvcvval  20602
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