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Theorem uvcfval 21822
Description: Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
uvcfval.u 𝑈 = (𝑅 unitVec 𝐼)
uvcfval.o 1 = (1r𝑅)
uvcfval.z 0 = (0g𝑅)
Assertion
Ref Expression
uvcfval ((𝑅𝑉𝐼𝑊) → 𝑈 = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
Distinct variable groups:   1 ,𝑗,𝑘   𝑅,𝑗,𝑘   𝑗,𝐼,𝑘   0 ,𝑗,𝑘
Allowed substitution hints:   𝑈(𝑗,𝑘)   𝑉(𝑗,𝑘)   𝑊(𝑗,𝑘)

Proof of Theorem uvcfval
Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uvcfval.u . 2 𝑈 = (𝑅 unitVec 𝐼)
2 elex 3499 . . 3 (𝑅𝑉𝑅 ∈ V)
3 elex 3499 . . 3 (𝐼𝑊𝐼 ∈ V)
4 df-uvc 21821 . . . . 5 unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)))))
54a1i 11 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟))))))
6 simpr 484 . . . . . 6 ((𝑟 = 𝑅𝑖 = 𝐼) → 𝑖 = 𝐼)
7 fveq2 6907 . . . . . . . . . 10 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
8 uvcfval.o . . . . . . . . . 10 1 = (1r𝑅)
97, 8eqtr4di 2793 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = 1 )
10 fveq2 6907 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
11 uvcfval.z . . . . . . . . . 10 0 = (0g𝑅)
1210, 11eqtr4di 2793 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = 0 )
139, 12ifeq12d 4552 . . . . . . . 8 (𝑟 = 𝑅 → if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
1413adantr 480 . . . . . . 7 ((𝑟 = 𝑅𝑖 = 𝐼) → if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
156, 14mpteq12dv 5239 . . . . . 6 ((𝑟 = 𝑅𝑖 = 𝐼) → (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟))) = (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))
166, 15mpteq12dv 5239 . . . . 5 ((𝑟 = 𝑅𝑖 = 𝐼) → (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)))) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
1716adantl 481 . . . 4 (((𝑅 ∈ V ∧ 𝐼 ∈ V) ∧ (𝑟 = 𝑅𝑖 = 𝐼)) → (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)))) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
18 simpl 482 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝑅 ∈ V)
19 simpr 484 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝐼 ∈ V)
20 mptexg 7241 . . . . 5 (𝐼 ∈ V → (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
2120adantl 481 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
225, 17, 18, 19, 21ovmpod 7585 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 unitVec 𝐼) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
232, 3, 22syl2an 596 . 2 ((𝑅𝑉𝐼𝑊) → (𝑅 unitVec 𝐼) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
241, 23eqtrid 2787 1 ((𝑅𝑉𝐼𝑊) → 𝑈 = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  ifcif 4531  cmpt 5231  cfv 6563  (class class class)co 7431  cmpo 7433  0gc0g 17486  1rcur 20199   unitVec cuvc 21820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-uvc 21821
This theorem is referenced by:  uvcval  21823  uvcff  21829  frlmdim  33639
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