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Theorem uvcfval 21206
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 3462 . . 3 (𝑅𝑉𝑅 ∈ V)
3 elex 3462 . . 3 (𝐼𝑊𝐼 ∈ V)
4 df-uvc 21205 . . . . 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 486 . . . . . 6 ((𝑟 = 𝑅𝑖 = 𝐼) → 𝑖 = 𝐼)
7 fveq2 6843 . . . . . . . . . 10 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
8 uvcfval.o . . . . . . . . . 10 1 = (1r𝑅)
97, 8eqtr4di 2791 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = 1 )
10 fveq2 6843 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
11 uvcfval.z . . . . . . . . . 10 0 = (0g𝑅)
1210, 11eqtr4di 2791 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = 0 )
139, 12ifeq12d 4508 . . . . . . . 8 (𝑟 = 𝑅 → if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
1413adantr 482 . . . . . . 7 ((𝑟 = 𝑅𝑖 = 𝐼) → if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
156, 14mpteq12dv 5197 . . . . . 6 ((𝑟 = 𝑅𝑖 = 𝐼) → (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟))) = (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))
166, 15mpteq12dv 5197 . . . . 5 ((𝑟 = 𝑅𝑖 = 𝐼) → (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)))) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
1716adantl 483 . . . 4 (((𝑅 ∈ V ∧ 𝐼 ∈ V) ∧ (𝑟 = 𝑅𝑖 = 𝐼)) → (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)))) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
18 simpl 484 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝑅 ∈ V)
19 simpr 486 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝐼 ∈ V)
20 mptexg 7172 . . . . 5 (𝐼 ∈ V → (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
2120adantl 483 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
225, 17, 18, 19, 21ovmpod 7508 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 unitVec 𝐼) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
232, 3, 22syl2an 597 . 2 ((𝑅𝑉𝐼𝑊) → (𝑅 unitVec 𝐼) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
241, 23eqtrid 2785 1 ((𝑅𝑉𝐼𝑊) → 𝑈 = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3444  ifcif 4487  cmpt 5189  cfv 6497  (class class class)co 7358  cmpo 7360  0gc0g 17326  1rcur 19918   unitVec cuvc 21204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-uvc 21205
This theorem is referenced by:  uvcval  21207  uvcff  21213  frlmdim  32363
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