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Theorem uvcfval 20482
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 3498 . . 3 (𝑅𝑉𝑅 ∈ V)
3 elex 3498 . . 3 (𝐼𝑊𝐼 ∈ V)
4 df-uvc 20481 . . . . 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 488 . . . . . 6 ((𝑟 = 𝑅𝑖 = 𝐼) → 𝑖 = 𝐼)
7 fveq2 6663 . . . . . . . . . 10 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
8 uvcfval.o . . . . . . . . . 10 1 = (1r𝑅)
97, 8eqtr4di 2877 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = 1 )
10 fveq2 6663 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
11 uvcfval.z . . . . . . . . . 10 0 = (0g𝑅)
1210, 11eqtr4di 2877 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = 0 )
139, 12ifeq12d 4470 . . . . . . . 8 (𝑟 = 𝑅 → if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
1413adantr 484 . . . . . . 7 ((𝑟 = 𝑅𝑖 = 𝐼) → if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
156, 14mpteq12dv 5138 . . . . . 6 ((𝑟 = 𝑅𝑖 = 𝐼) → (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟))) = (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))
166, 15mpteq12dv 5138 . . . . 5 ((𝑟 = 𝑅𝑖 = 𝐼) → (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)))) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
1716adantl 485 . . . 4 (((𝑅 ∈ V ∧ 𝐼 ∈ V) ∧ (𝑟 = 𝑅𝑖 = 𝐼)) → (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)))) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
18 simpl 486 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝑅 ∈ V)
19 simpr 488 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝐼 ∈ V)
20 mptexg 6977 . . . . 5 (𝐼 ∈ V → (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
2120adantl 485 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
225, 17, 18, 19, 21ovmpod 7297 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 unitVec 𝐼) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
232, 3, 22syl2an 598 . 2 ((𝑅𝑉𝐼𝑊) → (𝑅 unitVec 𝐼) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
241, 23syl5eq 2871 1 ((𝑅𝑉𝐼𝑊) → 𝑈 = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  ifcif 4450  cmpt 5133  cfv 6345  (class class class)co 7151  cmpo 7153  0gc0g 16715  1rcur 19253   unitVec cuvc 20480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-uvc 20481
This theorem is referenced by:  uvcval  20483  uvcff  20489  frlmdim  31077
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