MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uvcfval Structured version   Visualization version   GIF version

Theorem uvcfval 20339
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 3361 . . 3 (𝑅𝑉𝑅 ∈ V)
3 elex 3361 . . 3 (𝐼𝑊𝐼 ∈ V)
4 df-uvc 20338 . . . . 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 471 . . . . . 6 ((𝑟 = 𝑅𝑖 = 𝐼) → 𝑖 = 𝐼)
7 fveq2 6332 . . . . . . . . . 10 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
8 uvcfval.o . . . . . . . . . 10 1 = (1r𝑅)
97, 8syl6eqr 2822 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = 1 )
10 fveq2 6332 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
11 uvcfval.z . . . . . . . . . 10 0 = (0g𝑅)
1210, 11syl6eqr 2822 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = 0 )
139, 12ifeq12d 4243 . . . . . . . 8 (𝑟 = 𝑅 → if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
1413adantr 466 . . . . . . 7 ((𝑟 = 𝑅𝑖 = 𝐼) → if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
156, 14mpteq12dv 4865 . . . . . 6 ((𝑟 = 𝑅𝑖 = 𝐼) → (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟))) = (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))
166, 15mpteq12dv 4865 . . . . 5 ((𝑟 = 𝑅𝑖 = 𝐼) → (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)))) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
1716adantl 467 . . . 4 (((𝑅 ∈ V ∧ 𝐼 ∈ V) ∧ (𝑟 = 𝑅𝑖 = 𝐼)) → (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)))) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
18 simpl 468 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝑅 ∈ V)
19 simpr 471 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝐼 ∈ V)
20 mptexg 6627 . . . . 5 (𝐼 ∈ V → (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
2120adantl 467 . . . 4 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
225, 17, 18, 19, 21ovmpt2d 6934 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 unitVec 𝐼) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
232, 3, 22syl2an 575 . 2 ((𝑅𝑉𝐼𝑊) → (𝑅 unitVec 𝐼) = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
241, 23syl5eq 2816 1 ((𝑅𝑉𝐼𝑊) → 𝑈 = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  ifcif 4223  cmpt 4861  cfv 6031  (class class class)co 6792  cmpt2 6794  0gc0g 16307  1rcur 18708   unitVec cuvc 20337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  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 6795  df-oprab 6796  df-mpt2 6797  df-uvc 20338
This theorem is referenced by:  uvcval  20340  uvcff  20346
  Copyright terms: Public domain W3C validator