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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
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