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Theorem obsipid 21760
Description: A basis element has length one. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
obsipid.h , = (·𝑖𝑊)
obsipid.f 𝐹 = (Scalar‘𝑊)
obsipid.u 1 = (1r𝐹)
Assertion
Ref Expression
obsipid ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴𝐵) → (𝐴 , 𝐴) = 1 )

Proof of Theorem obsipid
StepHypRef Expression
1 eqid 2735 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 obsipid.h . . . 4 , = (·𝑖𝑊)
3 obsipid.f . . . 4 𝐹 = (Scalar‘𝑊)
4 obsipid.u . . . 4 1 = (1r𝐹)
5 eqid 2735 . . . 4 (0g𝐹) = (0g𝐹)
61, 2, 3, 4, 5obsip 21759 . . 3 ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴𝐵𝐴𝐵) → (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0g𝐹)))
763anidm23 1420 . 2 ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴𝐵) → (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0g𝐹)))
8 eqid 2735 . . 3 𝐴 = 𝐴
98iftruei 4538 . 2 if(𝐴 = 𝐴, 1 , (0g𝐹)) = 1
107, 9eqtrdi 2791 1 ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴𝐵) → (𝐴 , 𝐴) = 1 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  ifcif 4531  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301  ·𝑖cip 17303  0gc0g 17486  1rcur 20199  OBasiscobs 21740
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-sep 5302  ax-nul 5312  ax-pow 5371  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-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  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-fv 6571  df-ov 7434  df-obs 21743
This theorem is referenced by:  obsne0  21763
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