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Theorem obsipid 21144
Description: A basis element has length one. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
obsipid.h , = (Β·π‘–β€˜π‘Š)
obsipid.f 𝐹 = (Scalarβ€˜π‘Š)
obsipid.u 1 = (1rβ€˜πΉ)
Assertion
Ref Expression
obsipid ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 , 𝐴) = 1 )

Proof of Theorem obsipid
StepHypRef Expression
1 eqid 2733 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 obsipid.h . . . 4 , = (Β·π‘–β€˜π‘Š)
3 obsipid.f . . . 4 𝐹 = (Scalarβ€˜π‘Š)
4 obsipid.u . . . 4 1 = (1rβ€˜πΉ)
5 eqid 2733 . . . 4 (0gβ€˜πΉ) = (0gβ€˜πΉ)
61, 2, 3, 4, 5obsip 21143 . . 3 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐴 ∈ 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0gβ€˜πΉ)))
763anidm23 1422 . 2 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0gβ€˜πΉ)))
8 eqid 2733 . . 3 𝐴 = 𝐴
98iftruei 4494 . 2 if(𝐴 = 𝐴, 1 , (0gβ€˜πΉ)) = 1
107, 9eqtrdi 2789 1 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 , 𝐴) = 1 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ifcif 4487  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Scalarcsca 17141  Β·π‘–cip 17143  0gc0g 17326  1rcur 19918  OBasiscobs 21124
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-sep 5257  ax-nul 5264  ax-pow 5321  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-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  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-fv 6505  df-ov 7361  df-obs 21127
This theorem is referenced by:  obsne0  21147
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