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Theorem obsipid 21613
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 2726 . . . 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 2726 . . . 4 (0gβ€˜πΉ) = (0gβ€˜πΉ)
61, 2, 3, 4, 5obsip 21612 . . 3 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐴 ∈ 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0gβ€˜πΉ)))
763anidm23 1418 . 2 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0gβ€˜πΉ)))
8 eqid 2726 . . 3 𝐴 = 𝐴
98iftruei 4530 . 2 if(𝐴 = 𝐴, 1 , (0gβ€˜πΉ)) = 1
107, 9eqtrdi 2782 1 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 , 𝐴) = 1 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ifcif 4523  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  Scalarcsca 17207  Β·π‘–cip 17209  0gc0g 17392  1rcur 20084  OBasiscobs 21593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-obs 21596
This theorem is referenced by:  obsne0  21616
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