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

Theorem obsipid 21661
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 2727 . . . 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 2727 . . . 4 (0gβ€˜πΉ) = (0gβ€˜πΉ)
61, 2, 3, 4, 5obsip 21660 . . 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 2727 . . 3 𝐴 = 𝐴
98iftruei 4537 . 2 if(𝐴 = 𝐴, 1 , (0gβ€˜πΉ)) = 1
107, 9eqtrdi 2783 1 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 , 𝐴) = 1 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ifcif 4530  β€˜cfv 6551  (class class class)co 7424  Basecbs 17185  Scalarcsca 17241  Β·π‘–cip 17243  0gc0g 17426  1rcur 20126  OBasiscobs 21641
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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fv 6559  df-ov 7427  df-obs 21644
This theorem is referenced by:  obsne0  21664
  Copyright terms: Public domain W3C validator