Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-ghomOLD Structured version   Visualization version   GIF version

Definition df-ghomOLD 36393
Description: Obsolete version of df-ghm 19014 as of 15-Mar-2020. Define the set of group homomorphisms from 𝑔 to β„Ž. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
Assertion
Ref Expression
df-ghomOLD GrpOpHom = (𝑔 ∈ GrpOp, β„Ž ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran π‘”βŸΆran β„Ž ∧ βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)))})
Distinct variable group:   𝑓,𝑔,β„Ž,π‘₯,𝑦

Detailed syntax breakdown of Definition df-ghomOLD
StepHypRef Expression
1 cghomOLD 36392 . 2 class GrpOpHom
2 vg . . 3 setvar 𝑔
3 vh . . 3 setvar β„Ž
4 cgr 29480 . . 3 class GrpOp
52cv 1541 . . . . . . 7 class 𝑔
65crn 5638 . . . . . 6 class ran 𝑔
73cv 1541 . . . . . . 7 class β„Ž
87crn 5638 . . . . . 6 class ran β„Ž
9 vf . . . . . . 7 setvar 𝑓
109cv 1541 . . . . . 6 class 𝑓
116, 8, 10wf 6496 . . . . 5 wff 𝑓:ran π‘”βŸΆran β„Ž
12 vx . . . . . . . . . . 11 setvar π‘₯
1312cv 1541 . . . . . . . . . 10 class π‘₯
1413, 10cfv 6500 . . . . . . . . 9 class (π‘“β€˜π‘₯)
15 vy . . . . . . . . . . 11 setvar 𝑦
1615cv 1541 . . . . . . . . . 10 class 𝑦
1716, 10cfv 6500 . . . . . . . . 9 class (π‘“β€˜π‘¦)
1814, 17, 7co 7361 . . . . . . . 8 class ((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦))
1913, 16, 5co 7361 . . . . . . . . 9 class (π‘₯𝑔𝑦)
2019, 10cfv 6500 . . . . . . . 8 class (π‘“β€˜(π‘₯𝑔𝑦))
2118, 20wceq 1542 . . . . . . 7 wff ((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦))
2221, 15, 6wral 3061 . . . . . 6 wff βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦))
2322, 12, 6wral 3061 . . . . 5 wff βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦))
2411, 23wa 397 . . . 4 wff (𝑓:ran π‘”βŸΆran β„Ž ∧ βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)))
2524, 9cab 2710 . . 3 class {𝑓 ∣ (𝑓:ran π‘”βŸΆran β„Ž ∧ βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)))}
262, 3, 4, 4, 25cmpo 7363 . 2 class (𝑔 ∈ GrpOp, β„Ž ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran π‘”βŸΆran β„Ž ∧ βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)))})
271, 26wceq 1542 1 wff GrpOpHom = (𝑔 ∈ GrpOp, β„Ž ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran π‘”βŸΆran β„Ž ∧ βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)))})
Colors of variables: wff setvar class
This definition is referenced by:  elghomlem1OLD  36394
  Copyright terms: Public domain W3C validator