Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xrhval Structured version   Visualization version   GIF version

Theorem xrhval 29193
Description: The value of the embedding from the extended real numbers into a complete lattice. (Contributed by Thierry Arnoux, 19-Feb-2018.)
Hypotheses
Ref Expression
xrhval.b 𝐵 = ((ℝHom‘𝑅) “ ℝ)
xrhval.l 𝐿 = (glb‘𝑅)
xrhval.u 𝑈 = (lub‘𝑅)
Assertion
Ref Expression
xrhval (𝑅𝑉 → (ℝ*Hom‘𝑅) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))))
Distinct variable group:   𝑥,𝑅
Allowed substitution hints:   𝐵(𝑥)   𝑈(𝑥)   𝐿(𝑥)   𝑉(𝑥)

Proof of Theorem xrhval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 elex 3181 . 2 (𝑅𝑉𝑅 ∈ V)
2 fveq2 6085 . . . . . 6 (𝑟 = 𝑅 → (ℝHom‘𝑟) = (ℝHom‘𝑅))
32fveq1d 6087 . . . . 5 (𝑟 = 𝑅 → ((ℝHom‘𝑟)‘𝑥) = ((ℝHom‘𝑅)‘𝑥))
4 fveq2 6085 . . . . . . . 8 (𝑟 = 𝑅 → (lub‘𝑟) = (lub‘𝑅))
5 xrhval.u . . . . . . . 8 𝑈 = (lub‘𝑅)
64, 5syl6eqr 2658 . . . . . . 7 (𝑟 = 𝑅 → (lub‘𝑟) = 𝑈)
72imaeq1d 5368 . . . . . . . 8 (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = ((ℝHom‘𝑅) “ ℝ))
8 xrhval.b . . . . . . . 8 𝐵 = ((ℝHom‘𝑅) “ ℝ)
97, 8syl6eqr 2658 . . . . . . 7 (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = 𝐵)
106, 9fveq12d 6091 . . . . . 6 (𝑟 = 𝑅 → ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝑈𝐵))
11 fveq2 6085 . . . . . . . 8 (𝑟 = 𝑅 → (glb‘𝑟) = (glb‘𝑅))
12 xrhval.l . . . . . . . 8 𝐿 = (glb‘𝑅)
1311, 12syl6eqr 2658 . . . . . . 7 (𝑟 = 𝑅 → (glb‘𝑟) = 𝐿)
1413, 9fveq12d 6091 . . . . . 6 (𝑟 = 𝑅 → ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝐿𝐵))
1510, 14ifeq12d 4052 . . . . 5 (𝑟 = 𝑅 → if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))) = if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))
163, 15ifeq12d 4052 . . . 4 (𝑟 = 𝑅 → if(𝑥 ∈ ℝ, ((ℝHom‘𝑟)‘𝑥), if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)))) = if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵))))
1716mpteq2dv 4664 . . 3 (𝑟 = 𝑅 → (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑟)‘𝑥), if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))))) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))))
18 df-xrh 29192 . . 3 *Hom = (𝑟 ∈ V ↦ (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑟)‘𝑥), if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))))))
19 xrex 11658 . . . 4 * ∈ V
2019mptex 6365 . . 3 (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))) ∈ V
2117, 18, 20fvmpt 6173 . 2 (𝑅 ∈ V → (ℝ*Hom‘𝑅) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))))
221, 21syl 17 1 (𝑅𝑉 → (ℝ*Hom‘𝑅) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  Vcvv 3169  ifcif 4032  cmpt 4634  cima 5028  cfv 5787  cr 9788  +∞cpnf 9924  *cxr 9926  lubclub 16708  glbcglb 16709  ℝHomcrrh 29168  *Homcxrh 29191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-xr 9931  df-xrh 29192
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator