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Theorem xrhval 30606
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 3428 . 2 (𝑅𝑉𝑅 ∈ V)
2 fveq2 6432 . . . . . 6 (𝑟 = 𝑅 → (ℝHom‘𝑟) = (ℝHom‘𝑅))
32fveq1d 6434 . . . . 5 (𝑟 = 𝑅 → ((ℝHom‘𝑟)‘𝑥) = ((ℝHom‘𝑅)‘𝑥))
4 fveq2 6432 . . . . . . . 8 (𝑟 = 𝑅 → (lub‘𝑟) = (lub‘𝑅))
5 xrhval.u . . . . . . . 8 𝑈 = (lub‘𝑅)
64, 5syl6eqr 2878 . . . . . . 7 (𝑟 = 𝑅 → (lub‘𝑟) = 𝑈)
72imaeq1d 5705 . . . . . . . 8 (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = ((ℝHom‘𝑅) “ ℝ))
8 xrhval.b . . . . . . . 8 𝐵 = ((ℝHom‘𝑅) “ ℝ)
97, 8syl6eqr 2878 . . . . . . 7 (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = 𝐵)
106, 9fveq12d 6439 . . . . . 6 (𝑟 = 𝑅 → ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝑈𝐵))
11 fveq2 6432 . . . . . . . 8 (𝑟 = 𝑅 → (glb‘𝑟) = (glb‘𝑅))
12 xrhval.l . . . . . . . 8 𝐿 = (glb‘𝑅)
1311, 12syl6eqr 2878 . . . . . . 7 (𝑟 = 𝑅 → (glb‘𝑟) = 𝐿)
1413, 9fveq12d 6439 . . . . . 6 (𝑟 = 𝑅 → ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝐿𝐵))
1510, 14ifeq12d 4325 . . . . 5 (𝑟 = 𝑅 → if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))) = if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))
163, 15ifeq12d 4325 . . . 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 4967 . . 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 30605 . . 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 12108 . . . 4 * ∈ V
2019mptex 6741 . . 3 (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))) ∈ V
2117, 18, 20fvmpt 6528 . 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 1658  wcel 2166  Vcvv 3413  ifcif 4305  cmpt 4951  cima 5344  cfv 6122  cr 10250  +∞cpnf 10387  *cxr 10389  lubclub 17294  glbcglb 17295  ℝHomcrrh 30581  *Homcxrh 30604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pr 5126  ax-un 7208  ax-cnex 10307  ax-resscn 10308
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-xr 10394  df-xrh 30605
This theorem is referenced by: (None)
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