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Theorem xrhval 31146
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 3517 . 2 (𝑅𝑉𝑅 ∈ V)
2 fveq2 6666 . . . . . 6 (𝑟 = 𝑅 → (ℝHom‘𝑟) = (ℝHom‘𝑅))
32fveq1d 6668 . . . . 5 (𝑟 = 𝑅 → ((ℝHom‘𝑟)‘𝑥) = ((ℝHom‘𝑅)‘𝑥))
4 fveq2 6666 . . . . . . . 8 (𝑟 = 𝑅 → (lub‘𝑟) = (lub‘𝑅))
5 xrhval.u . . . . . . . 8 𝑈 = (lub‘𝑅)
64, 5syl6eqr 2878 . . . . . . 7 (𝑟 = 𝑅 → (lub‘𝑟) = 𝑈)
72imaeq1d 5925 . . . . . . . 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 6673 . . . . . 6 (𝑟 = 𝑅 → ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝑈𝐵))
11 fveq2 6666 . . . . . . . 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 6673 . . . . . 6 (𝑟 = 𝑅 → ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝐿𝐵))
1510, 14ifeq12d 4489 . . . . 5 (𝑟 = 𝑅 → if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))) = if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))
163, 15ifeq12d 4489 . . . 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 5158 . . 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 31145 . . 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 12379 . . . 4 * ∈ V
2019mptex 6984 . . 3 (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))) ∈ V
2117, 18, 20fvmpt 6764 . 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 1530  wcel 2106  Vcvv 3499  ifcif 4469  cmpt 5142  cima 5556  cfv 6351  cr 10528  +∞cpnf 10664  *cxr 10666  lubclub 17544  glbcglb 17545  ℝHomcrrh 31121  *Homcxrh 31144
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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-xr 10671  df-xrh 31145
This theorem is referenced by: (None)
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