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Theorem xrhval 29390
 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 3185 . 2 (𝑅𝑉𝑅 ∈ V)
2 fveq2 6103 . . . . . 6 (𝑟 = 𝑅 → (ℝHom‘𝑟) = (ℝHom‘𝑅))
32fveq1d 6105 . . . . 5 (𝑟 = 𝑅 → ((ℝHom‘𝑟)‘𝑥) = ((ℝHom‘𝑅)‘𝑥))
4 fveq2 6103 . . . . . . . 8 (𝑟 = 𝑅 → (lub‘𝑟) = (lub‘𝑅))
5 xrhval.u . . . . . . . 8 𝑈 = (lub‘𝑅)
64, 5syl6eqr 2662 . . . . . . 7 (𝑟 = 𝑅 → (lub‘𝑟) = 𝑈)
72imaeq1d 5384 . . . . . . . 8 (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = ((ℝHom‘𝑅) “ ℝ))
8 xrhval.b . . . . . . . 8 𝐵 = ((ℝHom‘𝑅) “ ℝ)
97, 8syl6eqr 2662 . . . . . . 7 (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = 𝐵)
106, 9fveq12d 6109 . . . . . 6 (𝑟 = 𝑅 → ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝑈𝐵))
11 fveq2 6103 . . . . . . . 8 (𝑟 = 𝑅 → (glb‘𝑟) = (glb‘𝑅))
12 xrhval.l . . . . . . . 8 𝐿 = (glb‘𝑅)
1311, 12syl6eqr 2662 . . . . . . 7 (𝑟 = 𝑅 → (glb‘𝑟) = 𝐿)
1413, 9fveq12d 6109 . . . . . 6 (𝑟 = 𝑅 → ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝐿𝐵))
1510, 14ifeq12d 4056 . . . . 5 (𝑟 = 𝑅 → if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))) = if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))
163, 15ifeq12d 4056 . . . 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 4673 . . 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 29389 . . 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 11705 . . . 4 * ∈ V
2019mptex 6390 . . 3 (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))) ∈ V
2117, 18, 20fvmpt 6191 . 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 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036   ↦ cmpt 4643   “ cima 5041  ‘cfv 5804  ℝcr 9814  +∞cpnf 9950  ℝ*cxr 9952  lubclub 16765  glbcglb 16766  ℝHomcrrh 29365  ℝ*Homcxrh 29388 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-xr 9957  df-xrh 29389 This theorem is referenced by: (None)
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