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Theorem xrhval 33997
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 3500 . 2 (𝑅𝑉𝑅 ∈ V)
2 fveq2 6904 . . . . . 6 (𝑟 = 𝑅 → (ℝHom‘𝑟) = (ℝHom‘𝑅))
32fveq1d 6906 . . . . 5 (𝑟 = 𝑅 → ((ℝHom‘𝑟)‘𝑥) = ((ℝHom‘𝑅)‘𝑥))
4 fveq2 6904 . . . . . . . 8 (𝑟 = 𝑅 → (lub‘𝑟) = (lub‘𝑅))
5 xrhval.u . . . . . . . 8 𝑈 = (lub‘𝑅)
64, 5eqtr4di 2794 . . . . . . 7 (𝑟 = 𝑅 → (lub‘𝑟) = 𝑈)
72imaeq1d 6075 . . . . . . . 8 (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = ((ℝHom‘𝑅) “ ℝ))
8 xrhval.b . . . . . . . 8 𝐵 = ((ℝHom‘𝑅) “ ℝ)
97, 8eqtr4di 2794 . . . . . . 7 (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = 𝐵)
106, 9fveq12d 6911 . . . . . 6 (𝑟 = 𝑅 → ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝑈𝐵))
11 fveq2 6904 . . . . . . . 8 (𝑟 = 𝑅 → (glb‘𝑟) = (glb‘𝑅))
12 xrhval.l . . . . . . . 8 𝐿 = (glb‘𝑅)
1311, 12eqtr4di 2794 . . . . . . 7 (𝑟 = 𝑅 → (glb‘𝑟) = 𝐿)
1413, 9fveq12d 6911 . . . . . 6 (𝑟 = 𝑅 → ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝐿𝐵))
1510, 14ifeq12d 4545 . . . . 5 (𝑟 = 𝑅 → if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))) = if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))
163, 15ifeq12d 4545 . . . 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 5242 . . 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 33996 . . 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 13025 . . . 4 * ∈ V
2019mptex 7241 . . 3 (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))) ∈ V
2117, 18, 20fvmpt 7014 . 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 1540  wcel 2108  Vcvv 3479  ifcif 4524  cmpt 5223  cima 5686  cfv 6559  cr 11150  +∞cpnf 11288  *cxr 11290  lubclub 18351  glbcglb 18352  ℝHomcrrh 33972  *Homcxrh 33995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pr 5430  ax-un 7751  ax-cnex 11207  ax-resscn 11208
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-xr 11295  df-xrh 33996
This theorem is referenced by: (None)
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